Correction: Flow and clogging of capillary droplets.

Correction for 'Flow and clogging of capillary droplets' by Yuxuan Cheng et al., Soft Matter, 2024, https://doi.org/10.1039/D4SM00752B.

Flow and clogging of capillary droplets.

Capillary droplets form due to surface tension when two immiscible fluids are mixed. We describe the motion of gravity-driven capillary droplets flowing through narrow constrictions and obstacle arrays in both simulations and experiments. Our new capillary deformable particle model recapitulates the shape and velocity of single oil droplets in water as they pass through narrow constrictions in microfluidic chambers. Using this experimentally validated model, we simulate the flow and clogging of single capillary droplets in narrow channels and obstacle arrays and find several important results. First, the capillary droplet speed profile is nonmonotonic as the droplet exits the narrow orifice, and we can tune the droplet properties so that the speed overshoots the terminal speed far from the constriction. Second, in obstacle arrays, we find that extremely deformable droplets can wrap around obstacles, which leads to decreased average droplet speed in the continuous flow regime and increased probability for clogging in the regime where permanent clogs form. Third, the wrapping mechanism causes the clogging probability in obstacle arrays to become nonmonotonic with surface tension Γ. At large Γ, the droplets are nearly rigid and the clogging probability is large since the droplets can not squeeze through the gaps between obstacles. With decreasing Γ, the clogging probability decreases as the droplets become more deformable. However, in the small-Γ limit, the clogging probability increases since the droplets are extremely deformable and wrap around the obstacles. The results from these studies are important for developing a predictive understanding of capillary droplet flows through complex and confined geometries.

Identifying topologically associating domains using differential kernels.

Chromatin is a polymer complex of DNA and proteins that regulates gene expression. The three-dimensional (3D) structure and organization of chromatin controls DNA transcription and replication. High-throughput chromatin conformation capture techniques generate Hi-C maps that can provide insight into the 3D structure of chromatin. Hi-C maps can be represented as a symmetric matrix [Formula: see text], where each element represents the average contact probability or number of contacts between chromatin loci i and j. Previous studies have detected topologically associating domains (TADs), or self-interacting regions in [Formula: see text] within which the contact probability is greater than that outside the region. Many algorithms have been developed to identify TADs within Hi-C maps. However, most TAD identification algorithms are unable to identify nested or overlapping TADs and for a given Hi-C map there is significant variation in the location and number of TADs identified by different methods. We develop a novel method to identify TADs, KerTAD, using a kernel-based technique from computer vision and image processing that is able to accurately identify nested and overlapping TADs. We benchmark this method against state-of-the-art TAD identification methods on both synthetic and experimental data sets. We find that the new method consistently has higher true positive rates (TPR) and lower false discovery rates (FDR) than all tested methods for both synthetic and manually annotated experimental Hi-C maps. The TPR for KerTAD is also largely insensitive to increasing noise and sparsity, in contrast to the other methods. We also find that KerTAD is consistent in the number and size of TADs identified across replicate experimental Hi-C maps for several organisms. Thus, KerTAD will improve automated TAD identification and enable researchers to better correlate changes in TADs to biological phenomena, such as enhancer-promoter interactions and disease states.

Computational modeling of the physical features that influence breast cancer invasion into adipose tissue.

Breast cancer invasion into adipose tissue strongly influences disease progression and metastasis. The degree of cancer cell invasion into adipose tissue depends on both biochemical signaling and the mechanical properties of cancer cells, adipocytes, and other key components of adipose tissue. We model breast cancer invasion into adipose tissue using discrete element method simulations of active, cohesive spherical particles (cancer cells) invading into confluent packings of deformable polyhedra (adipocytes). We quantify the degree of invasion by calculating the interfacial area At between cancer cells and adipocytes. We determine the long-time value of At vs the activity and strength of the cohesion between cancer cells, as well as the mechanical properties of the adipocytes and extracellular matrix in which adipocytes are embedded. We show that the degree of invasion collapses onto a master curve as a function of the dimensionless energy scale Ec , which grows linearly with the cancer cell velocity persistence time and fluctuations, is inversely proportional to the system pressure, and is offset by the cancer cell cohesive energy. When E c > 1 , cancer cells will invade the adipose tissue, whereas for E c < 1 , cancer cells and adipocytes remain de-mixed. We also show that At decreases when the adipocytes are constrained by the ECM by an amount that depends on the spatial heterogeneity of the adipose tissue.

Protein folding as a jamming transition.

Proteins fold to a specific functional conformation with a densely packed hydrophobic core that controls their stability. We develop a geometric, yet all-atom model for proteins that explains the universal core packing fraction of ϕc=0.55 found in experimental measurements. We show that as the hydrophobic interactions increase relative to the temperature, a novel jamming transition occurs when the core packing fraction exceeds ϕc. The model also recapitulates the global structure of proteins since it can accurately refold to native-like structures from partially unfolded states.

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.

Computational modeling of the physical features that influence breast cancer invasion into adipose tissue.

Breast cancer invasion into adipose tissue strongly influences disease progression and metastasis. The degree of cancer cell invasion into adipose tissue depends on numerous biochemical and physical properties of cancer cells, adipocytes, and other key components of adipose tissue. We model breast cancer invasion into adipose tissue as a physical process by carrying out simulations of active, cohesive spherical particles (cancer cells) invading into confluent packings of deformable polyhedra (adipocytes). We quantify the degree of invasion by calculating the interfacial area At between cancer cells and adipocytes. We determine the long-time value of At versus the activity and strength of the cohesion between cancer cells, as well as mechanical properties of the adipocytes and extracellular matrix (ECM) in which the adipocytes are embedded. We show that the degree of invasion collapses onto a master curve by plotting it versus a dimensionless energy scale Ec, which grows linearly with mean-square fluctuations and persistence time of the cancer cell velocities, is inversely proportional to the pressure of the system, and has an offset that increases with the cancer cell cohesive energy. The condition, Ec≫1, indicates that cancer cells will invade the adipose tissue, whereas for Ec≪1, the cancer cells and adipocytes remain demixed. We also show that constraints on adipocyte positions by the ECM decrease At relative to that obtained for unconstrained adipocytes. Finally, spatial heterogeneity in structural and mechanical properties of the adipocytes in the presence of ECM impedes invasion relative to adipose tissue with uniform properties.

Modeling the Effects of Varying the Ti Concentration on the Mechanical Properties of Cu-Ti Alloys.

The mechanical properties of CuTi alloys have been characterized extensively through experimental studies. However, a detailed understanding of why the strength of Cu increases after a small fraction of Ti atoms are added to the alloy is still missing. In this work, we address this question using density functional theory (DFT) and molecular dynamics (MD) simulations with the modified embedded atom method (MEAM) interatomic potentials. First, we performed calculations of the uniaxial tension deformations of small bicrystalline Cu cells using DFT static simulations. We then carried out uniaxial tension deformations on much larger bicrystalline and polycrystalline Cu cells by using MEAM MD simulations. In bicrystalline Cu, the inclusion of Ti increases the grain boundary separation energy and the maximum tensile stress. The DFT calculations demonstrate that the increase in the tensile stress can be attributed to an increase in the local charge density arising from Ti. MEAM simulations in larger bicrystalline systems have shown that increasing the Ti concentration decreases the density of the stacking faults. This observation is enhanced in polycrystalline Cu, where the addition of Ti atoms, even at concentrations as low as 1.5 atomic (at.) %, increases the yield strength and elastic modulus of the material compared to pure Cu. Under uniaxial tensile loading, the addition of small amounts of Ti hinders the formation of partial Shockley dislocations in the grain boundaries of Cu, leading to a reduced level of local deformation. These results shed light on the role of Ti in determining the mechanical properties of polycrystalline Cu and enable the engineering of grain boundaries and the inclusion of Ti to improve degradation resistance.

Designing the pressure-dependent shear modulus using tessellated granular metamaterials.

Jammed packings of granular materials display complex mechanical response. For example, the ensemble-averaged shear modulus 〈G〉 increases as a power law in pressure p for static packings of soft spherical particles that can rearrange during compression. We seek to design granular materials with shear moduli that can either increase or decrease with pressure without particle rearrangements even in the large-system limit. To do this, we construct tessellated granular metamaterials by joining multiple particle-filled cells together. We focus on cells that contain a small number of bidisperse disks in two dimensions. We first study the mechanical properties of individual disk-filled cells with three types of boundaries: periodic boundary conditions (PBC), fixed-length walls (FXW), and flexible walls (FLW). Hypostatic jammed packings are found for cells with FLW, but not in cells with PBC and FXW, and they are stabilized by quartic modes of the dynamical matrix. The shear modulus of a single cell depends linearly on p. We find that the slope of the shear modulus with pressure λ_{c}<0 for all packings in single cells with PBC where the number of particles per cell N≥6. In contrast, single cells with FXW and FLW can possess λ_{c}>0, as well as λ_{c}<0, for N≤16. We show that we can force the mechanical properties of multicell granular metamaterials to possess those of single cells by constraining the end points of the outer walls and enforcing an affine shear response. These studies demonstrate that tessellated granular metamaterials provide a platform for the design of soft materials with specified mechanical properties.

Hysteretic Behavior of Capillary Bridges between Flat Plates.

We studied the evolution of capillary bridges between nominally flat plates undergoing multiple cycles of compression and stretching in experiments and simulations. We varied the distance between the plates in small increments to study the full evolution of the bridge shape. Experiments show that contact angle hysteresis determines the shape of the bridge. In sliding drops, hysteresis can be modeled using a contact angle-dependent resistive force F̃R applied at the contact line. We developed a model that accurately captures the evolution of the bridge shape by combining F̃R and constrained energy minimization. Unlike previous work, this allows for both complete and partial contact line pinning. We also explored the effect of using nonparallel plates. The asymmetry in the bridge shape causes the movement of the center of mass of the bridge and can be explained by contact angle hysteresis. We find that even a slight misalignment between the flat plates can have a measurable effect.

Local and global measures of the shear moduli of jammed disk packings.

Strain-controlled isotropic compression gives rise to jammed packings of repulsive, frictionless disks with either positive or negative global shear moduli. We carry out computational studies to understand the contributions of the negative shear moduli to the mechanical response of jammed disk packings. We first decompose the ensemble-averaged, global shear modulus as 〈G〉=(1-F_{-})〈G_{+}〉+F_{-}〈G_{-}〉, where F_{-} is the fraction of jammed packings with negative shear moduli and 〈G_{+}〉 and 〈G_{-}〉 are the average values from packings with positive and negative moduli, respectively. We show that 〈G_{+}〉 and 〈|G_{-}|〉 obey different power-law scaling relations above and below pN^{2}∼1. For pN^{2}>1, both 〈G_{+}〉N and 〈|G_{-}|〉N∼(pN^{2})^{ß}, where ß∼0.5 for repulsive linear spring interactions. Despite this, 〈G〉N∼(pN^{2})^{ß^{'}} with ß^{'}≳0.5 due to the contributions from packings with negative shear moduli. We show further that the probability distribution of global shear moduli P(G) collapses at fixed pN^{2} and different values of p and N. We calculate analytically that P(G) is a Γ distribution in the pN^{2}≪1 limit. As pN^{2} increases, the skewness of P(G) decreases and P(G) becomes a skew-normal distribution with negative skewness in the pN^{2}≫1 limit. We also partition jammed disk packings into subsystems using Delaunay triangulation of the disk centers to calculate local shear moduli. We show that the local shear moduli defined from groups of adjacent triangles can be negative even when G>0. The spatial correlation function of local shear moduli C(r[over ⃗]) displays weak correlations for pn_{sub}^{2}<10^{-2}, where n_{sub} is the number of particles within each subsystem. However, C(r[over ⃗]) begins to develop long-ranged spatial correlations with fourfold angular symmetry for pn_{sub}^{2}≳10^{-2}.

Localized growth and remodelling drives spongy mesophyll morphogenesis.

The spongy mesophyll is a complex, porous tissue found in plant leaves that enables carbon capture and provides mechanical stability. Unlike many other biological tissues, which remain confluent throughout development, the spongy mesophyll must develop from an initially confluent tissue into a tortuous network of cells with a large proportion of intercellular airspace. How the airspace in the spongy mesophyll develops while the tissue remains mechanically stable is unknown. Here, we use computer simulations of deformable polygons to develop a purely mechanical model for the development of the spongy mesophyll tissue. By stipulating that cell wall growth and remodelling occurs only near void space, our computational model is able to recapitulate spongy mesophyll development observed in Arabidopsis thaliana leaves. We find that robust generation of pore space in the spongy mesophyll requires a balance of cell growth, adhesion, stiffness and tissue pressure to ensure cell networks become porous yet maintain mechanical stability. The success of this mechanical model of morphogenesis suggests that simple physical principles can coordinate and drive the development of complex plant tissues like the spongy mesophyll.

Hopper flows of deformable particles.

Numerous experimental and computational studies show that continuous hopper flows of granular materials obey the Beverloo equation that relates the volume flow rate Q and the orifice width w: Q ∼ (w/σavg - k)ß, where σavg is the average particle diameter, kσavg is an offset where Q ∼ 0, the power-law scaling exponent ß = d - 1/2, and d is the spatial dimension. Recent studies of hopper flows of deformable particles in different background fluids suggest that the particle stiffness and dissipation mechanism can also strongly affect the power-law scaling exponent ß. We carry out computational studies of hopper flows of deformable particles with both kinetic friction and background fluid dissipation in two and three dimensions. We show that the exponent ß varies continuously with the ratio of the viscous drag to the kinetic friction coefficient, λ = ζ/µ. ß = d - 1/2 in the λ → 0 limit and d - 3/2 in the λ → ∞ limit, with a midpoint λc that depends on the hopper opening angle θw. We also characterize the spatial structure of the flows and associate changes in spatial structure of the hopper flows to changes in the exponent ß. The offset k increases with particle stiffness until k ∼ kmax in the hard-particle limit, where kmax ∼ 3.5 is larger for λ → ∞ compared to that for λ → 0. Finally, we show that the simulations of hopper flows of deformable particles in the λ → ∞ limit recapitulate the experimental results for quasi-2D hopper flows of oil droplets in water.

Correction: The structural, vibrational, and mechanical properties of jammed packings of deformable particles in three dimensions.

Correction for 'The structural, vibrational, and mechanical properties of jammed packings of deformable particles in three dimensions' by Dong Wang et al., Soft Matter, 2021, 17, 9901-9915, DOI: 10.1039/D1SM01228B.

The structural, vibrational, and mechanical properties of jammed packings of deformable particles in three dimensions.

We investigate the structural, vibrational, and mechanical properties of jammed packings of deformable particles with shape degrees of freedom in three dimensions (3D). Each 3D deformable particle is modeled as a surface-triangulated polyhedron, with spherical vertices whose positions are determined by a shape-energy function with terms that constrain the particle surface area, volume, and curvature, and prevent interparticle overlap. We show that jammed packings of deformable particles without bending energy possess low-frequency, quartic vibrational modes, whose number decreases with increasing asphericity and matches the number of missing contacts relative to the isostatic value. In contrast, jammed packings of deformable particles with non-zero bending energy are isostatic in 3D, with no quartic modes. We find that the contributions to the eigenmodes of the dynamical matrix from the shape degrees of freedom are significant over the full range of frequency and shape parameters for particles with zero bending energy. We further show that the ensemble-averaged shear modulus 〈G〉 scales with pressure P as 〈G〉 ∼ Pß, with ß ≈ 0.75 for jammed packings of deformable particles with zero bending energy. In contrast, ß ≈ 0.5 for packings of deformable particles with non-zero bending energy, which matches the value for jammed packings of soft, spherical particles with fixed shape. These studies underscore the importance of incorporating particle deformability and shape change when modeling the properties of jammed soft materials.

Using delaunay triangularization to characterize non-affine displacement fields during athermal, quasistatic deformation of amorphous solids.

We investigate the non-affine displacement fields that occur in two-dimensional Lennard-Jones models of metallic glasses subjected to athermal, quasistatic simple shear (AQS). During AQS, the shear stress versus strain displays continuous quasi-elastic segments punctuated by rapid drops in shear stress, which correspond to atomic rearrangement events. We capture all information concerning the atomic motion during the quasi-elastic segments and shear stress drops by performing Delaunay triangularizations and tracking the deformation gradient tensor Fα associated with each triangle α. To understand the spatio-temporal evolution of the displacement fields during shear stress drops, we calculate Fα along minimal energy paths from the mechanically stable configuration immediately before to that after the stress drop. We find that quadrupolar displacement fields form and dissipate both during the quasi-elastic segments and shear stress drops. We then perform local perturbations (rotation, dilation, simple and pure shear) to single triangles and measure the resulting displacement fields. We find that local pure shear deformations of single triangles give rise to mostly quadrupolar displacement fields, and thus pure shear strain is the primary type of local strain that is activated by bulk, athermal quasistatic simple shear. Other local perturbations, e.g. rotations, dilations, and simple shear of single triangles, give rise to vortex-like and dipolar displacement fields that are not frequently activated by bulk AQS. These results provide fundamental insights into the non-affine atomic motion that occurs in driven, glassy materials.

Mechanical response of packings of nonspherical particles: A case study of two-dimensional packings of circulo-lines.

We investigate the mechanical response of jammed packings of circulo-lines in two spatial dimensions, interacting via purely repulsive, linear spring forces, as a function of pressure P during athermal, quasistatic isotropic compression. The surface of a circulo-line is defined as the collection of points that is equidistant to a line; circulo-lines are composed of a rectangular central shaft with two semicircular end caps. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power law, 〈G(P)〉∼P^{ß}, with ß∼0.5, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of 〈G(P)〉 over the same range of pressure for aspect ratios R≳1.2. However, the power-law scaling exponent ß∼0.8-0.9 is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose 〈G〉 into separate contributions from geometrical families, G_{f}, and from changes in the interparticle contact network, G_{r}, such that 〈G〉=〈G_{f}〉+〈G_{r}〉. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase and decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure. For this reason, the geometrical family contribution 〈G_{f}〉 is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent for 〈G(P)〉.

Static-state particle fabrication via rapid vitrification of a thixotropic medium.

Functional particles that respond to external stimuli are spurring technological evolution across various disciplines. While large-scale production of functional particles is needed for their use in real-life applications, precise control over particle shapes and directional properties has remained elusive for high-throughput processes. We developed a high-throughput emulsion-based process that exploits rapid vitrification of a thixotropic medium to manufacture diverse functional particles in large quantities. The vitrified medium renders stationary emulsion droplets that preserve their shape and size during solidification, and energetic fields can be applied to build programmed anisotropy into the particles. We showcase mass-production of several functional particles, including low-melting point metallic particles, self-propelling Janus particles, and unidirectionally-magnetized robotic particles, via this static-state particle fabrication process.

Shear response of granular packings compressed above jamming onset.

We investigate the mechanical response of jammed packings of repulsive, frictionless spherical particles undergoing isotropic compression. Prior simulations of the soft-particle model, where the repulsive interactions scale as a power law in the interparticle overlap with exponent α, have found that the ensemble-averaged shear modulus 〈G(P)〉 increases with pressure P as ∼P^{(α-3/2)/(α-1)} at large pressures. 〈G〉 has two key contributions: (1) continuous variations as a function of pressure along geometrical families, for which the interparticle contact network does not change, and (2) discontinuous jumps during compression that arise from changes in the contact network. Using numerical simulations, we show that the form of the shear modulus G^{f} for jammed packings within near-isostatic geometrical families is largely determined by the affine response G^{f}∼G_{a}^{f}, where G_{a}^{f}/G_{a0}=(P/P_{0})^{(α-2)/(α-1)}-P/P_{0}, P_{0}∼N^{-2(α-1)} is the characteristic pressure at which G_{a}^{f}=0, G_{a0} is a constant that sets the scale of the shear modulus, and N is the number of particles. For near-isostatic geometrical families that persist to large pressures, deviations from this form are caused by significant nonaffine particle motion. We further show that the ensemble-averaged shear modulus 〈G(P)〉 is not simply a sum of two power laws, but 〈G(P)〉∼(P/P_{c})^{a}, where a≈(α-2)/(α-1) in the P→0 limit and 〈G(P)〉∼(P/P_{c})^{b}, where b≳(α-3/2)/(α-1), above a characteristic pressure that scales as P_{c}∼N^{-2(α-1)}.

Contact network changes in ordered and disordered disk packings.

We investigate the mechanical response of packings of purely repulsive, frictionless disks to quasistatic deformations. The deformations include simple shear strain at constant packing fraction and at constant pressure, "polydispersity" strain (in which we change the particle size distribution) at constant packing fraction and at constant pressure, and isotropic compression. For each deformation, we show that there are two classes of changes in the interparticle contact networks: jump changes and point changes. Jump changes occur when a contact network becomes mechanically unstable, particles "rearrange", and the potential energy (when the strain is applied at constant packing fraction) or enthalpy (when the strain is applied at constant pressure) and all derivatives are discontinuous. During point changes, a single contact is either added to or removed from the contact network. For repulsive linear spring interactions, second- and higher-order derivatives of the potential energy/enthalpy are discontinuous at a point change, while for Hertzian interactions, third- and higher-order derivatives of the potential energy/enthalpy are discontinuous. We illustrate the importance of point changes by studying the transition from a hexagonal crystal to a disordered crystal induced by applying polydispersity strain. During this transition, the system only undergoes point changes, with no jump changes. We emphasize that one must understand point changes, as well as jump changes, to predict the mechanical properties of jammed packings.