Wang Ping, Song Chaoming, Briscoe Christopher, Makse Hernán A
Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Jun;77(6 Pt 1):061309. doi: 10.1103/PhysRevE.77.061309. Epub 2008 Jun 30.
We report experimental measurements of particle dynamics on slowly sheared granular matter in a three-dimensional Couette cell. A closely packed ensemble of transparent spherical beads is confined by an external pressure and filled with fluid to match both the density and refractive index of the beads. This allows us to track tracer particles embedded in the system and obtain three-dimensional trajectories [r(t),theta(t),z(t)] as a function of time. We study the probability distribution function of the vertical and radial displacements, finding Gaussian and exponential distributions, respectively. For slow shear rates, the mean-square fluctuations in all three directions are found to be dependent only on the angular displacement of the Couette cell, Delta theta e, (Delta z 2) approximately Delta theta e, (Delta r2) approximately Delta theta e alpha, Delta theta 2 approximately Delta theta e beta, where alpha and beta are constants. With Delta theta e proportional to the time between measurements, the values of the constants, alpha and beta , are found to be subdiffusive and superdiffusive, respectively. ThFe linear relation between (Delta z 2) and angular displacement implies a diffusive process, from which we can calculate an "effective temperature," T eff, in the vertical direction, through a fluctuation-dissipation relation. It is of interest to determine whether these systems can be described by analogous equilibrium statistical mechanics concepts such as "effective temperature" and "compactivity." By studying the dynamics of tracer particles, we find the effective temperature defined by the Stokes-Einstein relation to be independent of the tracer particle characteristic features, such as density and size, and dependent only on the packing density of the system. For slow shear rate, both the diffusivity and mobility of tracer particles are proportional to the shear rate, giving rise to a constant effective temperature, characteristic of the jammed system. We finally discuss the significance of the existence of T eff for a statistical mechanics formulation of granular matter.
我们报告了在三维库埃特流槽中对缓慢剪切颗粒物质的颗粒动力学进行的实验测量。紧密堆积的透明球形珠子集合体受到外部压力的限制，并充满流体以匹配珠子的密度和折射率。这使我们能够追踪嵌入系统中的示踪粒子，并获得作为时间函数的三维轨迹[r(t),θ(t),z(t)]。我们研究了垂直和径向位移的概率分布函数，分别发现高斯分布和指数分布。对于缓慢的剪切速率，发现所有三个方向上的均方涨落仅取决于库埃特流槽的角位移，Δθε，(Δz²)≈Δθε，(Δr²)≈Δθεα，Δθ²≈Δθεβ，其中α和β是常数。由于Δθε与测量之间的时间成正比，发现常数α和β的值分别是亚扩散和超扩散的。(Δz²)与角位移之间的线性关系意味着一个扩散过程，由此我们可以通过涨落耗散关系计算垂直方向上的“有效温度”Teff。确定这些系统是否可以用诸如“有效温度”和“压缩性”等类似的平衡统计力学概念来描述是很有意义的。通过研究示踪粒子的动力学，我们发现由斯托克斯 - 爱因斯坦关系定义的有效温度与示踪粒子的特征特性（如密度和大小）无关，仅取决于系统的堆积密度。对于缓慢的剪切速率，示踪粒子的扩散率和迁移率都与剪切速率成正比，从而产生一个恒定的有效温度，这是堵塞系统的特征。我们最后讨论了Teff的存在对于颗粒物质统计力学公式的意义。
