Central Force Interactions Research Articles

Rutile-mimic 3D metamaterials with simultaneously negative Poisson's ratio and negative compressibility

The origin of negative Poisson's ratio in rutile-type oxides and fluorides is explored through density functional theory (DFT) simulation. The study highlights the dominant role of nearest-neighbor central force interactions, thereby inspiring the design of a new three-dimensional (3D) mechanical metamaterial by mimicking the bond structure via elastic beams. Analytical expressions for the effective Poisson's ratio and compressibility are derived and validated by finite element computations. Numerical results indicate that the material can exhibit simultaneously negative Poisson's ratio and negative linear, area, or volume compressibility. The negative Poisson's ratio is conformed experimentally, and the parametric spaces leading to negative compressibilities are identified explicitly.

Rigidity and auxeticity transitions in networks with strong bond-bending interactions

.A widely studied model for gels or biopolymeric fibrous materials are networks with central force interactions, such as Hookean springs. Less commonly studied are materials whose mechanics are dominated by non-central force interactions such as bond-bending potentials. Inspired by recent experimental advancements in designing colloidal gels with tunable interactions, we study the micro- and macroscopic elasticity of two-dimensional planar graphs with strong bond-bending potentials, in addition to weak central forces. We introduce a theoretical framework that allows us to directly investigate the limit in which the ratio of characteristic central-force to bending stiffnesses vanishes. In this limit we show that a generic isostatic point exists at z_c = 4, coinciding with the isostatic point of frames with central-force interactions in two dimensions. We further demonstrate the emergence of a stiffening transition when the coordination is increased towards the isostatic point, which shares similarities with the strain-induced stiffening transition observed in biopolymeric fibrous materials, and coincides with an auxeticity transition above which the material’s Poisson’s ratio approaches -1 when bond-bending interactions dominate.Graphical abstract

Stress-stabilized subisostatic fiber networks in a ropelike limit.

The mechanics of disordered fibrous networks such as those that make up the extracellular matrix are strongly dependent on the local connectivity or coordination number. For biopolymer networks this coordination number is typically between 3 and 4. Such networks are sub-isostatic and linearly unstable to deformation with only central force interactions, but exhibit a mechanical phase transition between floppy and rigid states under strain. The introduction of weak bending interactions stabilizes these networks and suppresses the critical signatures of this transition. We show that applying external stress can also stabilize subisostatic networks with only tensile central force interactions, i.e., a ropelike potential. Moreover, we find that the linear shear modulus shows a power-law scaling with the external normal stress, with a non-mean-field exponent. For networks with finite bending rigidity, we find that the critical stain shifts to lower values under prestress.

Exact matrix elements for general two-body central-force interactions, expressed as sums of products

ABSTRACTIn a manner similar to but distinct from concurrent tensor efforts in electronic structure, it is shown that the Laplace transform can serve as a generator for a sum-of-products (SOP) form that allows one to write essentially any function of distance between two particles (i.e. any central force potential) as an exact two-body matrix. In particular, exact expressions for the Coulomb, Yukawa and long-range Ewald two-body operators are evaluated in a band-limited (Sinc function) basis. The resultant exact, full-basis, SOP representations for these interaction potentials – acting in conjunction with an external harmonic confining field – are validated via comparison with energy eigenstate solutions obtained via an independent calculation based on separation of variables. The new two-body matrix representations may have substantial impact in any of the many disciplines in which pair-wise central force interactions are relevant – especially, electronic structure and dynamics.

Strain-driven criticality underlies nonlinear mechanics of fibrous networks.

Networks with only central force interactions are floppy when their average connectivity is below an isostatic threshold. Although such networks are mechanically unstable, they can become rigid when strained. It was recently shown that the transition from floppy to rigid states as a function of simple shear strain is continuous, with hallmark signatures of criticality [Sharma etal., Nature Phys. 12, 584 (2016)1745-247310.1038/nphys3628]. The nonlinear mechanical response of collagen networks was shown to be quantitatively described within the framework of such mechanical critical phenomenon. Here, we provide a more quantitative characterization of critical behavior in subisostatic networks. Using finite-size scaling we demonstrate the divergence of strain fluctuations in the network at well-defined critical strain. We show that the characteristic strain corresponding to the onset of strain stiffening is distinct from but related to this critical strain in a way that depends on critical exponents. We confirm this prediction experimentally for collagen networks. Moreover, we find that the apparent critical exponents are largely independent of the spatial dimensionality. With subisostaticity as the only required condition, strain-driven criticality is expected to be a general feature of biologically relevant fibrous networks.

Nonlinear Mechanics of Athermal Branched Biopolymer Networks.

Naturally occurring biopolymers such as collagen and actin form branched fibrous networks. The average connectivity in branched networks is generally below the isostatic threshold at which central force interactions marginally stabilize the network. In the submarginal regime, for connectivity below this threshold, such networks are unstable toward small deformations unless stabilized by additional interactions such as bending. Here we perform a numerical study on the elastic behavior of such networks. We show that the nonlinear mechanics of branched networks is qualitatively similar to that of filamentous networks with freely hinged cross-links. In agreement with a recent theoretical study,1 we find that branched networks also exhibit nonlinear mechanics consistent with athermal critical phenomena controlled by strain. We obtain the critical exponents capturing the nonlinear elastic behavior near the critical point by performing scaling analysis of the stiffening curves. We find that the exponents evolve with the connectivity in the network. We show that the nonlinear mechanics of disordered networks, independent of the detailed microstructure, can be characterized by a strain-driven second-order phase transition, and that the primary quantitative differences among different architectures are in the critical exponents describing the transition.

Elastic regimes of subisostatic athermal fiber networks.

Athermal models of disordered fibrous networks are highly useful for studying the mechanics of elastic networks composed of stiff biopolymers. The underlying network architecture is a key aspect that can affect the elastic properties of these systems, which include rich linear and nonlinear elasticity. Existing computational approaches have focused on both lattice-based and off-lattice networks obtained from the random placement of rods. It is not obvious, a priori, whether the two architectures have fundamentally similar or different mechanics. If they are different, it is not clear which of these represents a better model for biological networks. Here, we show that both approaches are essentially equivalent for the same network connectivity, provided the networks are subisostatic with respect to central force interactions. Moreover, for a given subisostatic connectivity, we even find that lattice-based networks in both two and three dimensions exhibit nearly identical nonlinear elastic response. We provide a description of the linear mechanics for both architectures in terms of a scaling function. We also show that the nonlinear regime is dominated by fiber bending and that stiffening originates from the stabilization of subisostatic networks by stress. We propose a generalized relation for this regime in terms of the self-generated normal stresses that develop under deformation. Different network architectures have different susceptibilities to the normal stress but essentially exhibit the same nonlinear mechanics. Such a stiffening mechanism has been shown to successfully capture the nonlinear mechanics of collagen networks.

Local structure controls the nonaffine shear and bulk moduli of disordered solids.

Paradigmatic model systems, which are used to study the mechanical response of matter, are random networks of point-atoms, random sphere packings, or simple crystal lattices; all of these models assume central-force interactions between particles/atoms. Each of these models differs in the spatial arrangement and the correlations among particles. In turn, this is reflected in the widely different behaviours of the shear (G) and compression (K) elastic moduli. The relation between the macroscopic elasticity as encoded in G, K and their ratio, and the microscopic lattice structure/order, is not understood. We provide a quantitative analytical connection between the local orientational order and the elasticity in model amorphous solids with different internal microstructure, focusing on the two opposite limits of packings (strong excluded-volume) and networks (no excluded-volume). The theory predicts that, in packings, the local orientational order due to excluded-volume causes less nonaffinity (less softness or larger stiffness) under compression than under shear. This leads to lower values of G/K, a well-documented phenomenon which was lacking a microscopic explanation. The theory also provides an excellent one-parameter description of the elasticity of compressed emulsions in comparison with experimental data over a broad range of packing fractions.

Investigation of a Method to Efficiently Create Elastic Networks

Elastic Networks (EN) have proven useful in the study of proteins, where they have a wide range of applications, from modeling the normal mode fluctuations of a protein to amending coarse grained (CG) force fields in order to conserve secondary and tertiary structures. In this project, we investigate the effectiveness of a systematic method for coarse-graining a protein (or part of a protein) into an EN where a substantial fraction of the particles are removed. The method begins with the creation of a Gaussian Network Model (GN) from either a high resolution EN or an all atom protein reference conformation and force field. By virtue of the GN being quadratic, we can analytically integrate out any number of degrees of freedom to create a CG GN with the same free energy as the original GN. Unfortunately, we cannot exactly transform the CG GN back to an EN because the GN no longer represents only central force interactions between particles. However, we can approximate the GN using an EN where the accuracy of the EN depends on the choice of particles that are removed. We show that a judicious choice of the configuration can significantly improve the accuracy of the EN. We discuss the ability of different optimization processes to efficiently find an optimum choice of the remaining particles. Finally, we investigate how well the resulting EN can replicate observables from the original system.

Theory of the thermal properties ofδ-plutonium

The thermal properties of $\ensuremath{\delta}$-plutonium are studied in terms of the elementary electronic structure, and contrasted to those of aluminum. In both metals the thermal expansion can be understood as from the pressure of the gas of phonons, a negative expansion in the case of plutonium because of negative Gr\"uneisen constants for some transverse modes. Thermal expansion in aluminum is found positive, in agreement with experiments. This phonon gas also has a bulk modulus, negative for plutonium, which dominates the observed negative temperature dependence of the bulk modulus for plutonium, with a smaller negative contribution arising from the thermal expansion (in contrast to contraction for pure plutonium) of the alloys on which it is measured. For aluminum, the phonon-gas contribution is negligible and thermal expansion causes the bulk modulus to decrease with temperature, as in experiments. The Debye-Waller temperature, another measure of rigidity, contains a small positive contribution to the temperature dependence from thermal expansion in plutonium and a negative contribution in aluminum. There is also a much larger negative contribution to the temperature dependence of the Debye-Waller temperature in both (in accordance with experiments for plutonium) arising from anharmonicity of some transverse modes. The nearest-neighbor, central-force interactions used here inevitably lead to sizable errors in some elastic constants, and replacement of Brillouin-zone averages by averages along the $[100]$ and $[111]$ directions is very questionable for some properties. However, the simplification is major and the results appear to reveal which contributions are important.

Understanding the phonon spectrum of plutonium

The recently observed vibration spectrum of face-centered-cubic plutonium is interpreted in terms of a simple inclusion of electron correlations in a Friedel Model of the density of f-electron states, and an empty-core pseudopotential treatment of the three free electrons per atom. This model has given reasonable accounts of the volume-dependence of the total energy and of the experimental density of electronic states. It yields a volume-dependent energy, plus a two-body central-force interaction $V(d)$ between atoms, which enters the vibration spectrum only through the two parameters, ${V}^{(1)}=d\ensuremath{\partial}V/\ensuremath{\partial}d$ and ${V}^{(2)}{=d}^{2}{\ensuremath{\partial}}^{2}V/\ensuremath{\partial}{d}^{2},$ evaluated at the nearest-neighbor distance. Using existing parameters yields the three elastic constants in crude accord with experiment but peak vibrational frequencies too low and an instability against shear modes in the [111] and [110] directions. Moderate scaling of the two parameters brings the vibration spectrum into reasonable accord. It also produces the strong fluctuation in the [110] ${T}_{1}$ mode, suggesting it is not from Kohn anomalies, which are absent from the model.

Lattice-dynamical calculation of phonon scattering at a disordered interface

For an FCC crystal with central force interactions and separately for a scalar model on a square lattice, we compute exactly the phonon transmission coefficient T( ω) through a disordered planar interface between two identical semi-infinite leads. At high frequencies T( ω) exhibits a strong frequency dependence which is determined by the correlation length of the disorder.

Symplectic variable step size integration for N-body problems

Multiple time stepping can be applied to the leapfrog/Störmer/Verlet integrator so as to effect a variable step size algorithm. The strategy maintains the symplecticness, time-reversibility, and second-order accuracy of the leapfrog method. This method can be applied to 2-body central force interactions by partitioning them into distance classes and smoothly decomposing the potential energy into the sum of potential functions for the respective classes. The algorithm described here is much more efficient than leapfrog with very small step sizes and more accurate than leapfrog with larger step sizes.

A Maxwell Stress for Material Interactions

A Maxwell stress is identified to quantify the continuum average effect of molecules arising from two-particle central force interactions. The negative of the divergence of the Maxwell stress gives the body force due to such interactions. Such a stress was introduced previously but was interpreted as a surface traction to represent these effects (C. Argentoet al., J. Mech. Phys. Solids45, 1161 (1997)). The spatial scale of body force interactions among molecules is determined by the size of the bodies interacting and the size of the gaps between them, so that the Maxwell stress approach is necessary since forces of attraction cannot be guaranteed to isolate themselves to a vanishingly thin layer at the free surface. The Maxwell stress is directly useful in calculations of the total stress field within the body. A few examples for simple geometries are given.

Mechanisms of membrane rupture: From cracks to pores

The rupture kinetics under isotropic tension of solid membranes with central-force interactions are studied as a function of temperature and the range of the interparticle interactions. At zero temperature, rupture occurs homogeneously at the mechanical instability point. At low temperature, rupture is heterogeneous, mainly by the nucleation of a single crack, branching out into several cracks. This is accompanied by a homogeneous expansion, and the critical rupture tension drops very rapidly with increasing temperature. At high temperature, rupture involves the nucleation of dislocation dipoles, merging to form pores, and the critical rupture tension varies almost linearly with temperature. The temperature ${T}_{1}$, separating these two regimes, is higher the shorter the range of interaction, for potentials of equal depth. We find that under isotropic tension, the crack favors the path along the direction with lowest surface energy, in contrast to that observed under uniaxial tension. The identification of the rupture point is facilitated by the fact that the elastic constants, in particular the bulk modulus, usually show precursor effects. Transverse fluctuations can play an important role as evidenced by a one-dimensional model of self-assembled particles in a ring, pressurized by an ideal gas. These fluctuations are dominant at high temperature, where the rupture pressure is only weakly dependent on temperature.

Molecular dynamics simulation of sodium ions in one-dimensional tunnel structure of hollandite-type

Microscopic picture of Na ion behavior in the one-dimensional tunnel of hollandite-type has been studied by molecular dynamics simulations. The characteristic distribution of Na ions determined by the X-ray study has been satisfactorily reproduced by using a two-body central-force interaction between ions. Trajectories of Na ions are characteristic and not identical to each other. Alternative occupation of Cr(III) or Ti(IV) at the framework-metal position is an important factor for the trapping of Na ions which are more attracted toward CrO 6 octahedra than TiO 6.

Vacancy annealing kinetics in finite monolayer patches

The kinetics of the annealing of vacancies in finite monolayer patches of several thousand particles have been studied by molecular dynamics for three central force interactions; the nearest-neighbour piecewise linear force potential, the Lennard-Jones potential and the similar ( r −8 − r −4) potential. The simplest behaviour observed is the diffusion of the vacancies to the nearest basin of attraction; the edges or a larger concentration of vacancies, forming in the latter case voids. The voids in turn migrate to the next attractor, forming larger voids, until all vacancies are annealed. A rapid dislocation mediated healing (DMH), which removes most of the vacancies with the dislocations annealing out of the system, can be observed with a long-range interparticle interaction. Mobile vacancies and pressure also favour the phenomenon. DMH can occur well below the self-diffusion temperature of vacancies. A ‘void-dislocation dipole’ reversible conversion is observed with some small vacancy clusters in systems exhibiting both annealing processes. The relevance of this study to bulk annealing kinetics is discussed.

Anharmonic effects on the dynamics of Fe impurities in metallic hosts

The anharmonic effects on the dynamics of impurities in dilute alloys are evaluated explicitly for fcc crystals with nearest-neighbour central-force interactions and compared with the experimental results on 57Fe impurities in Cu, Pt and Pd. The incorporation of cubic and quartic anharmonicity with force constant and mass changes is found to yield results much closer to the reported values.

Some dynamical properties of the alloy disordered chain on a semi-infinite solid square lattice

The dynamics of an alloy disordered first surface layer of a simple cubic semi-infinite crystal is considered, where defect and host atoms are of equal masses. The host-defect and defect-defect interactions are unequal in general, and taken different from host-host interactions. The calculations are exemplified by the results for the surface phonon and resonance modes localized on a linear alloy disordered chain terminating a simple square solid lattice with nearest neighbour central force interactions. One phonon mode polarized parallel to the chain is decoupled from the bulk continuum. The mode polarized perpendicular to the chain is a resonance with signature characteristics depending on the model domains of the concentration of defect atoms in the chain, and on the host-defect interactions. This mode is also investigated outside the bulk continuum, where it is a surface mode whose frequency is dependent on the strength of host-defect interactions, and on the defect concentration. In mean field the mode disappears for sufficiently low concentrations.

Elastic constants of KMnF3 as functions of temperature and pressure.

We have measured the single-crystal elastic constants of ${\mathrm{KMnF}}_{3}$ and their first pressure derivatives from 200 to 430 K at a frequency of 20 MHz by means of the ultrasonic-pulse-superposition method. Both sets of data show cusplike anomalies above the improper ${O}_{h}$-${D}_{4h}$ ferroelastic transition at 186 K that extend up to 350 K. Above 350 K the temperature dependence is linear, making it possible to determine the bare harmonic and anharmonic constants by means of linear extrapolation to T=0. At room temperature the second pressure derivatives of the elastic constants were also measured. Analysis of the data on the basis of a model with Coulomb and short-range central force interactions indicates anomalously large fourth derivatives of the Mn-F and F-F pair potentials. A calculation of the temperature dependence of the elastic Gr\"uneisen parameter is consistent with models for the acoustic anomaly according to which coupling between strains and order-parameter fluctuations is limited to frequencies less than the relaxation rate of the soft ${R}_{25}$ mode.