Universal scaling of polygonal desiccation crack patterns

Decoding planetary surfaces by counting cracks

Planets are often covered with thin cracked shells. From mud films to lithospheres of rock or ice, fracture networks form two-dimensional (2D) tessellations of convex polygons whose geometry encodes their genesis. Here, we chart the geometry of 2D fracture mosaics across the solar system, and decode their formative conditions using a new dynamical crack model. We show that mosaics can be projected onto a Symbolic Ternary Diagram, where the relative proportions of "T," "X," and "Y" junctions are uniquely related to contributions from distinct modes of fracture. Most planetary mosaics cluster in a region associated with hierarchical fracture networks, where sequential cracking favors formation of T junctions. Exceptions to this rule may betray the presence of water. Europa's fracture networks stand apart due to the predominance of X junctions; this is a special feature of ice, where healing of cracks by refreezing of water allows new fractures to overprint older ones. Several fracture networks on Mars appear as outliers due to the high proportion of Y junctions. These patterns-previously interpreted as ancient mudcracks and frozen polar terrain, based on geological evidence-are consistent with the twisting of crack junctions by cyclic volume change. Our findings suggest that counting cracks could aid in the identification of other water-influenced planetary environments.

Regulable crack patterns for the fabrication of high-performance transparent EMI shielding windows

Crack pattern-based metal grid film is an ideal candidate material for transparent electromagnetic interference shielding optical windows. However, achieving crack patterns with narrow grid spacing, small wire width, and high connectivity remains challenging. Herein, an aqueous acrylic colloidal dispersion was developed as a crack precursor for preparing crack patterns. The ratio of hard monomers in the precursor, the coating thickness, and the drying mediation strategy were systematically varied to control the spacing and width of the crack patterns. The resulting dense and narrow crack patterns served as sacrificial templates for the fabrication of patterning metal grid films on transparent substrates, intended for optoelectronic applications. These films demonstrated excellent optoelectronic properties (82.7% transmission at 550 nm visible light, sheet resistance 4.1 Ω/sq) and strong EMI shielding effectiveness (average shielding effectiveness 33.6 dB at 1-18 GHz), showcasing their potential as a scalable and effective transparent EMI shielding solution.

Rheological Properties of Dense Particle Suspensions of Starches: Shear Thickening, Shear Jamming, and Shock Absorption Properties

Concentrated suspensions of Brownian and non-Brownian particles display distinctive rheological behavior highly dependent on shear rate and shear stress. Cornstarch suspensions, composed of starch particles from corn plants, served as a model for concentrated non-Brownian suspensions, demonstrating discontinuous shear thickening (DST) and dynamic shear jamming (SJ). However, starch particles from other plant sources have not yet been investigated, despite their different sizes and shapes. This study is focused on the evaluation of the effects of the structural parameters of starch particles by preparing concentrated suspensions of starch particles from 13 different plants at particle fractions of 25-50% and their rheological behavior through steady shear, pull-out, and ball-drop tests. Starch particles can be roughly classified as polygonal and ellipsoidal. The DST and SJ behavior typically reported for concentrated cornstarch suspensions were confirmed for other starch particles in both particle groups. The ball-drop test demonstrated excellent shock absorption properties for 11 concentrated suspensions of starch particles, except for sago palms. In the case of concentrated suspensions of starch particles, the particle fraction and shear applied were the dominant factors that significantly affected the rheological behavior, whereas the particle shape was not a primary contributor. The findings of this study drive further investigation on the effect of liquid and particle surface properties in concentrated particle suspensions on DST and SJ behaviors.

Formation mechanism of radial and circular cracks promoted by delamination in drying silica colloidal deposits

Cracks with radial and circular patterns are appealing in nature and industry. Although morphologies and propagation conditions of cracks are extensively studied, the formation mechanism of crack pattern by the interaction of channel fracture and interfacial delamination remains elusive. Here, we present the transition of radial to coexisting radial and circular crack patterns when the thickness of colloidal deposits on both hard and soft substrates exceeds a critical value, through the colloidal volume fraction dependence. In addition, a thickness-dependent phase diagram from radial crack to coexistence of radial and circular cracks was constructed with respect to the radius and the volume fractions of silica colloidal deposits. A phase-field fracture model is developed to elucidate how the formation of radial cracks is facilitated by simultaneous delamination. The warping-induced radial tensile stress at the bottom surface of the striped deposit is proportional to the thickness. It leads to subsequent nucleation and growth of circular cracks in thick deposits. This work provides insight into the formation mechanism of complex crack patterns in drying colloidal deposits and revolutionizes the design space of crack-based micro-nano structures.

Morphologies of electric-field-driven cracks in dried dispersions of ellipsoids

We report an experimental and theoretical study of the morphology of desiccation cracks formed in deposits of hematite ellipsoids dried in an externally applied alternating current (ac) electric field. A series of transitions in the crack morphology is observed by modulating the frequency and the strength of the applied field. We also found a clear transition in the morphology of cracks as a function of the aspect ratio of the ellipsoid. We show that these transitions in the crack morphology can be explained by a linear stability analysis of the equation describing the effective dynamics of an ellipsoid placed in an externally applied ac electric field.

Versatile Mesoporous Microblocks Prepared by Pattern-Induced Cracking of Colloidal Films

Mesoporous microparticles have the potential to be used in various fields, such as energy generation, sensing, and the environmental field. Recently, the process of making homogeneous microparticles in an economical and environmentally friendly way has gained much attention. Herein, rectangular mesoporous microblocks of various designs are produced by manipulating the fragmentation of colloidal films consisting of micropyramids while controlling the notch angles of pyramidal edges. During calcination of the colloidal films, cracks are generated in the valleys of micropyramids acting as notches, and the angle of notches can be controlled by the prepattern underneath the micropyramids. By changing the location of notches with sharp angles, the shape of microblocks can be controlled with excellent uniformity. After detaching the microblocks from substrates, mesoporous microparticles of various sizes with multiple functions are easily produced. This study demonstrates anti-counterfeiting functions by encoding the rotation angles of rectangular microblocks of various sizes. In addition, the mesoporous microparticles can be utilized for separating desired chemicals mixed with chemicals of different charges. The method of fabricating size-tunable functionalized mesoporous microblocks can be a platform technology to prepare special films and catalysts and for environmental applications.

Effect of Colloidal Surface Charge on Desiccation Cracks

We report the effect of polarity and surface charge density on the nucleation and growth kinetics of desiccation cracks in deposits of colloids formed by drying. We show that the average spacing between desiccation cracks and crack opening are higher for the deposit of positively charged colloids than that of negatively charged colloids. The temporal evolution of crack growth is found to be faster for positively charged particle deposits. The distinct crack patterns and their kinetics are understood by considering the spatial arrangement of particles in the deposit, which is strongly influenced by the substrate-particle and particle-particle interactions. Interestingly, the crack spacing, the crack opening, and the rate at which the crack widens are found to increase upon decreasing the surface charge of the colloids.

Crack patterns of drying dense bacterial suspensions

Drying of bacterial suspensions is frequently encountered in a plethora of natural and engineering processes. However, the evaporation-driven mechanical instabilities of dense consolidating bacterial suspensions have not been explored heretofore. Here, we report the formation of two different crack patterns of drying suspensions of Escherichia coli (E. coli) with distinct motile behaviors. Circular cracks are observed for wild-type E. coli with active swimming, whereas spiral-like cracks form for immotile bacteria. Using the elastic fracture mechanics and the poroelastic theory, we show that the formation of the circular cracks is determined by the tensile nature of the radial drying stress once the cracks are initiated by the local order structure of bacteria due to their collective swimming. Our study demonstrates the link between the microscopic swimming behaviors of individual bacteria and the mechanical instabilities and macroscopic pattern formation of drying bacterial films. The results shed light on the dynamics of active matter in a drying process and provide useful information for understanding various biological processes associated with drying bacterial suspensions.

Effect of the Shape of the Confining Boundary and Particle Shape Anisotropy on the Morphology of Desiccation Cracks

The control of the morphology of desiccation cracks is fascinating not only from the application point of view but also from the rich physics behind it. Here, we present a seemingly simple method to tailor the morphology of desiccation cracks by exploitation of the combined effect of particle shape anisotropy and the shape of the confining boundary. This allows us to make circular, square, and triangular-shaped desiccation cracks in the vicinity of the confining boundaries. As the colloidal dispersion dries in confined wells, a drying front appears at the center of the well. With further evaporation, the drying front recedes toward the boundary from the center of the well. We show that the temporal evolution of the drying front is strongly influenced by the shape of the well. Subsequently, desiccation cracks appear in the penultimate stage of drying, and the morphology of the cracks is governed by the shape of the drying front and hence by the shape of the wells. The spatial evolution of the crack pattern is quantified by estimation of the curvature of the cracks, which suggests that the influence of the confining boundary on crack formation is long-ranged. However, the cracks in the dried deposit consisting of spherical particles remain unaffected by the shape of the well, and the cracks are always radial. We establish a one-to-one correspondence between the shape of the drying front and the morphology of the crack pattern in the final dried deposit of ellipsoids.

Cracking of Colloidal Films to Generate Rectangular Fragments

Cracks are common in nature. Cracking is known as an irreversible and uncontrollable process. To control the cracking patterns, many researchers have proposed methods to prepare notches for stress localization on films. In this work, we investigate a method of controlling cracks by making microscale pyramid patterns that have notches between the pyramids. After preparing pyramid patterns consisting of colloidal particles with organic residue, we annealed them to induce volume shrinkage and cracking between the pyramids. We studied the effect of film thickness on cracking and the generation of rectangular fragments consisting of multiple pyramids. The area of rectangular fragments was in good agreement with the results of scaling analysis. The concept of controlling cracks by imprinting notches on a film and the relationship with the film thickness can guide the study of cracking phenomena.

The mechanics of brittle granular materials with coevolving grain size and shape

The influence of particle shape on the mechanics of sand is widely recognized, especially in mineral processing and geomechanics. However, most existing continuum theories for engineering applications do not encompass the morphology of the grains and its evolution during comminution. Similarly, the relatively few engineering models accounting for grain-scale processes tend to idealize particles as spheres, with their diameters considered as the primary and sole geometric descriptor. This paper inspires a new generation of constitutive laws for crushable granular continua with arbitrary, yet evolving, particle morphology. We explore the idea of introducing multiple grain shape descriptors into Continuum Breakage Mechanics (CBM), a theory originally designed to track changes in particle size distributions during confined comminution. We incorporate the influence of these descriptors on the elastic strain energy potential and treat them as dissipative state variables. In analogy with the original CBM, and in light of evidence from extreme fragmentation in nature, the evolution of the additional shape descriptors is postulated to converge towards an attractor. Comparisons with laboratory experiments, discrete element analyses and particle-scale fracture models illustrate the encouraging performance of the theory. The theory provides insights into the feedback among particle shape, compressive yielding and inelastic deformation in crushable granular continua. These results inspire new questions that should guide future research into crushable granular systems using particle-scale imaging and computations.

Prebiotic Aggregates (Tissues) Emerging from Reaction-Diffusion: Formation Time, Configuration Entropy and Optimal Spatial Dimension

For the formation of a proto-tissue, rather than a protocell, the use of reactant dynamics in a finite spatial region is considered. The framework is established on the basic concepts of replication, diversity, and heredity. Heredity, in the sense of the continuity of information and alike traits, is characterized by the number of equivalent patterns conferring viability against selection processes. In the case of structural parameters and the diffusion coefficient of ribonucleic acid, the formation time ranges between a few years to some decades, depending on the spatial dimension (fractional or not). As long as equivalent patterns exist, the configuration entropy of proto-tissues can be defined and used as a practical tool. Consequently, the maximal diversity and weak fluctuations, for which proto-tissues can develop, occur at the spatial dimension 2.5.

Time Evolution and Spatial Hierarchy of Crack Patterns

Cracks generated due to desiccation of wet colloidal systems are ubiquitous, examples being nanomaterial films, painted walls, cemented floors, mud fields, river beds, and even giant rocks. In all such cases, crack patterns are often appreciably similar but for the length and time scales, which can be widely differing. In this work, we have examined the crack formation more closely to see if there exists some generality with regard to the length scale of parameters and the formation time. Specifically, using a commonly used colloidal dispersion and optimized conditions to form polygonal network patterns rather than isolated cracks (films of subcritical thickness), we have studied the time evolution of the pattern parameters, the area occupied by the cracks, their lengths, and the widths. As is well known, initially, a network of cracks forms, which we term as the primary generation, followed by interconnecting cracks inside the polygonal regions (secondary) and, later, cracks spreading in local regions (tertiary). We find that the area and the width increase nearly linearly with time with the change in the slope corresponding to the change in the generation. When normalized with respect to the final values, the trends obtained for different film thicknesses overlap, the only exception being the pattern containing unconnected cracks. Thus, the time evolution of cracks is shown to be predictable based on width filtering. Including the angle between cracks as further input into the recursive model, the possibility of identifying the hierarchy of crack segments is also shown. The approach may be useful in determining the age, authenticity, and details of old paintings, understanding the stress profile of geological rocks, and analyzing various natural and manmade hierarchical structures.

Universal fluctuation of polygonal crack geometry in solidified lava

Outcrops of columnar joints made of solidified lava flows are often covered by semiordered polygonal cracks. The polygon diameters are fairly uniform at each outcrop, but their shapes largely vary in the number of sides and internal angles. Herein, we unveil that the statistical variation in the polygon shape follows an extreme value distribution class: the Gumbel distribution. The Gumbel law was found to hold for different columnar joints, regardless of the locality, lithologic composition, and typical diameter. A common distribution for columnar joints implies a universal class that may integrate the polygonal crack networks observed on the surface of various fractured brittle materials.

Ordered ring-shaped cracks induced by indentation in metal films on soft elastic substrates

Ordered crack patterns contain plentiful physical mechanisms and are useful for technological applications such as lithography, template, and biomimicry. Here we report on ordered multiple ring-shaped cracks induced by indentation in metal films on soft elastic polydimethylsiloxane (PDMS) substrates. It is shown that the indentation triggers the deformation of PDMS substrate and generates a radial tensile stress in the film, leading to the formation of ring-shaped cracks with a nearly uniform spacing. The morphological characteristics and evolution behaviors of the multiple ring-shaped cracks are revealed by optical microscopy, atomic force microscopy, and scanning electron microscopy. Their formation mechanisms are discussed by theoretical analysis based on the fracture mechanics. The report in this work can promote better understanding of the indentation-induced stress anisotropy and mode competition in rigid-film-soft-substrate systems and provide a facile strategy to control the crack patterns by simple mechanical loading.

Plato's cube and the natural geometry of fragmentation

Plato envisioned Earth's building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra-shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth's tectonic plates, has two attractors: "Platonic" quadrangles and "Voronoi" hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato's forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.

Narrowing Desiccating Crack Patterns by an Azeotropic Solvent for the Fabrication of Nanomesh Electrodes

Desiccation of a colloidal layer produces crack patterns because of stress arising out of solvent evaporation. Associated with it is the rearrangement of particles, while adhesion to the substrate resists such movements. The nature of solvent, which is often overlooked, plays a key role in the process as it dictates evaporation and wetting properties of the colloidal film. Herein, we study the crack formation process by using a mixture of solvents, water, and isopropyl alcohol (IPA). Among the various ratios, a water/IPA mixture (15:85 by volume) close to the azeotropic composition possesses unusual evaporation and wetting properties, leading to narrower cracks with widths down to ∼162 nm, uncommon among the known crackle patterns. The dense and narrow crack patterns have been used as sacrificial templates to obtain metal meshes on transparent substrates for optoelectronic applications.

Hierarchical crack patterns of metal films sputter deposited on soft elastic substrates

Controlled cracks are useful in a wide range of applications, including stretchable electronics, microfluidics, sensors, templates, biomimics, and surface engineering. Here we report on the spontaneous formation of hierarchical crack patterns in metal (nickel) films sputter deposited on soft elastic polydimethylsiloxane (PDMS) substrates. The experiment shows that the nickel film generates a high tensile stress during deposition, which is relieved by the formation of disordered crack networks (called primary cracks). Due to the strong interfacial adhesion and soft substrate, the cracks can penetrate into the PDMS substrate deeply. The width and depth of the primary cracks both increase with increasing film thickness, whereas the crack spacing is insensitive to the film thickness. The film pieces dividing by the primary cracks can fracture further when they are triggered by an external disturbance due to the residual tensile stress, resulting in the formation of fine crack networks (called secondary cracks). The width and spacing of the secondary cracks show different behaviors in comparison to the primary cracks. The morphological characteristics, growth behaviors, and formation mechanisms of the primary and secondary cracking modes have been discussed in detail. The report in this work could provide better understanding of two distinct cracking modes with different sizes and morphologies.

Scaling Law for Cracking in Shrinkable Granular Packings

Hydrated granular packings often crack into discrete clusters of grains when dried. Despite its ubiquity, an accurate prediction of cracking remains elusive. Here, we elucidate the previously overlooked role of individual grain shrinkage-a feature common to many materials-in determining crack patterning using both experiments and simulations. By extending classical Griffith crack theory, we obtain a scaling law that quantifies how cluster size depends on the interplay between grain shrinkage, stiffness, and size-applicable to a diverse array of shrinkable granular packings.

Influence of Bénard-Marangoni instability on the morphology of drying colloidal films

The drying of colloidal suspensions is a very complex process leading to a sol-gel transition induced by solvent evaporation. The resulting film can even crack and delaminate. In this study, we investigate the drying process of a colloidal suspension with a highly volatile solvent and we show for initially millimeter-thick layers that the resulting pattern of delaminated plates considerably differs from what is usually observed for aqueous colloidal suspensions. Visualization using an IR camera reveals that hexagonal convection cells can develop during the drying of suspensions with a highly volatile solvent and may persist until the film consolidation. This leads to the formation of non-homogeneous films presenting surface corrugations. Thus, we highlight the importance of the hydrodynamics during the first phase of strong solvent evaporation and its consequences for the following drying steps. A criterion predicting whether or not Bénard-Marangoni instability effectively occurs will be discussed. Finally, we report a non-classical delamination mode generating fragments with convex surfaces, whereas buckle-driven delamination usually results in concave shapes.