1.Centre for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China 2.College of Chemistry, Chemical Engineering and Materials Science, Soochow University, Suzhou 215123, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 11704269) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 17KJB140020)
Received Date:10 August 2020
Accepted Date:31 August 2020
Available Online:01 December 2020
Published Online:05 December 2020
Abstract:The diffusive transport in complex confined media is ubiquitous such as diffusions of micro- or nano-particles in glassy liquids and polymer solutions, protein diffusions under crowded conditions, and deliveries of drugs in the biological media. Therefore, the understanding of the diffusive transport arouses the great interest of researchers in the physics, materials science, and biology circles. Despite the fact that the shape of the colloidal particles acts as one of the important physical factors influencing their dynamic behaviors, the study of the anisotropic particles diffusing in confined media is still lacking. In this work, we propose a simple experimental model to investigate the confined diffusion of shape-anisotropic particles. The diffusion of an ellipsoid at different area fractions (?) of colloidal spheres is investigated through video microscopy. At low ?, ellipsoid exhibits a random trajectory and free diffusion in translational and rotational degree of freedom; while at high ?, the trajectory is in a small spatial range with a nearly constant orientation of the particle, indicating that the arrested diffusion takes place in translational and rotational degree of freedom. The translational and rotational mean square displacement decrease with the increase of ?. By power-law fitting (~tβ), it is found that β decreases from 1 to a small value at high ?, demonstrating that the ellipsoid experiences a transition from normal diffusion to sub-diffusion. Moreover, β for rotational motion decreases faster than that for translational motion at high ?, which signifies that the the rotational motion decouples from the translational motion with increasing ?. The results from the van Hove correlation function show that the translational displacement along the major axis of the ellipsoid is always larger than that along the minor axis, manifesting the ellipsoid prefers to diffuse along its major axis independent of ?. Significant non-Gaussian tail is observed in the distribution of the translational displacement along the major axis with increasing ?. However, the distribution of the translational displacement along the minor axis presents a nearly Gaussian behavior independent of ?. This indicates that the translational motion along the major axis decouples from the translational motion along the minor with increasing ?. For the rotational displacement, the non-Gaussian tail is only observed at the intermediate ?. These non-Gaussian behaviors are confirmed by calculating the non-Gaussian parameter (α2). Our experiments demonstrate that the confinements give rise to the anomalous diffusion behaviors of the anisotropic colloids, which is conducive to the understanding of transportations of anisotropic objects in complex environments. Keywords:colloids/ diffusion/ ellipsoid/ non-Gaussian distribution/ dynamics
3.实验结果与讨论为了直观展示椭球粒子扩散行为的运动特征, 绘制了椭球粒子在不同圆球面积分数下运动100 s的轨迹, 如图2所示. 在? = 0.57时, 椭球粒子的质心可以在很大的空间内移动, 平动运动明显, 表现出无规随机扩散; 粒子取向角度变化频繁, 其转动也呈现无规扩散行为. 然而在面积分数较大时(? = 0.81), 粒子运动的空间范围减小, 并且粒子的取向几乎不变, 这表明粒子的扩散在平动自由度和转动自由度都受到了抑制. 图 2 椭球在不同圆球浓度下运动100 s的轨迹: ? = 0.57(左)和? = 0.81(右). 椭球的不同时刻位置用空心椭圆表示, 其取向是椭球长轴方向和x轴方向的夹角, 用颜色表示 Figure2. 100 s trajectories of an ellipsoid at ? = 0.57 (left panel) and ? = 0.81 (right panel). The positions of the particle at different times are indicated by ellipses. The color indicates the orientation of the particle with respect to the x axis.
为了定量地分析椭球的动力学, 计算了不同圆球面积分数?下椭球的均方位移(mean square displacement, MSD). 得出椭球平动的MSD为$\langle {\Delta r^2(t)}\rangle$ = $\langle {|r(t)\!-\!r(0)|^2} \rangle$, 转动的MSD为$\langle {\theta r^2(t)}\rangle$=$\langle {|\theta(t)\!- \! \theta(0)|^2} \rangle$, 其中r(t) 和 θ(t) 分别表示t时刻椭球的中心位置和取向. 图3展示了椭球在不同圆球面积分数下均方位移. 随着圆球面积分数?的增大, 平动和转动均方位移均减小, 这源于椭球粒子的运动空间受到了周围圆球粒子的限制, 使其动力学减慢. 并且, 在高圆球面积分数下, 椭球的平动和转动均方位移都呈现出明显的次扩散行为. 为了更好反映这种次扩散行为的演化, 对平动和转动均方位移在1—20 s的时间范围内进行冥律(power-law)拟合: MSD~tβ, 得到了扩散指数β. 图4给出不同圆球面积分数?下平动和转动的扩散指数β. 当? = 0.57时, β值接近1, 表明圆球粒子的受限影响较小, 椭球表现出普通的扩散行为, 这与轨迹的结果相一致. 随着?增大, 平动和转动的扩散指数都开始减小, 说明椭球因为周围圆球的受限呈现次扩散行为. 此外, 在较高的圆球面积分数下(? > 0.75), 转动的扩散指数比平动的扩散系数下降得更多, 表明椭球的转动比平动受限严重. 说明椭球的转动扩散和平动扩散在比较强的受限情况下发生了解耦合(decoupling)行为, 这一结果与四面体、菱形等非球形粒子在浓密的胶体体系中的扩散行为一致[16,19], 表明非球形粒子在受限情况下的转动和平动扩散的解耦合行为具有一定的普遍性. 图 3 椭球在不同?下的平动均方位移(a)和转动均方位移(b), 实线是时间范围为1?20 s的幂律拟合, ~tβ Figure3. Translational mean square displacements (a) and rotational mean square displacements (b) of ellipsoids at different ?. Solid lines are the power-law fits: ~tβ in the time range of 1?20 s.
图 4 平动和转动扩散指数β随浓度的变化 Figure4. The ? dependent β for translational and rotational motions.
为了深入分析椭球粒子的扩散动力学, 本文还计算了范霍夫自关联函数(self-part of the van Hove correlation function), 用来表征椭球粒子运动位移的概率分布. 对于平动, 范霍夫自关联函数为 ${G_{\rm{s}}}(r, t) = \left\langle {\delta (r - \Delta {r_{}}(t))} \right\rangle$; 对于转动, ${G_{\rm{s}}}(\theta, t) = \left\langle {\delta (\theta - \Delta \theta (t))} \right\rangle $, 其中Δr(t)和Δθ(t)分别是椭球粒子经过t时间运动的平动位移和转动位移. 对平动进行更细致的分析, 将平动位移分解为两部分, 即沿着椭球长轴方向的位移和沿着短轴方向的位移. 图5(a),(b)分别给出了椭球粒子在4 s内沿着长轴方向和短轴方向的平动位移的概率分布. 结果表明, 在任意圆球面积分数?下, 沿椭球长轴方向的平动位移分布始终比沿短轴方向的平动位移分布要宽, 说明椭球粒子更倾向于沿其长轴方向进行扩散. 这与在没有受限情况下的椭球自由扩散行为一致[11], 表明这种各向异性的扩散行为与周围圆球的受限环境无关. 然而, 随着圆球面积分数?的增大, 沿椭球长轴方向的平动位移分布逐渐偏离高斯分布, 在高面积分数呈现非高斯行为; 而沿椭球短轴方向的平动位移始终接近高斯分布. 这说明随着?的增大, 沿椭球长轴方向和短轴方向的平动行为也发生了解耦合, 并非发生同样的演化行为. 图5(c)展示了对应的转动在不同圆球面积分数下的位移分布, 与前面两种平动位移分布的演化都不相同. 转动位移分布只在中间面积分数先呈现显著的非高斯行为. 这进一步说明了随着?的增大, 椭球粒子的平动和转动扩散演化不同步, 两种运动发生解耦合. 图 5 椭球在不同?运动4 s的位移分布 (a)沿长轴方向平动位移; (b)沿短轴方向平动位移; (c)转动位移. 实线是高斯拟合 Figure5. The distribution of the ellipsoid displacement for lag time of 4 s at different ?: (a) Translational displacement along the long axis of the ellipsoid; (b) translational displacement along the short axis, (c) rotational displacement. Solid lines are the best Gaussian fits.
为了进一步确认位移分布的非高斯行为, 计算了非高斯参量: α2 = 1/2$\langle {\Delta ^4(t)}\rangle/\langle {\Delta^2(t)}\rangle^2-1$ , 其中Δ(t)代表椭球粒子经过t时间运动的平动位移或转动位移. 图6给出了椭球粒子运动t = 4 s的平动位移非高斯参量和转动位移非高斯参量. 对于沿椭球长轴方向的平动位移, α2随着圆球面积分数?的增大而变大, 证实了其位移分布越来越偏离高斯行为. 这与胶体玻璃化转变中的结果相似, 即在体系浓度越来越高时, 粒子运动受到的牢笼效应越来越明显, 并出现逐渐增强的次扩散行为和非高斯行为[22]. 这里, 随着圆球面积分数增大, 椭球运动受到由圆球构成的牢笼限制, 从而导致了非高斯行为. 然而, 对于沿椭球短轴方向的平动位移, α2保持很小的值, 表明其接近高斯行为, 与位移分布的结果相一致. 可能的原因是椭球平动趋向于沿长轴方向扩散, 而沿短轴方向的扩散本身就很小, 因此牢笼效应主要影响沿长轴方向的运动, 引起非高斯行为; 而对短轴方向的扩散影响较小, 依然保持高斯行为. 对于转动位移, α2先增大后减小, 呈现非单调变化, 与转动位移分布结果一致. 这种增大是源于圆球的牢笼效应, 而减小则可能由于在很大的圆球面积分数下, 相比于平动, 椭球的转动已经被强烈限制[22], 均方位移的结果也说明了这一点. 因而导致了α2减小, 使转动位移趋向高斯位移分布. 这与圆球胶体体系发生玻璃化转变后的非高斯行为相似, 这种情况下粒子平动运动被冻结后, 非高斯行为会减弱[11]. 在本文的体系中, 由于椭球在高面积分数下, 其转动相比于平动先被冻结[22], 因而转动的非高斯行为会在高面积分数下减弱, 对应的位移分布又趋向高斯行为. 图 6 椭球运动4 s沿长轴平动位移 (r//), 沿短轴平动位移 (r⊥) 和转动位移 (θ) 的非高斯参量 Figure6. The non-Gauss parameter of the displacement of ellipsoid for lag time of 4 s: Translational displacement along the long axis of the ellipsoid (r//), translational displacement along the short axis (r⊥), and rotational displacement (θ).