Abstract

We present a dynamic synchrotron X-ray imaging study of the effective temperature $T_{\text{eff}}$ in a vibrated granular medium. By tracking the directed motion and the fluctuation dynamics of the tracers inside, we obtained $T_{\text{eff}}$ of the system using the Einstein relationship.

We found that as the system unjams with increasing vibration intensities $\Gamma$, the structural relaxation time $\tau$ increases substantially which can be fitted by an Arrhenius law using $T_{\text{eff}}$. And the characteristic energy scale of structural relaxation yielded by the Arrhenius fitting is $E = 0.20 \pm 0.02 pd^3$, where $p$ is the pressure and $d$ is the background particle diameter, which is consistent with those from hard sphere simulations in which the structural relaxation happens via the opening up of free volume against pressure.

我们介绍了对振动颗粒介质中有效温度 $T_{\text{eff}}$ 的动态同步辐射 $X$ 射线成像研究。通过跟踪内部示踪剂的定向运动和波动动态，我们利用爱因斯坦关系得到了系统的 $T_{\text{eff}}$。

我们发现，随着振动强度 $\Gamma$ 的增加，系统的弛豫时间 $\tau$ 会大幅增加，这可以用 $T_{\text{eff}}$ 的阿伦尼乌斯定律来拟合。阿伦尼乌斯拟合得到的结构弛豫特征能级为 $E = 0.20 \pm 0.02 pd^3$，其中 $p$ 为压力，$d$ 为背景颗粒直径，这与硬球模拟中的结构弛豫是通过打开自由体积对抗压力发生的结果一致。

Introduction

The effective temperature $T_{\text{eff}}$ has attracted a lot of interest in the study of out-of-equilibrium glassy systems like structural glass, colloids, foams, and granular materials.

有效温度 $T_{\text{eff}}$ 在结构玻璃、胶体、泡沫和颗粒材料等非平衡玻璃体系的研究中引起了广泛的兴趣。

The introduction of $T_{\text{eff}}$ can help the understanding of the aging and various transport phenomena in these systems. This is evidenced by the fact that the structural relaxation processes under shear in these out-of-equilibrium systems are controlled by the longtime scale $T_{\text{eff}}$ instead of the short-time scale kinetic temperature $T_k$.

引入 $T_{\text{eff}}$ 有助于理解这些系统中的老化和各种输运现象。事实证明，在这些失衡系统中，剪切作用下的结构弛豫过程是由长时间尺度 $T_{\text{eff}}$, 而非短时间尺度动力学温度 $T_k$ 控制的。

Additionally, the study of $T_{\text{eff}}$ can lead to a unified understanding of the jamming phase diagram and the plastic deformation of solids under shear.

The usefulness of $T_{\text{eff}}$ has also been validated by the fact that various definitions have yielded consistent values which makes it easy for experimental measurements. Recently, $T_{\text{eff}}$ has become one of the key concepts in the development of mesoscopic thermodynamic theories of amorphous solid plasticity and soft glassy rheology.

此外，通过对 $T_{\text{eff}}$ 的研究，可以统一理解固体在剪切力作用下的堵塞相图和塑性变形。

$T_{\text{eff}}$的实用性还通过以下事实得到了验证：各种定义都得出了一致的值，这使得它易于进行实验测量。最近，$T_{\text{eff}}$ 已成为发展非晶固体塑性和软玻璃流变的介观热力学理论的关键概念之一。

Granular systems are by nature out-of-equilibrium systems since they will simply come to rest without outside energy input. However, when agitated by shaking or shear, they can display gas-, fluid- and solid-like phases under different energy input strengths, which prompts possible thermodynamic descriptions of these phases.

Kinetic theory originally based on ideal gas has been quite successful in describing highly agitated dilute granular gases after taking into account the dissipation.

In the dilute limit, $T_{\text{eff}}$ based on the fluctuation-dissipation theorem can be defined, which justifies a thermodynamic approach despite energy non-equipartition and velocity non-Gaussian distribution.

颗粒系统本质上是失衡系统，因为它们在没有外部能量输入的情况下会简单地静止。然而，当受到振动或剪切力的作用时，它们会在不同的能量输入强度下呈现出气相、液相和固相。

最初以理想气体为基础的动力学理论，在考虑了耗散因素后，在描述高度搅拌的稀颗粒气体时相当成功。

在稀释极限中，可以定义基于波动-耗散定理的 $T_{\text{eff}}$，这就证明了热力学方法的合理性，尽管能量是非等分的，速度是非高斯分布的。

In this case, $T_{\text{eff}} \propto \Gamma^{2}$ where $\Gamma$ is the vibration intensity and $T_{\text{eff}}$ is on the same order of magnitude as the particle’s mean kinetic energy. At the other limit, when the outside energy input is absent, the granular system undergoes a jamming phase transition into a disordered solid phase.

在这种情况下，$T_{\text{eff}} \propto \Gamma^{2}$, 其中 $\Gamma$ 是振动强度，$T_{\text{eff}}$ 与粒子的平均动能处于同一数量级。在另一个极限上，当外部能量输入缺失时，颗粒系统会发生堵塞相变，进入无序固相。

Edwards first suggested that when granular packing is slowly sheared, the system explores the stable mechanical states which is the same as taking a flat average of the jammed configurations. The corresponding configurational temperature defined based on this ensemble turns out to be equivalent to the temperature $T_{\text{eff}}$ based on the fluctuation-dissipation theorem.

In practice, volume or stress instead of energy is normally considered as the conserved quantity and concepts similar to temperature like compactivity, angoricity or a combination of these two have been introduced and studied.

Edwards 首先提出，当粒状填料被缓慢剪切时，系统会探索稳定的机械状态，这等同于取卡住构型的平面平均值。根据这一集合定义的相应构型温度原来等同于基于波动-消散定理的温度 $T_{\text{eff}}$。

实际上，通常将体积或应力而不是能量视为守恒量，并引入和研究了与温度类似的概念，如密实度、角度或这两者的组合。

Experimentally, this thermodynamic approach has been adopted in granular compaction studies in which the vibration intensity $\Gamma$ has been interpreted as the temperaturelike parameter similar to the granular gas case.

Between the slowly sheared state and the highly agitated granular gas, the system is in a dense liquid state in which both particle collisions and perpetual contacts are important. This regime is important in many scientific and industrial applications and empirical constitutive relationships have been introduced to describe its rheological behaviour.

在实验中，粒状压实研究采用了这种热力学方法，其中振动强度 $\Gamma$ 被解释为类似于粒状气体情况下的温度参数。

在缓慢剪切状态和高度搅拌的颗粒气体之间，系统处于致密液态，其中颗粒碰撞和永久接触都很重要。这种状态在许多科学和工业应用中都很重要，人们已经引入了经验构成关系来描述其流变行为。

However, a more fundamental theory based on microscopic dynamics is needed to justify these models.

Therefore, it is interesting to see whether a thermodynamic theory is still valid to the different phases of a granular system when it is sheared or shaken with increasing strength as it evolves from a static packing into a granular gas, i.e., whether a valid and consistent $T_{\text{eff}}$ can be defined for all phases.

It is also important to understand what determines $T_{\text{eff}}$ and how does it influence the glassy transport properties and solid plasticity.

然而，要证明这些模型的合理性，还需要一个基于微观动力学的更基本的理论。

因此，当粒状系统从静态填料演变为粒状气体时，其受到的剪切或摇晃强度不断增加，这时热力学理论是否仍然适用于粒状系统的不同相，也就是说，可否为所有相定义一个有效且一致的 $T_{\text{eff}}$，这是一个有趣的问题。

同样重要的, 是要了解决定 $T_{\text{eff}}$ 的因素，以及它如何影响玻璃态传输特性和固体塑性。

In the current study, we investigate $T_{\text{eff}}$ inside a three-dimensional (3D) mechanically driven granular system when it evolves from a dense granular fluid to a granular gas.

By introducing tracers different from the background particles, we can monitor their trajectories inside the 3D granular medium non-invasively and dynamically using the synchrotron X-ray imaging technique.

We can observe the otherwise invisible tracer motions inside the granular medium with high spatial and temporal resolutions. We obtained $T_{\text{eff}}$ of the system using the Einstein relationship by tracking the directed motion and the fluctuation dynamics of the tracers inside.

It has been observed that as the system unjams, the structural relaxation time $\tau$ increases substantially and an Arrhenius fitting of $\tau$ versus $T_{\text{eff}}$ yields a characteristic activation dynamic energy scale similar to those from hard sphere simulations. It suggests that the structural relaxation in a dense granular fluid can be interpreted similar to those in hard spheres as opening up of free volume against background pressure.

在当前的研究中，我们研究了三维（3D）机械驱动颗粒系统内的 $T_{\text{eff}}$，当它从致密颗粒流体演变为颗粒气体时的情况。

通过引入不同于背景粒子的示踪剂，我们可以利用同步辐射 X 射线成像技术，非侵入式地动态监测它们在三维颗粒介质中的运动轨迹。

我们可以在高空间和时间分辨率下观察颗粒介质内部原本不可见的示踪剂运动。通过跟踪内部示踪剂的定向运动和波动动态，我们利用爱因斯坦关系得到了系统的 $T_{\text{eff}}$。

我们观察到，随着系统的松动，结构松弛时间 $\tau$ 大幅增加，并且 $\tau$ 与 $T_{\text{eff}}$ 的阿伦尼乌斯拟合得到了与硬球模拟相似的活化动能标度特征。这表明致密颗粒流体中的结构弛豫可以被解释为类似于硬球中的结构弛豫，即在背景压力下自由体积的开放。

Experimental setup and the imaging technique

The experiment was carried out at a 2BM beam line of the Advanced Photon Source of Argonne National Laboratory.

The brilliant unfiltered “pink” X-ray beam from the synchrotron ring is utilized for the dynamic X-ray studies. The imaging system consists of a fast LAG scintillator coupled to a high-speed Cooke Dimax CMOS camera ($11 \mu$m pixel, $2016 \times 1216 $ pixel array) via a $2\times$ microscope objective. The effective X-ray field-of-view is $11 \times 6$ mm$^2$.

实验在阿贡国家实验室先进光子源的 2BM 光束线进行。

同步辐射环发出的未经过滤的明亮 “粉红” X射线束被用于动态X射线研究。成像系统由一个快速LAG闪烁器和一个高速 Cooke Dimax CMOS 相机（像素为 $11\mu$m，像素阵列为 $1216\mu$m$\times 2016\mu$m）组成，相机通过一个 $2\times$ 的显微镜物镜。有效的 X 射线视场为 $11 \times 6$ mm$^2$。

The imaging system was placed $0.35$ m away from the sample to optimize the phase-contrast effects, which are very useful in the detection of the boundaries of the granular particles. A smooth acrylic container which has a square base with area $A_{\text{sys}} = 28\times 28 \text{ mm}^{2}$ was filled up to a height of $H = 10$ mm using polydisperse glass particles with diameter $d = 0.73 \pm 0.17$ mm and density $\rho = 2.7 \text{g cm}^{-3}$ (see schematic in Fig. 1(a)).

成像系统距离样品 0.35 米，以优化相对比效果，这对检测颗粒的边界非常有用。在一个面积为 $A_{\text{sys}} = 28\times 28 \text{ mm}^{2}$ 的正方形光滑丙烯酸容器中，使用直径为 $d = 0.73 \pm 0.17$ mm、密度为 $\rho = 2.7 \text{g cm}^{-3}$的多分散玻璃颗粒填充到高度为 $H = 10$ mm 的容器中（见图 1(a)中的示意图）。

Two steel tracer balls with diameter $d_{\text{tr}} = 4.1$ mm and density $\rho_{\text{tr}} = 7.9 \text{g cm}^{3}$ were buried near one corner of the container bottom.

Subsequently, the container was mounted on an electromagnetic exciter which vibrates sinusoidally at a fixed frequency $f = 50 \text{Hz}$.

The effective vibration intensity $\Gamma = A(2\pi f)^2$ was varied from $0.9g$ to $2.5g$ by changing the vibration amplitude $A$, where $g$ is the gravitational acceleration.

两个直径为 $d_{\text{tr}} = 4.1$ mm、密度为 $\rho_{\text{tr}} = 7.9 \text{g cm}^{3}$ 的钢制示踪球被埋在容器底部一角附近。

随后，容器被安装在一个电磁激振器上，该激振器以固定频率 $f = 50 \text{Hz}$ 正弦振动。

通过改变振幅 $A$，有效振动强度 $\Gamma = A(2\pi f)^2$ 从 $0.9g$ 到 $2.5g$，其中 $g$ 是重力加速度。

In Fig. 1(b) and (d), X-ray images of the granular medium (tracers within the X-ray field-of-view) before and after more than thirty minutes of vibration are shown. The small glass particles appear as a speckle background in the images due to the X-ray multiple-scattering effects.

The distinctive contrast between the tracers and the background is owing to the large X-ray absorption coefficient difference between steel and glass, which greatly facilitates the identification of the locations and speeds of the tracers.

图 1(b)和(d)显示了振动超过 30 分钟前后颗粒介质（X 射线视场内的示踪剂）的 X 射线图像。由于 X 射线的多重散射效应，小玻璃颗粒在图像中显示为斑点背景。

示踪剂与背景之间的明显对比是由于钢和玻璃之间的 X 射线吸收系数差异很大，这大大方便了对示踪球位置和速度的识别。

Convection and surface tilting

Macroscopic convection roll and surface tilting are both present in the system with the former serving as the major mechanism of the Brazil nut effect for the ascension movements of the tracers under shaking.

As the vibration intensity is gradually increased, the granular medium first develops a large surface tilting angle $\theta$ at $\Gamma = 0.9g$ without macroscopic flow (see Fig. 1(c)).

系统中同时存在宏观对流滚动和表面倾斜，前者是巴西果效应的主要机制，用于示踪子在振动下的上升运动。

巴西果效应(Brazil Nut Effect)

如果把两种颗粒的混合物置于容器中，然后由外部施加振荡，体积比较大的颗粒会上升到表层，而较小的颗粒会沉降到底部.

随着振动强度的逐渐增大，颗粒介质首先在 $\Gamma = 0.9g$ 时产生较大的表面倾斜角 $\theta$ 而不产生宏观流动（见图 1(c)）。

The granular bed is not fluidized and two tracers remain trapped at their original positions close to the container bottom.

As $\Gamma$ increases to $1.1g$, $\theta$ gradually decreases and a single convection roll develops, which drags the tracers to move upward. This is consistent with a previous study that the unjamming transition is concurrent with the appearance of convection and surface tilting.

颗粒床没有流体化，两个示踪剂仍被困在靠近容器底部的原始位置。

当 $\Gamma$ 增加到 $1.1g$ 时，$\theta$逐渐减小，并形成单个对流卷，拖曳示踪子向上移动。这与之前的研究一致，即非堵塞转变与对流, 表面倾斜的出现同时发生。

Unjamming Transition.

颗粒之间相互作用强烈时, 它们倾向于形成堵塞或者紧密排列的状态, 即"堵塞态", 类似于固体.

外部条件或者系统参数改变时, 颗粒从堵塞态变为非堵塞态, 此时颗粒之间排列相对松散,系统更容易流动, 从而类似于流体.

The convection speeds $V_{\text{conv}}$ are calculated by averaging the directed motion of background glass particles within the convection roll whose trajectories could be tracked at different $\Gamma$.

As shown in Fig.1(c), the convection speed reaches maximum at intermediate $\Gamma$. When $\Gamma$ is above $2.5g$, the granular medium turns into a granular gas and both surface tilting and convection disappear.

对流速度 $V_{\text{conv}}$ 是通过对对流辊内背景玻璃颗粒的定向运动进行平均计算得出的，这些玻璃颗粒的运动轨迹可以在不同的 $\Gamma$ 下进行跟踪。如图 1(c)所示，对流速度在中间 $\Gamma$ 时达到最大。当 $\Gamma$ 大于 $2.5g$ 时，颗粒介质变成颗粒气体，表面倾斜和对流都消失了。

Visual inspection from the top reveals that only rapid colliding and rattling motions of the granular particles can be seen which is different from the convective regime where the particles are in seemingly permanent contact with each other during the flow.

The progression of the phenomena suggests that as $\Gamma$ increases, the system first unjams, then turns into a dense granular fluid, and subsequently into a granular gas. The existence of convection will most likely introduce a volume fraction effect to influence $T_{\text{eff}}$.

从顶部目测，可以看到颗粒粒子只有快速碰撞和嘎嘎作响的运动，这与对流机制不同，在对流机制中，粒子在流动过程中似乎是相互永久接触的。

这些现象的发展表明，随着 $\Gamma$ 的增加，系统首先会解除堵塞，然后变成致密的粒状流体，最后变成粒状气体。对流的存在很可能会引入体积分数效应来影响 $T_{\text{eff}}$。

Volume Fraction Effect

多相混合系统中，不同组分的体积分数（即每种组分在混合物中所占的体积比例）对混合物性质和行为产生的影响。这个效应通常涉及到固体、液体或气体等不同相的混合物。

Motion characteristics of the tracers

We took X-ray images at an imaging speed of $150$ fps and tracked the tracers’ displacements along both $x$ and $z$ directions (see Fig. 1(a)) using an image processing routine. To avoid the artefacts brought by tracking the particle displacements in different phases of the vibration, we only analyse the images in the same phases which yields an effective imaging speed of $50$ fps.

Tracers’ motions under three typical $\Gamma$ are shown in the movies of the ESI.

We also checked the reproducibility of the vibration motion by monitoring objects fixed on the shaker and found that its position variance from different vibration cycles is negligible.

我们以 $150$ 帧/秒的成像速度拍摄 X 射线图像，并使用图像处理程序沿 $x$ 和 $z$ 两个方向跟踪示踪子的位移（见图 1(a)）。为了避免在振动的不同阶段跟踪粒子位移所带来的假象，我们只分析相同相的图像，这样可以获得 $50$ 帧/秒的有效成像速度。

示踪子在三种典型 $\Gamma$ 条件下的运动情况见 ESI 影片。

我们还通过监测固定在振动器上的物体来检查振动运动的重现性，发现不同振动周期的位置差异可以忽略不计。

Fig. 2(a) shows the trajectories of both tracers in the $x–z$ plane. Due to the initial position difference in the convection roll, the left tracer shows an almost $z$-direction motion while the right one has a large $x$-direction displacement component.

The terminal equilibrium positions of the tracers were recorded after more than thirty minutes of vibration and are marked by open symbols in Fig. 2(a).

图 2(a) 显示了两个示踪子在 $x-z$ 平面上的运动轨迹。由于对流辊中的初始位置差异，左侧示踪子几乎显示出 $z$ 方向的运动，而右侧示踪剂则有较大的 $x$ 方向位移分量。

示踪子的最终平衡位置是在振动超过 30 分钟后记录的，在图 2（a）中以空心符号标出。

Fig. 2(b) plots the left tracer’s vertical height $z$ $\textit{vs.}$ time $t$ for different $\Gamma$. Similar to the observation in a split-bottom Couette shear experiment, the tracers’ vertical trajectory $z(t)$ seems to satisfy an exponential behavior which suggests that a viscous-type of force is in action. By numerically differentiating the trajectory curves, we obtain the $v–t$ curves as shown in Fig.3, where $v = \mathrm{d}z/\mathrm{d}t$ is the tracer’s vertical velocity.

The curves can be fitted by an exponential law

$$ v = (v_{i} - v_{f})e^{-t/\tau_{0}} + v_{f} $$

where $v_{i} (v_{f})$ is the initial (final) velocity and $\tau$ is a characteristic time.

图 2(b)显示了在不同的 $\Gamma$ 条件下，左侧示踪子的垂直高度 $z$-时间 $t$。与在分底库埃特剪切实验中观察到的情况类似，示踪子的垂直轨迹 $z(t)$ 似乎满足指数行为，这表明有粘性类型的力在起作用。通过对轨迹曲线进行数值差分，我们得到了如图 3 所示的 $v-t$ 曲线，其中 $v = \mathrm{d}z/\mathrm{d}t$ 是示踪子的垂直速度。

这些曲线可以用指数定律来拟合:

$$ v = (v_{i} - v_{f})e^{-t/\tau_{0}} + v_{f} $$

其中，$v_{i} (v_{f})$ 是初始（最终）速度，$\tau$ 是特征时间。

Equation of motion and effective viscosity

In order to obtain $T_{\text{eff}}$ using the Einstein relationship $T_{\text{eff}} = 3\pi\eta d_{\text{tr}}D/k_{B}$, where $\eta$ is the viscosity and $D$ is the diffusion constant, we obtain $\eta$ and $D$ by measuring the viscous drag force on the tracer and its fluctuation dynamics respectively. There is still no generally accepted drag force model in the granular medium for both fluidized or nonfluidized granular systems despite the long-term study.

为了利用 爱因斯坦关系 $T_{\text{eff}} = 3\pi\eta d_{\text{tr}}D/k_{B}$ 得到 $T_{\text{eff}}$，其中 $\eta$ 是粘度，$D$ 是扩散常数，我们分别通过测量示踪子的粘性阻力和其波动动力学来获得 $\eta$ 和 $D$。尽管已经进行了长期研究，但在颗粒介质中，无论是流化还是非流化的颗粒系统，仍然没有普遍接受的阻力模型。

The empirical constitutive relationship using the dynamic friction coefficient based on the inertial number has been quite successful in describing the dense flow regime.

A recent study has also found that the effective friction coefficient increases with the drag velocity in addition to a linear depth dependency.39 Similarly, it has been observed that a granular system under shear behaves very similarly to a simple fluid which satisfies typical Archimedes’ law and has well-defined effective viscosity.

基于惯性数的动态摩擦系数的经验构成关系已经相当成功地描述了致密流动区域。

最近的一项研究还发现，除了线性深度依赖性外，有效摩擦系数还随着阻力速度的增加而增加。同样，已经观察到，在剪切作用下的颗粒系统的行为与简单流体非常相似，它满足典型的阿基米德定律，并具有明确定义的有效粘度。

In the following, we adopt a force model to account for the aforementioned exponential law observed. We separate the drag force into a frictional term and a “viscous” term. We assume that there are three forces in action: namely, the effective gravity of tracer $M*g$, the Coulomb friction force $F_c$, and the viscous drag force $F_{\eta}$ due to convection.

下面，我们采用一个力模型来解释上述观察到的指数规律。我们将阻力分为摩擦力和 “粘性 “力。我们假设有三种力在起作用：即示踪子的有效重力 $M^{*}g$、库仑摩擦力 $F_c$ 和对流产生的粘性阻力 $F_{\eta}$。

The tracers’ movements are over-damped, so the inertial effects can be neglected. i.e., the sum of the Coulomb friction force and the viscous drag force will balance the tracer’s effective gravity,

$$ F_{c} + F_{\eta} - M^{*}g = 0 $$

跟踪器的运动阻尼过大, 因此惯性效应可以忽略不计。库仑摩擦力和粘性阻力之和将平衡示踪子的有效重力:

$$ F_{c} + F_{\eta} - M^{*}g = 0 $$

It is well-known that the Coulomb friction force $F_{c}$ has a linear depth-dependency with the formula $F_{c} = \mu^{*}\frac{\pi}{4}d_{\text{tr}}^{2}\rho_{g}g(z_{\text{surf}-z})$, where $m^{*}$ is the effective friction coefficient, here we assume that $m^{*}$ is a constant without any drag velocity dependency, $\rho_{g}$ is the effective density of the granular medium, $z_{\text{surf}}$ is the surface height, and $z = z(t)$ is the tracer’s height at time $t$.

众所周知，库仑摩擦力 $F_{c}$ 具有线性深度依赖性，公式为 $F_{c} = \mu^{*}\frac{pi}{4}d_{\text{tr}}^{2}\rho_{g}g(z_{text{surf}-z})$ ，其中 $m^{*}$ 是有效摩擦系数. 这里我们假设 $m^{*}$ 是一个常数，与阻力速度无关; $\rho_{g}$ 是颗粒介质的有效密度；$z_{\text{surf}}$ 是表面高度；$z = z(t)$ 是示踪子在时间 $t$ 时的高度。

We also adopt a Stokes-type viscous drag force $F_{\eta} = \eta d_{\text{tr}}\left(V_{\text{conv}} - \frac{\mathrm{d}z}{\mathrm{d}t}\right)$. We insert the expressions of $F_{c}$ and $F_{\eta}$ into eqn (2) and simplify it to have the formula

$$ z = -C_{1}\eta\frac{\mathrm{d}z}{\mathrm{d}t} + C_{2} $$

where $C_1 = 4/(\mu^{*}\pi d_{\text{tr}}\rho_{g}g)$ and $C_{2} = z_{\text{surf}} + 4(\eta d_{\text{tr}} V_{\text{conv}} - M^{*}g)/(\mu^{*}\pi d_{\text{tr}}^{2}\rho_{g}g)$ are constants, $M^{*}g = (\rho_{\text{tr}} - \rho_{g})\pi d_{\text{tr}}^{3}/6$.

我们还采用斯托克斯型粘性阻力 $F_{\eta} = \eta d_{\text{tr}}\left(V_{text{conv}} - \frac{\mathrm{d}z}{\mathrm{d}t}\right)$ 。我们将 $F_{c}$ 和 $F_{\eta}$ 的表达式插入公式 (2)，并将其简化为以下公式

$$ z = -C_{1}\eta\frac{\mathrm{d}z}{\mathrm{d}t} + C_{2} $$

其中 $C_1 = 4/(\mu^{*}\pi d_{\text{tr}}\rho_{g}g)$ 和 $C_{2} = z_{\text{surf}} + 4(\eta d_{\text{tr}} V_{\text{conv}} - M^{*}g)/(\mu^{*}\pi d_{\text{tr}}^{2}\rho_{g}g)$ 是常数，$M^{*}g = (\rho_{\text{tr}} - \rho_{g})\pi d_{\text{tr}}^{3}/6$。

It is obvious that there exists two unknown parameters in the equation, including both $\mu^{*}$ and $\eta$. However, the magnitude of these two parameters can be determined since they are proportional to each other as expressed in eqn (4) when we try to match the time constant of eqn (3) with the experimentally measured $\tau_{0}$.

Additionally, the two forces have to balance the tracer’s gravity as expressed in eqn (5), which specifies the force balance equation the tracer satisfied at $t = 2\tau_{0}$. In the current study, we obtain $m^{*}$ and $\eta$ by solving these two equations,

$$ \tau_{0} = C_{1}\eta = \frac{4\eta}{\mu^{*}\pi d_{\text{tr}}\rho_{g}g},\\ \eta d_{\text{tr}}\left(V_{\text{conv}} - v|_{t=2\tau_{0}}\right) = M^{*}g - \mu^{*}\frac{\pi}{4}d_{\text{tr}}^{2}\rho_{g}g(z_{\text{surf}} - z|_{t=2\tau_{0}}) $$

很明显，方程中存在两个未知参数，包括 $\mu^{*}$ 和 $\eta$ 。不过，这两个参数的大小是可以确定的，因为当我们试图将公式 (3) 中的时间常数与实验测量的 $\tau_{0}$ 匹配时，它们是成正比的，如公式 (4) 所示。

此外，这两种力还必须平衡示踪剂的重力，如式（5）所示，该式规定了示踪子在 $t = 2\tau_{0}$ 时的力平衡方程。在当前的研究中，我们通过求解这两个方程得到了 $m^{*}$ 和 $\eta$:

$$ \tau_{0} = C_{1}\eta = \frac{4\eta}{\mu^{*}\pi d_{\text{tr}}\rho_{g}g},\\ \eta d_{\text{tr}}\left(V_{\text{conv}} - v|_{t=2\tau_{0}}\right) = M^{*}g - \mu^{*}\frac{\pi}{4}d_{\text{tr}}^{2}\rho_{g}g(z_{\text{surf}} - z|_{t=2\tau_{0}}) $$

One thing to note is that we have assumed that both $\eta$ and $V_{\text{conv}}$ are constant within the narrow $z$ range investigated. The solution yields $\mu^{*} = 1.5 \pm 0.2$ as shown in the inset of Fig. $3$, and relative values used in calculation are listed in Table $1$. The rather constant value of $m^{*}$ for different $\Gamma$ suggests the consistency of our force model.

Interestingly, the value of $\mu^{*}$ is larger than the static frictional one $0.45$ as determined by a repose angle measurement which is also observed in previous measurements.

We notice that in our system, the tracers in most cases do not rise all the way to the top of the surface and the calculated magnitudes of $\mu^{*}$ and $\eta$ are consistent with this observation, $\textit{e.g.}$, when $\Gamma = 1.7g$, in the early stage, the frictional force is $17.9\times 10^{-4}$ N, accounting for about $80$% of the effective gravity while the viscous drag force is $4.3 \times 10^{-4}$ N, which accounts for the rest $20$%.

需要注意的一点是，我们假设 $\eta$ 和 $V_{\text{conv}}$ 在所研究的较窄的 $z$ 范围内都是常数。如图 $3$ 的插图所示，求解得到 $\mu^{*} = 1.5 \pm 0.2$，计算中使用的相对值列于表 $1$。在不同的 $\Gamma$ 条件下，$m^{*}$ 的值相当恒定，这表明我们的力模型是一致的。

有趣的是，$\mu^{*}$ 的值大于静摩擦力值 $0.45$，这是由俯仰角测量确定的，在以前的测量中也观察到过。

我们注意到，在我们的系统中，示踪子在大多数情况下不会一直上升到表面顶端，$\mu^{*}$ 和 $\eta$ 的计算值与这一观察结果是一致的，比如当 $\Gamma = 1.7g$ 时, 在早期阶段，摩擦力为 $17.9\times 10^{-4}$ N，占有效重力的约 $80$%，而粘性阻力为 $4.3 \times 10^{-4}$ N，占剩余 $20$%。

As the tracer rises to the equilibrium position when the drag velocity is maximum, the viscous drag force increases to $6.6 \times 10^{-4}$ N. However, it still cannot balance the gravity alone.

The resulting $\tau_{0}$ and the the corresponding $\eta$ are shown in Fig. $4$. It is observed that as $\Gamma$ decreases, $\eta$ increases by about a decade.

当示踪子上升到平衡位置，阻力速度达到最大值时，粘性阻力增加到 $6.6 × 10^{-4}$ N。

得出的 $\tau_{0}$ 和相应的 $\eta$ 如图 $4$ 所示。可以看出，随着 $\Gamma$ 的减小，$\eta$ 会增加约 10%。

Fluctuation and diffusion dynamics

We studied the tracers’ diffusion dynamics along the $z$-direction. To obtain the fluctuating dynamics only, we subtract trajectories along the $z$-direction by their smoothed counterparts using exponential fits. The corresponding mean square displacement (MSD) as a function of $t$ under different $\Gamma$ is shown in Fig. 5(a) and (b).

The diffusive dynamics is clearly established and from each curve we can extract a diffusion constant $D$. We also define the structural relaxation time $\tau$ as the corresponding time scale when the MSD of each curve equals $\frac{d^{{2}}}{3}$. These two parameters are plotted in Fig. 5(c) and (d) respectively as a function of $\Gamma$. Overall, the diffusion constant $D$ increases as $\Gamma$ increases initially and saturates at large $\Gamma$, while $\tau$ has a decreasing trend as $\Gamma$ increases and also saturates at large $\Gamma$.

我们研究了示踪子沿 $z$ 方向的扩散动力学。为了只得到波动动态，我们用指数拟合法减去沿 $z$ 方向的平滑轨迹。图 5(a)和(b)显示了在不同的 $\Gamma$ 条件下，相应的均方位移（MSD）与 $t$ 的函数关系。

我们可以从每条曲线中提取出扩散常数 $D$，从而清晰地建立起扩散动力学。我们还将结构弛豫时间 $\tau$ 定义为每条曲线的 MSD 等于 $\frac{d^{2}}{3}$ 时的相应时间尺度。

图 5(c)和(d)分别绘制了这两个参数与 $\Gamma$ 的函数关系。总的来说，扩散常数 $D$ 随着 $\Gamma$ 的增加而增加，并在 $\Gamma$ 较大时达到饱和，而 $\tau$ 随着 $\Gamma$ 的增加呈下降趋势，并在 $\Gamma$ 较大时达到饱和。

Effective temperature and activation energy

The measured $k_{B}T_{\text{eff}}$ remains above $10^{-9}$J for the wide range of \Gamma studied as shown in Fig. 6(a) and is listed in Table $1$. This energy is on the same scale as $mgd \approx 4\times 10^{-9}$ where $m$ is the glass particle mass, which is reminiscent of the measurement using a Couette cell under constant shear. To understand the significance of $T_{\text{eff}}$, we studied how structural relaxation time $\tau$ is dependent on $T_{\text{eff}}$.

如图 6(a)所示，测得的 $k_{B}T_{\text{eff}}$ 在所研究的宽 $\Gamma$ 范围内保持在 $10^{-9}$J 以上，并列于表 $1$ 中。这个能量与 $mgd \approx 4\times 10^{-9}$J 处于同一量级，其中 $m$ 是玻璃颗粒质量，这让人想起了在恒定剪切下使用 Couette 单元进行的测量。为了理解 $T_{\text{eff}}$ 的意义，我们研究了结构弛豫时间 $\tau$ 如何取决于 $T_{\text{eff}}$。

A similar approach has been adopted to study the compaction process of granular packings under tapping, where $\Gamma$ is used instead of $T_{\text{eff}}$ and an Arrhenius law has been adopted. In the following, we follow the same strategy and compare our results with those from a hard sphere simulation in a more quantitative fashion. To be consistent with the dimensionless results from the hard sphere simulation, we normalize $T_{\text{eff}}$ and $\tau$ with the typical energy scale $pd^{3}$ and the time scale $\sqrt{pd/M}$.

采用了类似的方法来研究攻丝下颗粒填料的压实过程，其中使用了 $\Gamma$ 代替 $T_{\text{eff}}$，并采用了阿伦尼乌斯定律。在下面，我们遵循相同的策略，并以更定量的方式将我们的结果与硬球模拟的结果进行比较。为了与硬球模拟的无量纲结果保持一致，我们将 $T_{\text{eff}}$ 和 $\tau$ 与典型能量尺度 $pd^{3}$ 和时间尺度 $\sqrt{pd/M}$ 进行归一化。

Here we use a $z$-dependent pressure $p = Nmg/A_{\text{sys}}$, where $N = \phi A_{\text{sys}}(z_{\text{surf}}-z)/(\pi d^{3}/6)$ is the approximate number of glass particles above the tracer in the system with packing fraction $\phi \approx 0.60$. The values are also listed in Table 1.

这里我们使用依赖于 $z$ 的压强 $p = Nmg/A_{\text{sys}}$，其中 $N = \phi A_{\text{sys}}(z_{\text{surf}}-z)/(\pi d^{3}/6)$ 是系统中示踪子上方玻璃颗粒的近似数量，其填料分数为 $\phi \approx 0.60$。这些值也列于表 1 中。

Once $p$ is known, we obtain the dimensionless relationship between $\tau$ and $T_{\text{eff}}$ as shown in Fig. 6(b). The divergence of $\tau$ towards jamming is fitted using a simple Arrhenius law

$$ y = y_{c}e^{\frac{E}{x}}, $$

where $x = k_{\text{B}}T_{\text{eff}}/(pd^{3}), y = \tau\sqrt{pd/M}$ and $y_{c} = \tau_{c}\sqrt{pd/M}$ with $\tau_{c}$ a constant.

The fitting result yields $y_{c} = e^{6.38}$ and an activation dynamic energy scale $E = 0.20 \pm 0.02 pd^{3}$, which was found to be consistent with that of the out-of-equilibrium hard sphere fluid under shear where $E = 0.11 - 0.22pd^{3}$ and that of the thermal hard sphere fluid where $E = 0.18-0.25pd^{3}$. This suggests that the structural relaxation energy scale in vibrated dense granular medium is similar to those in hard sphere systems where structural relaxation happens by opening up of free volume against pressure.

一旦知道了 $p$，我们就可以得到 $\tau$ 和 $T_{\text{eff}}$ 之间的无量纲关系，如图 6(b) 所示。使用简单的阿伦尼乌斯定律可以拟合 $\tau$ 向干扰方向发散的情况:

$$ y = y_{c}e^{\frac{E}{x}} $$

其中，$x = k_{\text{B}}T_{\text{eff}}/(pd^{3})，y = \tau/\sqrt{pd/M}$，$y_{c} = \tau_{c}/\sqrt{pd/M}$，$\tau_{c}$ 为常数。

拟合结果得到 $y_{c} = e^{6.38}$ 和活化动能尺度 $E = 0.20 \pm 0.02 pd^{3}$，这与剪切力作用下的非平衡硬球流体的活化动能尺度 $E = 0.11 - 0.22pd^{3}$ 和热硬球流体的活化动能尺度 $E = 0.18-0.25pd^{3}$ 是一致的。

这表明，振动致密颗粒介质的结构弛豫能级与硬球系统中的结构弛豫能级相似，在硬球系统中，结构弛豫是通过打开自由体积对抗压力来实现的。

A temperature-like parameter is only very useful when it satisfies the thermodynamic zeroth law among different subsystems of a big system. In the current study, we also tested whether $T_{\text{eff}}$ satisfies this law by measuring $T_{\text{eff}}$ at different $z$.

In the following, we focus on $\Gamma = 1.1g$. As shown in Fig. 7 and Table 2, we measured the self-diffusion curves of the left tracer at several $z$. Since $T_{\text{eff}}$ is small as compared to activation energy $E = 0.78 \times 10^{-8}$ J at $\Gamma = 1.1g$, the $z$-dependent pressure $p$ has a quite dramatic effect in $D$ and the corresponding $\tau$, which is to be compared with that at large $\Gamma$, when $T_{\text{eff}}$ is much larger than $E$, the slight $\tau$ variation due to pressure change is overwhelmed by experimental uncertainties.

只有当类温度参数满足一个大系统不同子系统之间的热力学零律时，它才会非常有用。在目前的研究中，我们还通过测量不同 $z$ 下的 $T_{\text{eff}}$ 来检验 $T_{\text{eff}}$ 是否满足这一定律。

下面，我们重点讨论 $\Gamma = 1.1g$。如图 7 和表 2 所示，我们测量了左侧示踪子在多个 $z$ 条件下的自扩散曲线。由于 $T_{\text{eff}}$ 在 $\Gamma = 1.1g$时，与 $z$ 有关的压力 $p$ 对 $D$ 和相应的 $\tau$ 有相当大的影响，与之相比，当 $T_{\text{eff}}$ 远大于 $E$ 时，压力变化引起的微小 $\tau$ 变化会被实验不确定性所淹没。

Particularly, by assuming that $\tau$ satisfies eqn (6), we obtain a fairly constant $T_{\text{eff}}$ at different $z$ as listed in Table 2.

Despite the fact that the left tracer has a limited range of movement at $\Gamma = 1.1g$, the results suggest that at least in a small $z$ range, $T_{\text{eff}}$ satisfied the zeroth law for different subsystems. This observation also naturally explains why a shaken sand pile is dense and glass-like in the bottom and gas- or fluid-like at the top, assuming that a putative thermal equilibrium is reached by $T_{\text{eff}}$.

特别是，通过假设 $\tau$ 满足公式 (6)，我们可以在不同的 $z$ 条件下得到相当恒定的 $T_{\text{eff}}$，如表 2 所列。

尽管左侧示踪子在 $\Gamma = 1.1g$ 时的运动范围有限，但结果表明，至少在很小的 $z$ 范围内，$T_{\text{eff}}$ 满足不同子系统的热力学第零定律。这一观察结果也很自然地解释了为什么摇晃的沙堆在底部是致密和玻璃状的，而在顶部则是气体或流体状的，前提是$T_{\text{eff}}$达到了假定的热平衡。

Kinetic temperature

We also define a lower bound estimate of the kinetic temperature $k_{\text{B}}T_{\text{k}}^{\text{low}} = M\langle\delta v^{2}\rangle$ of the tracers using the mean square fluctuating velocity along the $z$-direction during a time period of $0.02$ s.

This corresponds to our shortest time resolution in resolving the particle displacements. The reason $T_{\text{k}}^{\text{low}}$ is a lower bound is due to the fact that even in dense flows, ballistic motions of granular particles can still be present and will lead the real $T_{\text{k}}$ higher than $T_{\text{k}}^{\text{low}}$.

The measured $T_{\text{k}}^{\text{low}}$ is shown in Fig. 6(a).

我们还定义了示踪子动力学温度的下限估计值：$k_{\text{B}}T_{\text{k}}^{\text{low}} = M\langle\delta v^{2}\rangle$ ，使用的是 $0.02$ s时间段内沿 $z$ 方向的均方波动速度。

这相当于我们解析颗粒位移的最短时间分辨率。之所以$T_{\text{k}}^{text{low}}$是一个下限，是因为即使在稠密的流动中，颗粒的弹道运动仍然可能存在，这将导致实际的$T_{\text{k}}$高于$T_{\text{k}}^{\text{low}}$。

测得的 $T_{\text{k}}^{\text{low}}$ 如图 6(a) 所示。

It is clear that $T_{\text{k}}^{\text{low}}$ is is substantially lower than $T_{\text{eff}}$ over the whole $\Gamma$ range and only reaches the same order of magnitude with our measured $T_{\text{eff}}$ when the system turns into a gas state at $\Gamma = 2.5g$.

At this $\Gamma$, the dominating particle motions are colliding motions. This contrasts with the small $\Gamma$ regime when the particles are seemingly in frictional motions against each other. When $\Gamma = 2.5g$, $T_{\text{k}}^{\text{low}}$ is on the same order of magnitude as $T_{\text{eff}}$ which also suggests that the system reaches thermal equilibrium between its short- and long-time dynamics when it turns into a granular gas.

很明显，在整个 $\Gamma$ 范围内，$T_{\text{k}}^{text{low}}$ 都大大低于 $T_{text{eff}}$，只有当系统在 $\Gamma = 2.5g$ 时进入气态时，才会达到与我们测量的 $T_{\text{eff}}$ 相同的数量级。

在这种 $\Gamma$ 条件下，主要的粒子运动是碰撞运动。这与小 $\Gamma$ 时粒子之间似乎是摩擦运动形成了鲜明对比。当 $\Gamma = 2.5g$ 时，$T_{\text{k}}^{\text{low}}$ 与 $T_{\text{eff}}$ 的数量级相同，这也表明当系统变成粒状气体时，其短时和长时动力学达到了热平衡。

Conclusions

In the current study, we have examined the complex behaviour of a dense granular system under vibration very close to the jamming density. It is found that close to jamming, the system does not show a simple $T_{\text{eff}} \propto \Gamma^{2}$ in highly agitated granular medium where the $T_{\text{eff}}$ is on the same order of magnitude as the tracers’ kinetic temperature $T_{\text{k}}$.

Instead, we observe that $T_{\text{eff}}$ Instead, we observe that $T_{\text{eff}}$ is controlled by pressure $p$ (ref. 45) which is owing to the possible universal structural relaxation mechanism similar to hard spheres. We also notice that a similar energy scale could be relevant for plastic deformation. How $T_{\text{eff}}$ is related to STZ or SGR theory for a future study.

在当前的研究中，我们考察了致密颗粒系统在非常接近干扰密度的振动下的复杂行为。研究发现，在接近堵塞态时，系统并不表现出简单的 $T_{\text{eff}}$ 在高度搅拌的颗粒介质中，$T_{text{eff}}$ 与示踪剂的动力学温度 $T_{\text{k}}$ 处于同一数量级。

相反，我们观察到 $T_{\text{eff}}$ 是由压力 $p$ 控制的（参考文献 45），这是由于可能存在与硬球类似的普遍结构松弛机制。我们还注意到，类似的能级可能与塑性变形有关。$T_{\text{eff}}$ 与 STZ 或 SGR 理论的关系有待今后研究。

Acknowledgements & References

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