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Register a new userEnergetically favoured defects in dense packings of particles on spherical surfaces
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The dense packing of interacting particles on spheres has proved to be a useful model for virus capsids and colloidosomes. Indeed, icosahedral symmetry observed in virus capsids corresponds to potential energy minima that occur for magic numbers of, e.g., 12, 32 and 72 identical Lennard-Jones particles, for which the packing has exactly the minimum number of twelve five-fold defects. It is unclear, however, how stable these structures are against thermal agitation. We investigate this property by means of basin-hopping global optimisation and Langevin dynamics for particle numbers between ten and one hundred. An important measure is the number and type of point defects, that is, particles that do not have six nearest neighbours. We find that small icosahedral structures are the most robust against thermal fluctuations, exhibiting fewer excess defects and rearrangements for a wide temperature range. Furthermore, we provide evidence that excess defects appearing at low non-zero temperatures lower the potential energy at the expense of entropy. At higher temperatures defects are, as expected, thermally excited and thus entropically stabilised. If we replace the Lennard-Jones potential by a very short-ranged (Morse) potential, which is arguably more appropriate for colloids and virus capsid proteins, we find that the same particle numbers give a minimum in the potential energy, although for larger particle numbers these minima correspond to different packings. Furthermore, defects are more difficult to excite thermally for the short-ranged potential, suggesting that the short-ranged interaction further stabilises equilibrium structures.
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The mechanical response of packings of purely repulsive, spherical particles to athermal, quasistatic simple shear near jamming onset is highly nonlinear. Previous studies have shown that, at small pressure $p$, the ensemble-averaged static shear modulus $langle G-G_0 rangle$ scales with $p^alpha$, where $alpha approx 1$, but above a characteristic pressure $p^{**}$, $langle G-G_0 rangle sim p^beta$, where $beta approx 0.5$. However, we find that the shear modulus $G^i$ for an individual packing typically decreases linearly with $p$ along a geometrical family where the contact network does not change. We resolve this discrepancy by showing that, while the shear modulus does decrease linearly within geometrical families, $langle G rangle$ also depends on a contribution from discontinuous jumps in $langle G rangle$ that occur at the transitions between geometrical families. For $p > p^{**}$, geometrical-family and rearrangement contributions to $langle G rangle$ are of opposite signs and remain comparable for all system sizes. $langle G rangle$ can be described by a scaling function that smoothly transitions between the two power-law exponents $alpha$ and $beta$. We also demonstrate the phenomenon of {it compression unjamming}, where a jammed packing can unjam via isotropic compression.
We investigate the mechanical response of jammed packings of circulo-lines, interacting via purely repulsive, linear spring forces, as a function of pressure $P$ during athermal, quasistatic isotropic compression. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power-law, $langle G(P) rangle sim P^{beta}$, with $beta sim 0.5$, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of $langle G(P)rangle$ over the same range of pressure for aspect ratios ${cal R} gtrsim 1.2$. However, the power-law scaling exponent $beta sim 0.8$-$0.9$ is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose $langle Grangle$ into separate contributions from geometrical families, $G_f$, and from changes in the interparticle contact network, $G_r$, such that $langle G rangle = langle G_frangle + langle G_r rangle$. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase {it 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 $langle G_f rangle$ 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.
The study of hard-particle packings is of fundamental importance in physics, chemistry, cell biology, and discrete geometry. Much of the previous work on hard-particle packings concerns their densest possible arrangements. By contrast, we examine kinetic effects inevitably present in both numerical and experimental packing protocols. Specifically, we determine how changing the compression/shear rate of a two-dimensional packing of noncircular particles causes it to deviate from its densest possible configuration, which is always periodic. The adaptive shrinking cell (ASC) optimization scheme maximizes the packing fraction of a hard-particle packing by first applying random translations and rotations to the particles and then isotropically compressing and shearing the simulation box repeatedly until a possibly jammed state is reached. We use a stochastic implementation of the ASC optimization scheme to mimic different effective time scales by varying the number of particle moves between compressions/shears. We generate dense, effectively jammed, monodisperse, two-dimensional packings of obtuse scalene triangle, rhombus, curved triangle, lens, and ice cream cone (a semicircle grafted onto an isosceles triangle) shaped particles, with a wide range of packing fractions and degrees of order. To quantify these kinetic effects, we introduce the kinetic frustration index $K$, which measures the deviation of a packing from its maximum possible packing fraction. To investigate how kinetics affect short- and long-range ordering in these packings, we compute their spectral densities and characterize their contact networks. We find that kinetic effects are most significant when the particles have greater asphericity, less curvature, and less rotational symmetry. This work may be relevant to the design of laboratory packing protocols.
We report numerical results of effective attractive forces on the packing properties of two-dimensional elongated grains. In deposits of non-cohesive rods in 2D, the topology of the packing is mainly dominated by the formation of ordered structures of aligned rods. Elongated particles tend to align horizontally and the stress is mainly transmitted from top to bottom, revealing an asymmetric distribution of local stress. However, for deposits of cohesive particles, the preferred horizontal orientation disappears. Very elongated particles with strong attractive forces form extremely loose structures, characterized by an orientation distribution, which tends to a uniform behavior when increasing the Bond number. As a result of these changes, the pressure distribution in the deposits changes qualitatively. The isotropic part of the local stress is notably enhanced with respect to the deviatoric part, which is related to the gravity direction. Consequently, the lateral stress transmission is dominated by the enhanced disorder and leads to a faster pressure saturation with depth.
The formation of quasi-spherical cages from protein building blocks is a remarkable self-assembly process in many natural systems, where a small number of elementary building blocks are assembled to build a highly symmetric icosahedral cage. In turn, this has inspired synthetic biologists to design de novo protein cages. We use simple models, on multiple scales, to investigate the self-assembly of a spherical cage, focusing on the regularity of the packing of protein-like objects on the surface. Using building blocks, which are able to pack with icosahedral symmetry, we examine how stable these highly symmetric structures are to perturbations that may arise from the interplay between flexibility of the interacting blocks and entropic effects. We find that, in the presence of those perturbations, icosahedral packing is not the most stable arrangement for a wide range of parameters; rather disordered structures are found to be the most stable. Our results suggest that (i) many designed, or even natural, protein cages may not be regular in the presence of those perturbations, and (ii) that optimizing those flexibilities can be a possible design strategy to obtain regular synthetic cages with full control over their surface properties.
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