Sticky Matters: Jamming and Rigid Cluster Statistics with Attractive Particle Interactions

Jamming transition and normal modes of polydispersed soft particle packing

The jamming transition of soft particles characterized by narrow size distributions has been well studied by physicists. However, polydispersed systems are more relevant to engineering, and the influence of polydispersity on jamming phenomena is still unexplored. Here, we numerically investigate jamming transitions of polydispersed soft particles in two dimensions. We find that polydispersity strongly influences contact forces, local coordination, and the jamming transition density. In contrast, the critical scaling of pressure and elastic moduli is not affected by the particle size distribution. Consistent with this observation, we find that the vibrational density of states is also insensitive to the polydispersity. Our results suggest that, regardless of particle size distributions, both mechanical and vibrational properties of soft particle packings near jamming are governed by the distance to jamming.

Void Connectivity and Criticality in the Compression-Induced Gel-to-Glass Transition of Short-Range Attractive Colloids

Gels and attractive glasses-both dynamically arrested states formed through short-range attraction-are commonly found in a range of soft matter systems, including colloids, emulsions, proteins, and wet granular materials. Previous studies have revealed intriguing similarities and distinctions in their structural, dynamic, vibrational, and mechanical properties. However, the microstructural mechanisms underlying the gel-to-glass transition remain elusive. To address this, we investigate uniaxial compression-induced gel collapse using confocal microscopy, which provides experimental access to the relationship between mechanical stress and microstructure. Together with Brownian dynamics simulations, our study reveals two sequential transitions in void structure: from gels with percolating voids to isolated voids, and ultimately to voidless glasses. These transitions are closely linked to a shift from superlinear power-law scaling to explosive divergence toward the packing limit in both normal and deviatoric stresses as a function of volume fraction, particularly at lower temperatures. Understanding these mechanically self-organized transitions and their associated criticality deepens our insight into disordered solids, enabling better control over mechanical properties, interfacial characteristics, and transport behavior in porous materials.

Thinning by cluster breaking: Active matter and shear flows share thinning mechanisms

Among many unexpected phenomena of active matter is the recently observed superfluid-like thinning (viscosity drop) behavior of bacteria suspensions. Understanding this peculiar self-propelled thinning by active matter is of theoretical and practical importance. Here, we find that, although distinct in driving mechanisms, active matter and shear flows exhibit similar thinning behaviors upon the increase of self-propulsion and shear forces, respectively. Our structural characterizations reveal that they actually share the same cluster-breaking mechanism of thinning. How fast and how shattered the cluster is broken determines the (dis)continuity of the thinning. This explains why adding active particles to Newtonian fluids can cause thinning, in which rotation of active particles play a key role in breaking clusters. Our work proposes a mechanism of self-propelled thinning and further establishes the underlying connections between active matter and shear flows.

Connecting polymer collapse and the onset of jamming

Previous studies have shown that the interiors of proteins are densely packed, reaching packing fractions that are as large as those found for static packings of individual amino-acid-shaped particles. How can the interiors of proteins take on such high packing fractions given that amino acids are connected by peptide bonds and many amino acids are hydrophobic with attractive interactions? We investigate this question by comparing the structural and mechanical properties of collapsed attractive disk-shaped bead-spring polymers to those of three reference systems: static packings of repulsive disks, of attractive disks, and of repulsive disk-shaped bead-spring polymers. We show that the attractive systems quenched to temperatures below the glass transition T≪T_{g} and static packings of both repulsive disks and bead-spring polymers possess similar interior packing fractions. Previous studies have shown that static packings of repulsive disks are isostatic at jamming onset, i.e., the number of interparticle contacts N_{c} matches the number of degrees of freedom, which strongly influences their mechanical properties. We find that repulsive polymer packings are hypostatic at jamming onset (i.e., with fewer contacts than degrees of freedom) but are effectively isostatic when including stabilizing quartic modes, which give rise to quartic scaling of the potential energy with displacements along these modes. While attractive disk and polymer packings are often considered hyperstatic with excess contacts over the isostatic number, we identify a definition for interparticle contacts for which they can also be considered as effectively isostatic. As a result, we show that the mechanical properties (e.g., scaling of the potential energy with excess contact number and low-frequency contribution to the density of vibrational modes) of weakly attractive disk and polymer packings are similar to those of isostatic repulsive disk and polymer packings. Our results demonstrate that static packings generated via attractive collapse or compression of repulsive particles possess similar structural and mechanical properties.

Properties of packings and dispersions of superellipse sector particles

Superellipse sector particles (SeSPs) are segments of superelliptical curves that form a tunable set of hard-particle shapes for granular and colloidal systems. SeSPs allow for continuous parametrization of corner sharpness, aspect ratio, and particle curvature; rods, circles, rectangles, and staples are examples of shapes SeSPs can model. We compare three computational processes: pair-wise Monte Carlo simulations that explore particle-particle geometric constraints, Monte Carlo simulations that reveal how these geometric constraints play out over dispersions of many particles, and Molecular Dynamics simulations that form random loose and close packings. We investigate the dependence of critical random loose and close packing fractions on particle parameters, finding that both values increase with opening aperture and decrease with increasing corner sharpness. The identified packing fractions are compared with the mean-field prediction of the random contact model; we find deviations from the model's prediction due to correlations between particle orientations. The complex interaction of spatial proximity and orientational alignment is also explored with a generalized spatioorientational distribution area (SODA) plot, which shows how higher density packings are achieved through particles assuming a small number of preferred configurations that depend sensitively on particle shape and system preparation.

The combined effect of cohesion and finite size on the collapse of wet granular columns

The collapse of low-saturation liquid-containing granular materials is prevalent in nature and industrial processes, and understanding the associated transient dynamics is extremely important for exploring such complex flow processes. In this paper, the collapse of a finite-size wet granular column is systematically studied and the determinants affecting its dynamics are analyzed based on the discrete element model for wet particles and the corresponding small-scale experiments. With the aid of parametric analysis, the dimensionless cohesion parameter containing the system size and grain-scale bond number is proposed, and its relevance in characterizing column stability and collapse dynamics of wet granular materials is further confirmed. For the collapse of wet granular columns with a fixed aspect ratio, the initial height contained in the cohesion parameter is verified to be a manifestation of the finite size effect, which is present in a wet granular collapse and is coupled with the cohesive effect. Such a coupling effect is taken into account in our proposed scaling laws that can be applied to uniformly describe the deposit morphology of wet granular columns after collapse.

Stimuli-responsive viscosity modifiers

Stimuli responsive viscosity modifiers entail an important class of materials which allow for smart material formation utilizing various stimuli for switching such as pH, temperature, light and salinity. They have seen applications in the biomedical space including tissue engineering and drug delivery, wherein stimuli responsive hydrogels and polymeric vessels have been extensively applied. Applications have also been seen in other domains like the energy sector and automobile industry, in technologies such as enhanced oil recovery. The chemistry and microstructural arrangements of the aqueous morphologies of dissolved materials are usually sensitive to the aforementioned stimuli which subsequently results in rheological sensitivity as well. Herein, we overview different structures capable of viscosity modification as well as go over the rheological theory associated with classical systems studied in literature. A detailed analysis allows us to explore correlations between commonly discussed models such as molecular packing parameter, tube reptation and stress relaxation with structural and rheological changes. We then present five primary mechanisms corresponding to stimuli responsive viscosity modification: (i) packing parameter modification via functional group conditioning and (ii) via dynamic bond formation, (iii) mesh formation by interlinking of network nodes, (iv) viscosity modification by chain conformation changes and (v) viscosity modification by particle jamming. We also overview several recent examples from literature that employ the concepts discussed to create novel classes of intriguing stimuli responsive structures and their corresponding rheological properties. Furthermore, we also explore systems that are responsive to multiple stimuli which can provide enhanced functionality and versatility by providing multi-level and precise actuation. Such systems have been used for programmed site-specific drug delivery.

Discontinuous rigidity transition associated with shear jamming in granular simulations

We investigate the rigidity transition associated with shear jamming in frictionless, as well as frictional, disk packings in the quasi-static regime and at low shear rates. For frictionless disks, the transition under quasi-static shear is discontinuous, with an instantaneous emergence of a system spanning rigid clusters at the jamming transition. For frictional systems, the transition appears continuous for finite shear rates, but becomes sharper for lower shear rates. In the quasi-static limit, it is discontinuous as in the frictionless case. Thus, our results show that the rigidity transition associated with shear jamming is discontinuous, as demonstrated in the past for isotropic jamming of frictionless particles, and therefore a unifying feature of the jamming transition in general.

Heterogeneous Dynamics of Sheared Particle-Laden Fluid Interfaces with Janus Particle Doping

The formation of particle clusters can substantially modify the dynamics and mechanical properties of suspensions in both two and three dimensions. While it has been well established that large network-spanning clusters increase the rigidity of particle systems, it is still unclear how the presence of localized nonpercolating clusters affects the dynamics and mechanical properties of particle suspensions. Here, we introduce self-assembled localized particle clusters at a fluid-fluid interface by mixing a fraction of Janus particles in a monolayer of homogeneous colloids. Each Janus particle binds to a few nearby homogeneous colloids, resulting in numerous small clusters uniformly distributed across the interface. Using a custom magnetic rod interfacial stress rheometer, we apply linear oscillatory shear to the particle-laden fluid interface. By analyzing the local affine deformation of particles from optical microscopy, we show that particles in localized clusters experience substantially lower shear-induced stretching than their neighbors outside clusters. We hypothesize that such heterogeneous dynamics induced by particle clusters increase the effective surface coverage of particles, which in turn enhances the shear moduli of the interface, as confirmed by direct interfacial rheological measurements. Our study illustrates the microscopic dynamics of small clusters in a shear flow and reveals their profound effects on the macroscopic rheology of particle-laden fluid interfaces. Our findings open an avenue for designing interfacial materials with improved mechanical properties via the control of formation of localized particle clusters.

Ultraslow Settling Kinetics of Frictional Cohesive Powders

Using discrete element method simulations, we show that the settling of frictional cohesive grains under ramped-pressure compression exhibits strong history dependence and slow dynamics that are not present for grains that lack either cohesion or friction. Systems prepared by beginning with a dilute state and then ramping the pressure to a small positive value P_{final} over a time τ_{ramp} settle at packing fractions given by an inverse-logarithmic rate law, ϕ_{settled}(τ_{ramp})=ϕ_{settled}(∞)+A/[1+Bln(1+τ_{ramp}/τ_{slow})]. This law is analogous to the one obtained from classical tapping experiments on noncohesive grains, but crucially different in that τ_{slow} is set by the slow dynamics of structural void stabilization rather than the faster dynamics of bulk densification. We formulate a kinetic free-void-volume theory that predicts this ϕ_{settled}(τ_{ramp}), with ϕ_{settled}(∞)=ϕ_{ALP} and A=ϕ_{settled}(0)-ϕ_{ALP}, where ϕ_{ALP}≡.135 is the "adhesive loose packing" fraction found by Liu et al. [Equation of state for random sphere packings with arbitrary adhesion and friction, Soft Matter 13, 421 (2017)SMOABF1744-683X10.1039/C6SM02216B].

Self-assembly and percolation in two dimensional binary magnetic colloids

We study the self-assembly of branching-chain networks and crystals in a binary colloidal system with tunable interactions. The particle positions are extracted from microscopy images and order parameters are extracted by image processing and statistical analysis. With these, we construct phase diagrams with respect to particle density, ratio and interaction. In order to draw a more complete picture, we complement the experiments with computer simulations. We establish a region in the phase diagram, where bead ratios and interactions are symmetric, promoting percolated structures.

Structural, mechanical, and vibrational properties of particulate physical gels

Our lives are surrounded by a rich assortment of disordered materials. In particular, glasses are well known as dense, amorphous materials, whereas gels exist in low-density, disordered states. Recent progress has provided a significant step forward in understanding the material properties of glasses, such as mechanical, vibrational, and transport properties. In contrast, our understanding of particulate physical gels is still highly limited. Here, using molecular dynamics simulations, we study a simple model of particulate physical gels, the Lennard-Jones (LJ) gels, and provide a comprehensive understanding of their structural, mechanical, and vibrational properties, all of which are markedly different from those of LJ glasses. First, the LJ gels show sparse, heterogeneous structures, and the length scale ξ_s of the structures grows as the density is lowered. Second, the LJ gels are extremely soft, with both shear G and bulk K moduli being orders of magnitude smaller than those of LJ glasses. Third, many low-frequency vibrational modes are excited, which form a characteristic plateau with the onset frequency ω_* in the vibrational density of states. Structural, mechanical, and vibrational properties, characterized by ξ_s, G, K, and ω_*, respectively, show power-law scaling behaviors with the density, which establishes a close relationship between them. Throughout this work, we also reveal that LJ gels are multiscale, solid-state materials: (i) homogeneous elastic bodies at long lengths, (ii) heterogeneous elastic bodies with fractal structures at intermediate lengths, and (iii) amorphous structural bodies at short lengths.

Protein microparticles visualize the contact network and rigidity onset in the gelation of model proteins

Protein aggregation into gel networks is of immense importance in diverse areas from food science to medical research; however, it remains a grand challenge as the underlying molecular interactions are complex, difficult to access experimentally, and to model computationally. Early stages of gelation often involve protein aggregation into protein clusters that later on aggregate into a gel network. Recently synthesized protein microparticles allow direct control of these early stages of aggregation, decoupling them from the subsequent gelation stages. Here, by following the gelation of protein microparticles directly at the particle scale, we elucidate in detail the emergence of a percolating structure and the onset of rigidity as measured by microrheology. We find that the largest particle cluster, correlation length, and degree of polymerization all diverge with power laws, while the particles bind irreversibly indicating a nonequilibrium percolation process, in agreement with recent results on weakly attractive colloids. Concomitantly, the elastic modulus increases in a power-law fashion as determined by microrheology. These results give a consistent microscopic picture of the emergence of rigidity in a nonequilibrium percolation process that likely underlies the gelation in many more systems such as proteins, and other strongly interacting structures originating from (bio)molecules.

Rigidity percolation uncovers a structural basis for embryonic tissue phase transitions

Embryo morphogenesis is impacted by dynamic changes in tissue material properties, which have been proposed to occur via processes akin to phase transitions (PTs). Here, we show that rigidity percolation provides a simple and robust theoretical framework to predict material/structural PTs of embryonic tissues from local cell connectivity. By using percolation theory, combined with directly monitoring dynamic changes in tissue rheology and cell contact mechanics, we demonstrate that the zebrafish blastoderm undergoes a genuine rigidity PT, brought about by a small reduction in adhesion-dependent cell connectivity below a critical value. We quantitatively predict and experimentally verify hallmarks of PTs, including power-law exponents and associated discontinuities of macroscopic observables. Finally, we show that this uniform PT depends on blastoderm cells undergoing meta-synchronous divisions causing random and, consequently, uniform changes in cell connectivity. Collectively, our theoretical and experimental findings reveal the structural basis of material PTs in an organismal context.

Spongelike Rigid Structures in Frictional Granular Packings

We show how rigidity emerges in experiments on sheared two-dimensional frictional granular materials by using generalizations of two methods for identifying rigid structures. Both approaches, the force-based dynamical matrix and the topology-based rigidity percolation, agree with each other and identify similar rigid structures. As the system becomes jammed, at a critical contact number z_{c}=2.4±0.1, a rigid backbone interspersed with floppy, particle-filled holes of a broad range of sizes emerges, creating a spongelike morphology. While the pressure within rigid structures always exceeds the pressure outside the rigid structures, they are not identified with the force chains of shear jamming. These findings highlight the need to focus on mechanical stability arising through arch structures and hinges at the mesoscale.

Mechanical disorder of sticky-sphere glasses. I. Effect of attractive interactions

Recent literature indicates that attractive interactions between particles of a dense liquid play a secondary role in determining its bulk mechanical properties. Here we show that, in contrast with their apparent unimportance to the bulk mechanics of dense liquids, attractive interactions can have a major effect on macro- and microscopic elastic properties of glassy solids. We study several broadly applicable dimensionless measures of stability and mechanical disorder in simple computer glasses, in which the relative strength of attractive interactions-referred to as "glass stickiness"-can be readily tuned. We show that increasing glass stickiness can result in the decrease of various quantifiers of mechanical disorder, on both macro- and microscopic scales, with a pair of intriguing exceptions to this rule. Interestingly, in some cases strong attractions can lead to a reduction of the number density of soft, quasilocalized modes, by up to an order of magnitude, and to a substantial decrease in their core size, similar to the effects of thermal annealing on elasticity observed in recent works. Contrary to the behavior of canonical glass models, we provide compelling evidence indicating that the stabilization mechanism in our sticky-sphere glasses stems predominantly from the self-organized depletion of interactions featuring large, negative stiffnesses. Finally, we establish a fundamental link between macroscopic and microscopic quantifiers of mechanical disorder, which we motivate via scaling arguments. Future research directions are discussed.

Viscosity of cohesive granular flows

Cohesive granular materials such as wet sand, snow, and powders can flow like a viscous liquid. However, the elementary mechanisms of momentum transport in such athermal particulate fluids are elusive. As a result, existing models for cohesive granular viscosity remain phenomenological and debated. Here we use discrete element simulations of plane shear flows to measure the viscosity of cohesive granular materials, while tuning the intensity of inter-particle adhesion. We establish that two adhesion-related, dimensionless numbers control their viscosity. These numbers compare the force and energy required to break a bond to the characteristic stress and kinetic energy in the flow. This progresses the commonly accepted view that only one dimensionless number could control the effect of adhesion. The resulting scaling law captures strong, non-Newtonian variations in viscosity, unifying several existing viscosity models. We then directly link these variations in viscosity to adhesion-induced modifications in the flow micro-structure and contact network. This analysis reveals the existence of two modes of momentum transport, involving either grain micro-acceleration or balanced contact forces, and shows that adhesion only affects the latter. This advances our understanding of rheological models for granular materials and other soft materials such as emulsions and suspensions, which may also involve inter-particle adhesive forces.

Nonequilibrium master kinetic equation modeling of colloidal gelation

We present a detailed study of the kinetic cluster growth process during gelation of weakly attractive colloidal particles by means of experiments on critical Casimir attractive colloidal systems, simulations, and analytical theory. In the experiments and simulations, we follow the mean coordination number of the particles during the growth of clusters to identify an attractive-strength independent cluster evolution as a function of mean coordination number. We relate this cluster evolution to the kinetic attachment and detachment rates of particles and particle clusters. We find that single-particle detachment dominates in the relevant weak attractive-strength regime, while association rates are almost independent of the cluster size. Using the limit of single-particle dissociation and size-independent association rates, we solve the master kinetic equation of cluster growth analytically to predict power-law cluster mass distributions with exponents -3/2 and -5/2 before and after gelation, respectively, which are consistent with the experimental and simulation data. These results suggest that the observed critical Casimir-induced gelation is a second-order nonequilibrium phase transition (with broken detailed balance). Consistent with this scenario, the size of the largest cluster is observed to diverge with power-law exponent according to three-dimensional percolation on approaching the critical mean coordination number.

Nonequilibrium continuous phase transition in colloidal gelation with short-range attraction

The dynamical arrest of attractive colloidal particles into out-of-equilibrium structures, known as gelation, is central to biophysics, materials science, nanotechnology, and food and cosmetic applications, but a complete understanding is lacking. In particular, for intermediate particle density and attraction, the structure formation process remains unclear. Here, we show that the gelation of short-range attractive particles is governed by a nonequilibrium percolation process. We combine experiments on critical Casimir colloidal suspensions, numerical simulations, and analytical modeling with a master kinetic equation to show that cluster sizes and correlation lengths diverge with exponents  ~1.6 and 0.8, respectively, consistent with percolation theory, while detailed balance in the particle attachment and detachment processes is broken. Cluster masses exhibit power-law distributions with exponents  -3/2 and  -5/2 before and after percolation, as predicted by solutions to the master kinetic equation. These results revealing a nonequilibrium continuous phase transition unify the structural arrest and yielding into related frameworks.

Two-Scale Scenario of Rigidity Percolation of Sticky Particles

In the presence of attraction, the jamming transition of packings of frictionless particles corresponds to the rigidity percolation. When the range of attraction is long, the distribution of the size of rigid clusters, P(s), is continuous and shows a power-law decay. For systems with short-range attractions, however, P(s) appears discontinuous. There is a power-law decay for small cluster sizes, followed by a low probability gap and a peak near the system size. We find that this appearing "discontinuity" does not mean that the transition is discontinuous. In fact, it signifies the coexistence of two distinct length scales, associated with the largest cluster and smaller ones, respectively. The comparison between the largest and second largest clusters indicates that their growth rates with system size are rather different. However, both cluster sizes tend to diverge in the large system size limit, suggesting that the jamming transition of systems with short-range attractions is still continuous. In the framework of the two-scale scenario, we also derive a generalized hyperscaling relation. With robust evidence, our work challenges the former single-scale view of the rigidity percolation.

Depletion attraction impairs the plasticity of emulsions flowing in a constriction

We study the elasto-plastic behavior of dense attractive emulsions under a mechanical perturbation. The attraction is introduced through non-specific depletion interactions between the droplets and is controlled by changing the concentration of surfactant micelles in the continuous phase. We find that such attractive forces are not sufficient to induce any measurable modification on the scalings between the local packing fraction and the deformation of the droplets. However, when the emulsions are flowed through 2D microfluidic constrictions, we uncover a measurable effect of attraction on their elasto-plastic response. Indeed, we measure higher levels of deformation inside the constriction for attractive droplets. In addition, we show that these measurements correlate with droplet rearrangements that are spatially delayed in the constriction for higher attraction forces.

Correlated Rigidity Percolation and Colloidal Gels

Rigidity percolation (RP) occurs when mechanical stability emerges in disordered networks as constraints or components are added. Here we discuss RP with structural correlations, an effect ignored in classical theories albeit relevant to many liquid-to-amorphous-solid transitions, such as colloidal gelation, which are due to attractive interactions and aggregation. Using a lattice model, we show that structural correlations shift RP to lower volume fractions. Through molecular dynamics simulations, we show that increasing attraction in colloidal gelation increases structural correlation and thus lowers the RP transition, agreeing with experiments. Hence, the emergence of rigidity at colloidal gelation can be understood as a RP transition, but occurs at volume fractions far below values predicted by the classical RP, due to attractive interactions which induce structural correlation.

A broader view on jamming: from spring networks to circle packings

Jamming occurs when objects like grains are packed tightly together (e.g. grain silos). It is highly cooperative and can lead to phenomena like earthquakes, traffic jams, etc. In this paper we point out the paramount importance of the underlying contact network for jammed systems; the network must have one contact in excess of isostaticity and a finite bulk modulus. Isostatic means that the number of degrees of freedom is exactly balanced by the number of constraints. This defines a large class of networks that can be constructed without the necessity of packing particles together compressively (either in the lab or computationally). One such construction, which we explore here, involves setting up the Delaunay triangulation of a Poisson disk sampling and then removing edges to maximize the bulk modulus until the isostatic plus one edge is reached. This construction works in any dimensions and here we give results in 2D where we also show how such networks can be transformed into disk packs.

The thermal jamming transition of soft harmonic disks in two dimensions

By exploring the properties of the energy landscape of a bidisperse system of soft harmonic disks in two dimensions we determine the thermal jamming transition. To be specific, we study whether the ground state of the system where the particles do not overlap can be reached within a reasonable time. Starting with random initial configurations, the energy landscape is probed by energy minimization steps as in case of athermal jamming and in addition steps where an energy barrier can be crossed with a small but non-zero probability. For random initial conditions we find that as a function of packing fraction the thermal jamming transition, i.e. the transition from a state where all overlaps can be removed to an effectively non-ergodic state where one cannot get rid of the overlaps, occurs at a packing fraction of [Formula: see text], which is smaller than the transition packing fraction of athermal jamming at [Formula: see text]. Furthermore, we show that the thermal jamming transition is in the universality class of directed percolation and therefore is fundamentally different from the athermal jamming transition.

Universal scaling of polygonal desiccation crack patterns

Polygonal desiccation crack patterns are commonly observed in natural systems. Despite their quotidian nature, it is unclear whether similar crack patterns which span orders of magnitude in length scales share the same underlying physics. In thin films, the characteristic length of polygonal cracks is known to monotonically increase with the film thickness; however, existing theories that consider the mechanical, thermodynamic, hydrodynamic, and statistical properties of cracking often lead to contradictory predictions. Here we experimentally investigate polygonal cracks in drying suspensions of micron-sized particles by varying film thickness, boundary adhesion, packing fraction, and solvent. Although polygonal cracks were observed in most systems above a critical film thickness, in cornstarch-water mixtures, multiscale crack patterns were observed due to two distinct desiccation mechanisms. Large-scale, primary polygons initially form due to capillary-induced film shrinkage, whereas small-scale, secondary polygons appear later due to the deswelling of the hygroscopic particles. In addition, we find that the characteristic area of the polygonal cracks, A_{p}, obeys a universal power law, A_{p}=αh^{4/3}, where h is the film thickness. By quantitatively linking α with the material properties during crack formation, we provide a robust framework for understanding multiscale polygonal crack patterns from microscopic to geologic scales.