Size-topology relations in packings of grains, emulsions, foams and biological cells K. Newhall, 1 L. L. Pontani, 2 I. Jorjadze, 2 S. Hilgenfeldt, 3 and J. Brujic 2 1 New York University, Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY, 10012, USA 2 New York University, Department of Physics and Center for Soft Matter Research, 4 Washington Place, New York, NY, 10003, USA 3 Mechanical Science and Engineering, University of Illinois at Urbana Champaign, 1206 W GreenSt, Illinois, IL, 61801, USA (Dated: May 17, 2012) Particulate packings in 3D are used to study the effects of compression and polydispersity on the geometry of the tiling in these systems. We find that the dependence of the neighbor number on cell size is quasi-linear in the monodisperse case and becomes nonlinear above a threshold polydispersity, independent of the method of creation of the tiling. These size-topology relations can be described by a simple analytical theory, which quantifies the effects of positional disorder in the monodisperse case and those of size disorder in the polydisperse case, and is applicable in two and three dimensions. The theory thus gives a unifying framework for a wide range of amorphous systems, ranging from biological tissues, foams and bidisperse disks, to compressed emulsions and granular matter. Systems that tile space range from tessellations of particulate packings [1–4] to soap foams [5–10], biolog- ical tissues [11–13] and even mathematically generated tilings [14]. In each of these systems, the size distribu- tion, topology, and dimensionality of the constituent cells determine the relation between the number of neighbor- ing cells and the cell size [15, 16]. For example, this relationship is linear in the case of the empirical Lewis law [11] observed in epithelial tissues from organisms as diverse as the Drosophila wing, Xenopus tadpole tail, Hydra vulgaris or the cucumber epidermis, but it fol- lows a power law in the case of foam tilings [9, 10, 17]. While many theoretical approaches have been devised to predict a specific size-topology trend [17–22], none have been able to explain the origin of the discrepancy be- tween systems nor quantify the limits of validity of dif- ferent size-topology relations. Another open question is how these relations are affected by the dimensionality of the system, given that structural information inside 3D tilings is difficult to access. To this end, we confocally im- age transparent, fluorescent particulate packings ranging from monodisperse poly-methyl-methacrylate (PMMA) particles to compressed emulsion droplets with varying polydispersity. The individual cells in the mosaic are con- structed either by tessellating space around spheres using the navigation map [23, 24] or by physical compression of the particles themselves. We find that the monodis- perse particles follow a different size-topology relation to the polydisperse emulsions, independent of the method of creation of the tiling. Building on the work in [22], we derive analyti- cal results from simplified versions of the granocentric model [1, 25] for the size-topology relations to explain the diverse data in 2D and in 3D. This idealized model is based on creating individual cells in the tiling by sur- rounding a central particle with a first shell of neighbors. Variations in the particle sizes introduce disorder arising from polydispersity, while fluctuations in the surface-to- surface distance between the neighbors mimic positional disorder in the packing. This model allows us to decou- ple the two effects and distinguish between systems where the particles are all of the same or similar size but the distance between them can vary (i.e. positional disorder) and those where the size-distribution of the particles (i.e. polydispersity) dominates the disorder in the tiling. We find that the analytical predictions of the model capture experimental trends in systems as diverse as biological tissues, bubbles, droplets and grains, and thus classify them according to the dominant source of disorder in their packing. Figure 1 depicts examples of amorphous tilings rang- ing from epithelial cells and bidisperse disks in 2D, to the confocal slices of compressed and relaxed emulsions in 3D. The emulsion packings are visualized in 3D by re- fractive index matching the droplets with the continuous phase and dying the particles using fluorescent Nile Red dye. The image analysis, based on a Fourier transform algorithm, identifies the particle positions and radii [26]. Note that the cells are either defined by the space fill- ing of deformable particles as in Fig. 1(A, C) or by the navigation map tessellation of space [23, 24] around circu- lar/spherical particles as shown in Fig. 1(B, D). The nav- igation map tessellation attributes each point in space to the particle whose surface is closest to it. In the monodis- perse case, this method reduces to the Voronoi tessella- tion. In the polydisperse case, this method results in individual cell volumes separated by hyperbolic surfaces. In either case, the number of neighbors n is defined as the number of interfaces that a cell shares with other cells. In 2D, the cellular tissues and jammed disks follow the linear law proposed by Lewis, while the foam data follow a nonlinear increase with the cell area, as shown in Fig. 2. A similar distinction is observed for the dependence of the average cell volume V on neighbor number in packings of monodisperse PMMA particles and polydisperse emul- sions in 3D, as shown in Fig. 3(A). Interestingly, varying the level of polydispersity p (where p is the coefficient of variation of the droplet radii) from 11% to 42% in 3D