Simulation Study of Seemingly Fickian but Heterogeneous Dynamics of Two Dimensional Colloids Jeongmin Kim, 1 Chanjoong Kim, 2 and Bong June Sung 1 1 Department of Chemistry, Sogang University, Seoul 121-742, Republic of Korea 2 Chemical Physics Interdisciplinary Program and Liquid Crystal Institute, Kent State University, Kent, OH 44242, USA (Received 4 October 2012; revised manuscript received 19 November 2012; published 24 January 2013) A two-dimensional (2D) solid lacks long-range positional order and is diffusive by means of the cooperative motion of particles. We find from molecular dynamics simulations of hard discs that 2D colloids in solid and hexatic phases show seemingly Fickian but strongly heterogeneous dynamics. Beyond translational relaxation time, the mean-square displacement is linear with time, t, implying that discs would undergo Brownian diffusion and the self-part of the van Hove correlation function [G s ðr; tÞ] might be Gaussian. But dynamics is still heterogeneous and G s ðr; tÞ is exponential at large r and oscillatory with multiple peaks at intermediate length. We attribute the existence of several such peaks to the observation that there are several clusters of discs with discretized mobility. The cluster of marginally mobile discs grows with time and begins to percolate around translational relaxation time while clusters of fast discs emerge in the middle of the marginally mobile cluster. DOI: 10.1103/PhysRevLett.110.047801 PACS numbers: 61.20.Lc, 05.40.a, 64.70.dj, 66.30.h Dynamics in supercooled liquids [1–3], gels [4], and porous materials [5] is complex and depends on both time and length scales. Three well-defined regimes of dynamics have been proposed based on simulations and experiments [6,7]. At short time and length scales, a bal- listic motion is a dominant mechanism for diffusion. At very large spatiotemporal scales beyond translational re- laxation time (  ), the distribution of the particle displace- ment [the self-part of the van Hove correlation function, G s ðr; tÞ] is Gaussian and the dynamics is Fickian; i.e., the mean-square displacement (MSD) is linear with time t. At an intermediate scale, particles may show subdiffusive behavior with MSD  t b and b< 1. G s ðr; tÞ is not Gaussian and often divides into two parts that fit well with a Gaussian function for small r and an exponen- tial function for large r, respectively [8]. And Pðr; tÞ [  2rG s ðr; tÞ] often shows the second peak due to mobile particles that undergo hopping motions [9,10]. It has been supposed for decades that such a non-Gaussian G s ðr; tÞ would result in a non-Fickian diffusion at corre- sponding time scales. However, recent studies revealed that G s ðr; tÞ could be exponential instead of being Gaussian while Fickian diffusion is still observed with MSD  t [8,9,11]. In such cases, intermediate and large spatiotem- poral scales are not always separated sharply; for example, Wang et al. showed that colloid beads on phospholipid bilayer tubes or in entangled actin suspensions entered the regime of Fickian diffusion while G s ðr; tÞ is non-Gaussian with an exponential tail at large r [11]. They suggested that Fickian diffusion with such a non-Gaussian distribution could be observed in various complex physicochemical and socioeconomic systems. In this Letter, we report an important case of two- dimensional (2D) colloids where dynamics becomes Fickian soon after   , even though G s ðr; tÞ at intermediate length is neither Gaussian nor exponential but has several peaks for t  12  . We show that even in a simple system of 2D colloids the separation of intermediate and large spatiotemporal scales is unclear even after more than an order of magnitude times   . This implies that 2D colloids in solid and hexatic phases undergo exceptionally corre- lated and collective motions. We also find that in a liquid phase, a caging time ( c ), during which a particle is in a cage of neighbor particles before it escapes, is smaller than a noncaging time ( nc ) that a particle has to wait before a cage forms to trap the particle. Near freezing transition,  c   nc . Cage formation and hopping motion are sensi- tive to the thermodynamic phase transition. 2D solids might look as if they are simpler than 3D systems. However, due to the lack of a thermodynami- cally stable crystal and a long-range positional order in a solid phase [12], their dynamics is quite complex as well. According to the celebrated Kosterlitz-Thouless-Halperin- Nelson-Young theory, 2D solids would melt via an inter- mediate phase called a hexatic phase that does not exist in three-dimensional solids [13–15]. This has drawn great attention regarding the existence of the hexatic phase and the nature of transitions among solid, hexatic and liquid phases. A recent simulation study by Bernard and Krauth showed that the hexatic-to-solid transition was continuous and the hexatic-to-liquid transition was a first order phase transition [16]. There were relatively few studies on the 2D colloid dynamics near the melting transition [17]. Zangi and Rice found from a simulation study for a quasi-two- dimensional liquid that dynamics became strongly hetero- geneous and the cooperative motions of particles were generated by instantaneous normal mode vibrations [18]. And dynamic criteria were introduced to identify 2D PRL 110, 047801 (2013) PHYSICAL REVIEW LETTERS week ending 25 JANUARY 2013 0031-9007= 13=110(4)=047801(5) 047801-1 Ó 2013 American Physical Society