The dynamics of two-dimensional fluids confined within a random matrix of obstacles is investigated using both colloidal model experiments and molecular dynamics simulations. By varying fluid and matrix area fractions in the experiment, we find delocalized tracer particle dynamics at small matrix area fractions and localized motion of the tracers at high matrix area fractions. In the delocalized region, the dynamics is subdiffusive at intermediate times, and diffusive at long times, while in the localized regime, trapping in finite pockets of the matrix is observed. These observations are found to agree with the simulation of an ideal gas confined in a weakly correlated matrix. Our results show that Lorentz gas systems with soft interactions are exhibiting a smoothening of the critical dynamics and consequently a rounded delocalization-to-localization transition.

The generic mechanisms of anomalous transport in porous media are investigated by computer simulations of two-dimensional model systems. In order to bridge the gap between the strongly idealized Lorentz model and realistic models of porous media, two models of increasing complexity are considered: a cherry-pit model with hard-core correlations as well as a soft-potential model. An ideal gas of tracer particles inserted into these structures is found to exhibit anomalous transport which extends up to several decades in time. Also, the self-diffusion of the tracers becomes suppressed upon increasing the density of the systems. These phenomena are attributed to an underlying percolation transition. In the soft potential model the transition is rounded, since each tracer encounters its own critical density according to its energy. Therefore, the rounding of the transition is a generic occurrence in realistic, soft systems.

We introduce fractal liquids by generalizing classical liquids of integer dimensions d ¼ 1; 2; 3 to a noninteger dimension d l . The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here, we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semianalytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results. DOI: 10.1103/PhysRevLett.115.097801 PACS numbers: 61.20.Gy, 61.20.Ja, 61.43.Hv The liquid state, an intermediate between gas and solid, exhibits short-ranged particle pair correlations in isotropic shells around a tagged particle [1]. While the shell structure is lost for gases, solids exhibit long-ranged correlations and anisotropy. Particle correlations are accessible by experiments [2,3] and contain valuable information about the particle interactions. A fundamental task in theory and computer simulation of the liquid state is to predict particle correlations for given interactions. In this respect, one particularly successful approach is liquid integral equation theory [1,4].Molecular and colloidal liquids can be restricted to one or two spatial dimensions [5,6] by confining them on substrates [7], at interfaces [8], between plates [9,10], or in channeled matrices [11] or optical landscapes [12]. The thermodynamic properties change accordingly [13] and the systems are called one-dimensional or two-dimensional (2D) liquids. While these classical examples of confinement involve integer-dimensional configuration spaces, there are also cases where liquids are confined in porous media [14][15][16][17], or along quenched polymer coils [18], and where the configuration space exhibits noninteger (fractal) dimension at suitable length scales. Much work has been devoted to understanding the limiting cases of low and high density-namely, the motion of a single particle on a fractal [19][20][21][22][23] and the structure of a fractal aggregate itselfcorresponding to an arrested high-density particulate system [24][25][26][27][28]. In all of these situations, particles interact in the embedding integer-dimensional space and interaction energies depend on Euclidean particle distance.In this Letter, we break new ground by considering fractal particles in a fractal configuration space, both of the same noninteger dimension. An alternative model in which the particle dimension differs from the configuration-space dimension is briefly mentioned near the end of the Letter. Figure 1 is a snapshot from one of our Monte Carlo (MC) simulations. The interaction between two particles is not described by a pair potential in Euclidean spa...

Contact inhibition plays a crucial role in cell motility, wound healing, and tumour formation. By mimicking the mechanical motion of cells crawling on a substrate, we constructed a minimal model of migrating cells that naturally gives rise to contact inhibition of locomotion (CIL). The model cell consists of two disks, a front disk (a pseudopod) and a back disk (cell body), which are connected by a finite extensible spring. Despite the simplicity of the model, the collective behaviour of the cells is highly non-trivial and depends on both the shape of the cells and whether CIL is enabled. Cells with a small front disk (i.e., a narrow pseudopod) form immobile colonies. In contrast, cells with a large front disk (e.g., a lamellipodium) exhibit coherent migration without any explicit alignment mechanism in the model. This result suggests that crawling cells often exhibit broad fronts because this helps facilitate alignment. After increasing the density, the cells develop density waves that propagate against the direction of cell migration and finally stop at higher densities.Directional collective motion of cells is fundamentally important for embryogenesis, wound healing and tumour invasion [1][2][3][4][5] . Cells move in clusters, strands or sheets to cover empty areas 6 , grow or invade tissues. The manner in which the cells coordinate and control their motion is the subject of ongoing research. At the level of a single cell, it is well-established that a cell's motion is intricately linked to its shape. The shape of crawling cells is highly variable and depends on the type of cell, the substrate and aspects of the migration process itself [7][8][9][10] . To move, a cell needs to break symmetry 8 , as a circular cell does not move. While there is evidence that shape has a strong influence on scattering and can lead to clustering and collective directed motion in the case of active swimming particles 11,12 , less is known regarding the role of cell shape in the organization of collective crawling. It has been shown in simulations that inelastic collisions between crawling cells, e.g., due to deformation, can lead to coherent migration [13][14][15][16][17] , which suggests that deformation is important for collective cell behaviour. When crawling cells come into contact, their protrusions are inhibited, which tends to change their shape and orientation 18,19 . It was shown that this effect, which is called contact inhibition of locomotion (CIL), enables cells to follow chemical gradients more effectively by aligning them 20,21 . In growing colonies, CIL leads to a slowdown of the motility of individual cells when the density of their environment crosses a certain threshold 22 . Thus, CIL is believed to play a crucial role in the control of collective tissue migration 15,20,[23][24][25][26] , tissue growth 22,27 , morphogenesis, wound healing and tumour development 28 . The behaviour of cells undergoing CIL depends on many factors, such as the presence of cell adhesion molecules and receptors, and different types o...