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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&#95;{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&#95;{c}>0, as well as λ&#95;{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.

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&#95;{-})〈G&#95;{+}〉+F&#95;{-}〈G&#95;{-}〉, where F&#95;{-} is the fraction of jammed packings with negative shear moduli and 〈G&#95;{+}〉 and 〈G&#95;{-}〉 are the average values from packings with positive and negative moduli, respectively. We show that 〈G&#95;{+}〉 and 〈|G&#95;{-}|〉 obey different power-law scaling relations above and below pN^{2}∼1. For pN^{2}>1, both 〈G&#95;{+}〉N and 〈|G&#95;{-}|〉N∼(pN^{2})^{β}, where β∼0.5 for repulsive linear spring interactions. Despite this, 〈G〉N∼(pN^{2})^{β^{'}} with β^{'}≳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&#95;{sub}^{2}<10^{-2}, where n&#95;{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&#95;{sub}^{2}≳10^{-2}.

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 / σ avg - k ) β , where σ avg is the average particle diameter, kσ avg is an offset where Q ∼ 0, the power-law scaling exponent β = 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 β . 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 β varies continuously with the ratio of the viscous drag to the kinetic friction coefficient, λ = ζ / μ . β = d - 1/2 in the λ → 0 limit and d - 3/2 in the λ → ∞ limit, with a midpoint λ c that depends on the hopper opening angle θ w . We also characterize the spatial structure of the flows and associate changes in spatial structure of the hopper flows to changes in the exponent β . The offset k increases with particle stiffness until k ∼ k max in the hard-particle limit, where k max ∼ 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.

The Integrated Graduate Program in Physical and Engineering Biology (IGPPEB) at Yale University brings together Ph.D. students from the physical, engineering, and biological sciences. The main goals of this program are for students to become comfortable working in an interdisciplinary and collaborative research environment and adept at communicating with scientists and nonscientists. To fill a student-identified learning gap in engaging in inclusive discussions, IGPPEB students developed a communication workshop to improve skills in visual engagement, citing specific content, constructive conversation entrances, and encouragement of peers. Based on short- and long-term assessment of the workshop, 100% of students reported that it should be offered to future cohorts and 63% of students perceived it to be personally helpful. Additionally, 92% of participants reported using one or more of the core skills beyond the course, with skills in "Encouraging peers" and "Constructive conversation entrances" rated the highest in perceived improvement. Based on the highest average rating of 76 ± 24 (on a scale of 0-100), students agreed that the workshop made them feel more welcome in the IGPPEB community. With a rating of 68 ± 13, they also agreed that the workshop had a positive impact on their graduate school experience. Participants provided suggestions for future improvements, such as increasing student involvement in leading discussions of course material. This study demonstrates that a student-led workshop can improve perceived discussion skills and build community across an interdisciplinary program in the sciences., Competing Interests: The authors have stated explicitly that there are no conflicts of interest in connection with this article., (©2022 The Authors FASEB BioAdvances published by The Federation of American Societies for Experimental Biology.)

Numerous studies have investigated the differences and similarities between protein structures determined by solution NMR spectroscopy and those determined by X-ray crystallography. A fundamental question is whether any observed differences are due to differing methodologies or to differences in the behavior of proteins in solution versus in the crystalline state. Here, we compare the properties of the hydrophobic cores of high-resolution protein crystal structures and those in NMR structures, determined using increasing numbers and types of restraints. Prior studies have reported that many NMR structures have denser cores compared with those of high-resolution X-ray crystal structures. Our current work investigates this result in more detail and finds that these NMR structures tend to violate basic features of protein stereochemistry, such as small non-bonded atomic overlaps and few Ramachandran and sidechain dihedral angle outliers. We find that NMR structures solved with more restraints, and which do not significantly violate stereochemistry, have hydrophobic cores that have a similar size and packing fraction as their counterparts determined by X-ray crystallography at high resolution. These results lead us to conclude that, at least regarding the core packing properties, high-quality structures determined by NMR and X-ray crystallography are the same, and the differences reported earlier are most likely a consequence of methodology, rather than fundamental differences between the protein in the two different environments., (© 2022 The Protein Society.)

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 〉 ∼ P β , with β ≈ 0.75 for jammed packings of deformable particles with zero bending energy. In contrast, β ≈ 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.

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 F α associated with each triangle α . To understand the spatio-temporal evolution of the displacement fields during shear stress drops, we calculate F α 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.

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^{β}, with β∼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 β∼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&#95;{f}, and from changes in the interparticle contact network, G&#95;{r}, such that 〈G〉=〈G&#95;{f}〉+〈G&#95;{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&#95;{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)〉.

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.

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&#95;{a}^{f}, where G&#95;{a}^{f}/G&#95;{a0}=(P/P&#95;{0})^{(α-2)/(α-1)}-P/P&#95;{0}, P&#95;{0}∼N^{-2(α-1)} is the characteristic pressure at which G&#95;{a}^{f}=0, G&#95;{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&#95;{c})^{a}, where a≈(α-2)/(α-1) in the P→0 limit and 〈G(P)〉∼(P/P&#95;{c})^{b}, where b≳(α-3/2)/(α-1), above a characteristic pressure that scales as P&#95;{c}∼N^{-2(α-1)}.

Many experiments over the past half century have shown that, for a range of protocols, granular materials compact under pressure and repeated small disturbances. A recent experiment on cyclically sheared spherical grains showed significant compaction via homogeneous crystallization (Rietz et al., 2018). Here we present numerical simulations of frictionless, purely repulsive spheres undergoing cyclic simple shear via Newtonian dynamics with linear viscous drag at fixed vertical load. We show that for sufficiently small strain amplitudes, cyclic shear gives rise to homogeneous crystallization at a volume fraction ϕ=0.646±0.001. This result indicates that neither friction nor gravity is essential for homogeneous crystallization in driven granular media. Understanding how crystal formation is initiated within a homogeneous disordered state gives key insights into the old open problem of glass formation in fluids.

In 2008, we established the Integrated Graduate Program in Physical and Engineering Biology (IGPPEB) at Yale University. Our goal was to create a comprehensive graduate program to train a new generation of scientists who possess a sophisticated understanding of biology and who are capable of applying physical and quantitative methodologies to solve biological problems. Here we describe the framework of the training program, report on its effectiveness, and also share the insights we gained during its development and implementation. The program features co-teaching by faculty with complementary specializations, student peer learning, and novel hands-on courses that facilitate the seamless blending of interdisciplinary research and teaching. It also incorporates enrichment activities to improve communication skills, engage students in science outreach, and foster a cohesive program cohort, all of which promote the development of transferable skills applicable in a variety of careers. The curriculum of the graduate program is integrated with the curricular requirements of several Ph.D.-granting home programs in the physical, engineering, and biological sciences. Moreover, the wide-ranging recruiting activities of the IGPPEB serve to enhance the quality and diversity of students entering graduate school at Yale. We also discuss some of the challenges we encountered in establishing and optimizing the program, and describe the institution-level changes that were catalyzed by the introduction of the new graduate program. The goal of this article is to serve as both an inspiration and as a practical "how to" manual for those who seek to establish similar programs at their own institutions. © 2016 by The International Union of Biochemistry and Molecular Biology, 44(6):537-549, 2016., (© 2016 The Authors Biochemistry and Molecular Biology Education published by Wiley Periodicals, Inc. on behalf of International Union of Biochemistry and Molecular Biology.)

We construct a statistical framework for static assemblies of deformable grains which parallels that of equilibrium statistical mechanics but with a conservation principle based on the mechanical stress tensor. We define a state function that has all the attributes of entropy. In particular, maximizing this function leads to a well-defined granular temperature and the equivalent of the Boltzman distribution for ensembles of grain packings. Predictions of the ensemble are verified against simulated packings of frictionless, deformable disks.

Using simulations of glassy systems under steady-state shear, we compare effective temperatures obtained from static linear response with those from time-dependent fluctuation-dissipation relations. Although these two definitions are not expected to agree, we show that they yield the same answer over two and a half decades of effective temperature. This suggests that a more complete conceptual framework is necessary for effective temperatures in steady-state driven systems.

We have studied how two- and three-dimensional systems made up of particles interacting with finite range, repulsive potentials jam (i.e., develop a yield stress in a disordered state) at zero temperature and zero applied stress. At low packing fractions phi, the system is not jammed and each particle can move without impediment from its neighbors. For each configuration, there is a unique jamming threshold phi(c) at which particles can no longer avoid each other, and the bulk and shear moduli simultaneously become nonzero. The distribution of phi(c) values becomes narrower as the system size increases, so that essentially all configurations jam at the same packing fraction in the thermodynamic limit. This packing fraction corresponds to the previously measured value for random close packing. In fact, our results provide a well-defined meaning for "random close packing" in terms of the fraction of all phase space with inherent structures that jam. The jamming threshold, point J, occurring at zero temperature and applied stress and at the random-close-packing density, has properties reminiscent of an ordinary critical point. As point J is approached from higher packing fractions, power-law scaling is found for the divergence of the first peak in the pair correlation function and in the vanishing of the pressure, shear modulus, and excess number of overlapping neighbors. Moreover, near point J, certain quantities no longer self-average, suggesting the existence of a length scale that diverges at J. However, point J also differs from an ordinary critical point: the scaling exponents do not depend on dimension but do depend on the interparticle potential. Finally, as point J is approached from high packing fractions, the density of vibrational states develops a large excess of low-frequency modes. Indeed, at point J, the density of states is a constant all the way down to zero frequency. All of these results suggest that point J is a point of maximal disorder and may control behavior in its vicinity-perhaps even at the glass transition.