Hard Spheres Research Articles

How to Compute Density Fluctuations at the Nanoscale.

The standard definition of particle number fluctuations based on point-like particles neglects the excluded volume effect. This leads to a large and systematic finite-size scaling and an unphysical surface term in the isothermal compressibility. We correct these errors by introducing a modified pair distribution function that takes account of the finite size of the particles. For the hard sphere fluid in one-dimension, we show that the compressibility is strictly size-independent, and we reproduce this result from the number fluctuations calculated with the new theory. In general, the present method eliminates the leading finite-size effect, which makes it possible to compute density fluctuations accurately in very small sampling volumes, comparable to a single particle size. These findings open the way for obtaining the local compressibility from fluctuation theory at the nanometer scale.

High‐Crystallinity Wrinkled Hard Carbon via Molecular‐Level Anchoring Glucose Molecules with Cooperative‐Assembly of Intermolecular Hydrogen Bonds for Sodium‐Ion Batteries

AbstractTuning of micro‐ and macro‐structures of hard carbon is important for expanding its properties. However, achieving controlled and integrated modulation of the micro‐ and macro‐structures of such materials remains a challenge. An intermolecular hydrogen bond‐driven interfacial cooperative‐assembly and then polymerization strategy is reported herein: micelles‐glucose molecules are used for the synthesis of wrinkled hard carbon spheres with precisely tunable crystallinity. As intermolecular hydrogen is positioned bonds to achieve an anchoring effect, which facilitates immobilization of glucose molecules at the molecular level during the assembly and subsequent dehydration‐aromatization. In virtue of controlling the micelle concentration, the density of intermolecular hydrogen bonding interactions is precisely regulated. As a result, the yielded hard carbon spheres exhibited excellent sodium‐ion storage performance because of their high crystallinity and exposed surfaces. By establishing a simple micelle–based interfacial reaction strategy, the study leverages optimized intermolecular hydrogen bond anchoring effects to tailor the properties of materials at a molecular level.

ON THE IONIC RADIUS OF METHYLAMMONIUM IN HYBRID ORGANIC-INORGANIC PEROVSKITE PHASES

Hybrid organic-inorganic perovskites ABX3 usually contain an organic cation A and an inorganic matrix formed from coordination octahedra [BX6], where B is a metal and X is a halogen. Predicting the formation of hybrid perovskite phases and modeling their structural parameters requires basic knowledge of the effective sizes (ionic radii) of the constituent components; however, the previously proposed approach to estimate the effective ionic radius of a particular organic cation A based on its molecular geometry apparently does not correspond to the hard spheres model, within the framework of which the vast majority of ionic radii systems and the main criteria for assessing the stability of perovskite structures were developed. To independently estimate the effective ionic radius of methylammonium CH3NH3+, which is a component of a number of perovskite structures, a scheme for modeling the lattice parameter of halide cubic perovskites ABX3 was developed by using multiple linear regression analysis based on the effective ionic radii of inorganic components with the corresponding coordination numbers intrinsic for perovskites. Using the found direct and inverse correlations between the effective ionic radius of the cation A and the cubic lattice parameter of perovskite, as well as numerical data of the lattice parameters of the known cubic hybrid perovskite phases (CH3NH3)PbCl3 and (CH3NH3)PbBr3, the value of the effective ionic radius of methylammonium CH3NH3+ was first estimated as ~1.9 Å, which is significantly different from the value of 2.17 Å obtained within the framework of the approach based on the molecular geometry.

Impact of mixing and configurational entropies on the entropy-driven potential of mean force

Using classical density functional theory (cDFT) with the fundamental measure functional (FMF) approximation, we examine how the entropic force and its integration, the potential of mean force (PMF), between two hard walls in a polydisperse hard-body solvent vary with monomer size polydispersity (whether the solvent consists of spherical monomers or chain molecules formed by freely linked monomers) or the number of monomers in the chain solvent. Molecular dynamics simulations validate the FMF's accuracy in predicting density distributions for five- and seven-component hard sphere fluids in slit geometries, though improvements are possible at high packing fractions. The PMF calculations reveal that increasing monomer size polydispersity or the number of monomers in a chain molecule reduce entropic attraction strength at contact and lower the main potential barrier height. This reduction is due to the system's limited capacity to further increase entropy when both mixing and conformational entropies are already high. For monomer or dimer solvent, the main potential barrier position shows either monotonic or non-monotonic dependence on monomer size polydispersity, depending on bulk packing fractions, while the PMF range shortens with increasing monomer size polydispersity or chain molecule monomer number. These findings could inform the design of tunable materials.

Simple Fluctuations in Simple Glass Formers.

Critical single-particle fluctuations associated with particle displacements are inherent to simple glass-forming liquids in the limit of large dimensions and leave a pseudocritical trace across all finite dimensions. This characteristic could serve as a crucial test for distinguishing between theories of glass formation. We here examine these critical fluctuations, as captured by the well-established non-Gaussian parameter, within both mode-coupling theory (MCT) and dynamical mean-field theory (DMFT) across dimensions for hard sphere fluids and for the minimally structured Mari-Kurchan model. We establish general scaling laws relevant to any liquid dynamics theory in large dimensions and show that the dimensional scalings predicted by MCT are inconsistent with those from DMFT. Simulation results for hard sphere fluids in moderately high dimensions align with the DMFT scenario, reinforcing the relevance of mean-field theory for capturing glass physics in finite dimensions. We identify potential adjustments to MCT to account for certain mean-field physics. Our findings also highlight that local structure and spatial dimensionality can affect single-particle critical fluctuations in nontrivial ways.

SAXS, DLS, and MD studies of the Rg/Rh ratio for swollen and collapsed dendrimers.

The radius of gyration, Rg, and the hydrodynamic radius, Rh, are the main experimental parameters that characterize the size of linear and branched macromolecules. In the case of dendrimers in solution, the ratio Rg/Rh, depending on the global conformation, varies from 1 (for a Gaussian soft sphere) to 3/5 (for a hard sphere). However, for high-generation dendrimers, this ratio may be less than the limiting value for a hard sphere. To understand the reasons of the low Rg/Rh value (<0.77), we have studied the second-generation peptide dendrimer containing pH-sensitive histidine amino acid residues (Lys-2His dendrimer) using small-angle x-ray (SAXS) and dynamic light scattering (DLS) experiments, as well as molecular dynamics simulations. The Lys-2His dendrimer takes a swollen conformation at pH = 2 and a collapsed 1 at pH = 7. Our results show that the Rg/Rh ratio for the considered dendrimer decreases from ≈3/5 at pH = 2 to 0.5 at pH = 7. We have found that the very low Rg/Rh value is due to (1) the formation of a dense impenetrable core (i.e., the transformation of the dendrimer from a Gaussian soft sphere into a sphere with a dense core) and (2) the presence of a larger number of solvent molecules in the dendrimer corona than in a typical macromolecule. In addition, in this work, we have directly confirmed in the experiments for the first time, the collapse of the Lys-2His dendrimer with increasing pH.

Numerical framework for the development of atmosphere-breathing electric propulsion for Earth and Mars atmosphere

Atmosphere-breathing electric propulsion systems provide a competitive advantage for the lower orbit altitudes since the propellant is collected directly from the atmosphere. The effectiveness of this technology depends on crucial aspects such as the collection and compression performance characterization, as well as the drag estimation and compensation. In the first part of this study, the lower Mars and Earth atmospheric characterization is derived based on current models and mission data. This characterization is a reliable dataset for the boundary conditions for the simulations carried out in the second part of this study. The proposed computational framework based on the Direct Simulation Monte Carlo method aims to investigate the collection and compression performances and to estimate the drag. The numerical comparison with a literature case validates the numerical setup presented in this study. The effect of different gas-surface interaction models is investigated by comparing the results yielded by the Maxwellian model (fully specular and partially diffuse reflection) and the Cercignani-Lampis-Lord model. Since the intermolecular collisions can become more relevant at the inlet of the ionization stage, both the variable hard and variable soft sphere models are briefly examined, as well as the inclusion of gas-phase reactions. Finally, the simulation results of the two cases for the low Mars orbit (150 and 140 km) are compared to the Earth case (180 km).

Molecular dynamics study of diffusion coefficient in Uniformly Heated Hard Sphere Granular Gas in three dimensions

This study presents a molecular dynamics simulation of an inelastic gas system using an event-driven algorithm. The model incorporates a coefficient of restitution of less than one to govern molecular collisions and employs a thermostat mechanism introducing Gaussian white noise to maintain system temperature. We investigate the behavior of a Uniformly Heated Hard Sphere Granular Gas, focusing on the relationship between the coefficient of restitution and the self-diffusion coefficient, as well as the temporal evolution of Mean Square Displacement. Initially, the system’s kinetic energy decreases but eventually reaches a non-equilibrium steady state. Our findings reveal that higher restitution coefficients result in slightly larger particle displacements, likely due to enhanced energy retention post-collision. Counterintuitively, we observe that the diffusion coefficient decreases as the restitution coefficient increases. This research contributes to our understanding of granular gas dynamics and non-equilibrium thermodynamics.

Medium-Range Structural Order as the Driver of Activated Dynamics and Complexity Reduction in Glass-Forming Liquids.

We analyze in depth the Elastically Collective Nonlinear Langevin Equation theory of activated dynamics in metastable liquids to establish that the predicted inter-relationships between the alpha relaxation time, local cage and collective elastic barriers, dynamic localization length, and shear modulus are causally related within the theory to the medium range order (MRO) static correlation length. The latter grows exponentially with density for metastable hard sphere fluids and as a nonuniversal inverse power law with temperature for supercooled liquids under isobaric conditions. The physical origin of predicted connections between the alpha time and other metrics of cage order and the thermodynamic inverse dimensionless compressibility is fully established. It is discovered that although kinetic constraints from the real space first coordination shell are important for the alpha time, they are of secondary importance compared to the consequences of the more universal MRO correlations in both the modestly and deeply metastable regimes. This understanding sheds new light on the theoretical basis for, and prior successes of, the predictive mapping of chemically complex thermal liquids to effective hard sphere fluids based on matching their dimensionless compressibilities, a scheme we call "complexity reduction". In essence, the latter is equivalent to the physical requirement that the thermal liquid MRO correlation equals that of its effective hard sphere analog. The mapping alone is shown to provide a remarkable level of quantitative predictive power for the glass transition temperature Tg of 21 molecular and polymer liquids. Predictions for the chemically specific absolute magnitude and growth with cooling of the MRO correlation length are obtained and lie in the window of 2-6 nm at Tg. Dynamic heterogeneity, elastic facilitation, and beyond pair structure issues are briefly discussed. Future opportunities to theoretically analyze the equilibrated deep glass regime are outlined.

Increase in Auxeticity Due to the Presence of a Disordered Crystalline Phase of Hard Dumbbells Within the Nanolayer-Nanochannel Inclusion Introduced to the f.c.c. Hard Sphere Crystal.

To obtain materials or metamaterials with desired elastic properties that are tailor-made for a particular application, it is necessary to design a new material or composite (which may be cumbersome) or to modify the structure of existing materials in order to change their properties in the desired direction. The latter approach, although also not easy, seems favourable with respect to parameters like costs and time-to-market. Despite the fact that elastic properties are one of the oldest studied physical parameters of matter, our understanding of the processes at the microstructural level, that are behind these properties, is still far from being complete. The present work, with the help of Monte Carlo computer simulations, aims to broaden this knowledge. The previously studied model crystal of hard spheres, containing a combined nanolayer and nanochannel inclusions, is revisited. This periodic model crystal has been extended to include a degree of disorder in the form of degenerate crystalline phase by introducing a degenerate crystalline phase within its structure. The inclusion has been transformed (without changes to its shape, size, or orientation) by randomly connecting the neighbouring spheres into di-atomic molecules (dumbbells). The impact of this modification on elastic properties has been investigated with the help of the Parrinello-Rahman approach in the isothermal-isobaric ensemble (NpT). It has been shown, that the presence of the degenerate crystalline phase of hard dumbbells in the system leads to a significant decrease in the Poisson's ratio in [110]-direction (ν=-0.235) and an overall enhancement of the auxetic properties.

Fine‐Tuning of Elastic Properties by Modifying Ordering of Parallel Nanolayer Inclusions in Hard Sphere Crystals

It is known that changes on the microscopic, structural level of materials exert changes on their macroscopic properties, in particular elastic ones. This approach can be used to alter the elastic properties of materials. However, it is not easy to predict the impact of a particular change in the structure of crystals. The recent studies show that depending on the size, shape, and orientation of inclusions used, the resulting elastic properties may differ. Inclusions may enhance, weaken, or eliminate the auxetic properties in a hard sphere model. This study is focused on the influence of the spatial distribution of planar inclusions within the hard sphere crystal and its impact on its elastic properties. Periodic systems containing two nanolayer inclusions in the representative volume element, oriented orthogonally to ‐direction and in various spatial ordering, are considered. It has been shown that introducing layer inclusions causes the deterioration of auxetic properties in the ‐direction. However, the latter weakly depends on the spatial ordering of the inclusion layers. Moreover, changes in the size of the inclusion particles, combined with different ordering of the inclusion layers, can be used to coarsely and finely tune the elastic properties of the model crystal.

Statistical mechanics of crystal nuclei of hard spheres.

In the study of crystal nucleation via computer simulations, hard spheres are arguably the most extensively explored model system. Nonetheless, even in this simple model system, the complex thermodynamics of crystal nuclei can sometimes give rise to counterintuitive results, such as the recent observation that the pressure inside a critical nucleus is lower than that of the surrounding fluid, seemingly clashing with the strictly positive Young-Laplace pressure we would expect in liquid droplets. Here, we re-derive many of the founding equations associated with crystal nucleation and use the hard-sphere model to demonstrate how they give rise to this negative pressure difference. We exploit the fact that, in the canonical ensemble, a nucleus can be in a (meta)stable equilibrium with the fluid and measure the surface stress for both flat and curved interfaces. Additionally, we explain the effect of defects on the chemical potential inside the crystal nucleus. Finally, we present a simple, fitted thermodynamic model to capture the properties of the nucleus, including the work required to form critical nuclei.

Ion-Specific Surface Tension of Aqueous Electrolyte Solutions: Analytical Insights from a Restricted Primitive Model.

The ion-specific surface tension of aqueous electrolyte solutions is of fundamental importance in physical chemistry, solution chemistry, electrochemistry, and biochemistry, yet it remains a challenge to be qualitatively predicted. In this work, an analytical theory of ion-specific surface tension is developed. By modeling the solution to a restricted primitive model (RPM), the surface tension increment of the electrolyte solution reduces to the surface tension of the RPM, which is further determined analytically by using the integral equation theory and the morphological thermodynamics theory. According to our formula, the surface tension increment of the electrolyte solution consists of a dominant and positive hard sphere contribution, which depends on the salt concentration, and a secondary electrostatic contribution, which depends on the inverse Debye length and the salt concentration. Our theory is applied to 1:1, 2:2, 1:2, 2:1, 3:1, and 3:2 electrolyte solutions with typical salt concentrations up to several mol/L and compared with experimental data. Without introducing any adjustable parameters, our theory leads to a good prediction of the surface tension increment of more than 50 kinds of aqueous solutions. Such a good agreement demonstrates the great potential of our theory for a fundamental understanding of specific ion effects in a variety of electrolyte solutions.

Comparative analysis of technological properties and microstructure of titanium powders of different classes of alloys

The results of a study of the technological properties of titanium powders of pseudo-α and pseudo-β alloys are presented. A comparative analysis of the characteristics of powder materials obtained by centrifugal spraying has been carried out. The results of the granulometric composition of the powders were obtained, and the calculation of the spatial packing of particles was applied within the framework of the hard sphere model for various fractional compositions. To compare the microstructure and mechanical properties, test samples from pseudo-α and pseudo-β titanium alloys were obtained using the hot isostatic pressing method according to selected modes.

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

We consider a gas of bosons interacting through a hard-sphere potential with radius a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa and shows that corrections are smaller than Cρa(ρa3)1/2, for a sufficiently large constant C>0 0$$\\end{document}]]>. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order ρa(ρa3)1/2, in agreement with the Lee–Huang–Yang prediction.

Structural analysis of superparamagnetic colloid aggregates confined in spherical cavities under external magnetic field

In this study, we investigated the behaviour of aggregates composed of superparamagnetic colloids under the influence of an external magnetic field, confined within several spherical cavities. The study included systems of two and three cavities in contact, as well as two cavities separated by a distance d along the direction of the external field. Monte Carlo computer simulations employing a model of hard spheres with central dipoles were realised to explore the behaviour of the magnetic particles, constrained to move within pairs (and trios) of spherical cavities. The primary focus of this investigation was to elucidate the structures formed by magnetic particles within systems presented here. Chains or ribbons primarily form within the central region of the spherical cavities. The aggregates within each cavity are drawn towards one another until they make contact at the closest points of the cavities. This behaviour is replicated when there is a separation distance between the spherical cavities, including cases where three cavities are in contact. This phenomenon may explain the observed attraction between magnetoliposomes as reported in the literature [García-Jimeno et al. Nanoscale. 9 (39), 15131–15143 (2017)].

Long ranged stress correlations in the hard sphere liquid.

The smooth emergence of shear elasticity is a hallmark of the liquid to glass transition. In a liquid, viscous stresses arise from local structural rearrangements. In the solid, Eshelby has shown that stresses around an inclusion decay as a power law r-D, where D is the dimension of the system. We study glass-forming hard sphere fluids by simulation and observe the emergence of the unscreened power-law Eshelby pattern in the stress correlations of the isotropic liquid state. By a detailed tensorial analysis, we show that the fluctuating force field, viz., the divergence of the stress field, relaxes to zero with time in all states, while the shear stress correlations develop spatial power-law structures inside regions that grow with longitudinal and transverse sound propagation. We observe the predicted exponents r-D and r-D-2. In Brownian systems, shear stresses relax diffusively within these regions, with the diffusion coefficient determined by the shear modulus and the friction coefficient.

Interpreting the power spectral density of a fluctuating colloidal current.

The transport of molecules through biological and synthetic nanopores is governed by multiple stochastic processes that lead to noisy, fluctuating currents. Disentangling the characteristics of different noise-generating mechanisms is central to better understanding molecular transport at a fundamental level but is extremely challenging in molecular systems due to their complexity and relative experimental inaccessibility. Here, we construct a colloidal model microfluidic system for the experimental measurement of particle currents, where the governing physical properties are directly controllable and particle dynamics directly observable, unlike in the molecular case. Currents of hard spheres fluctuate due to the random arrival times of particles into the channel and the distribution of particle speeds within the channel, which results in characteristic scalings in the power spectral density. We rationalize these scalings by quantitatively comparing to a model for shot noise with a finite transit time, extended to include the distribution of particle speeds. Particle velocity distributions sensitively reflect the confining geometry, and we interpret and model these in terms of the underlying fluid flow profiles. Finally, we explore the extent to which details of these distributions govern the form of the resulting power spectral density, thereby establishing concrete links between the power spectral density and underlying mechanisms for this experimental system. This paves the way for establishing a more systematic understanding of the links between characteristics of transport fluctuations and underlying molecular mechanisms in driven systems such as nanopores.

A scaling relationship between thermodynamic and hydrodynamic interactions in protein solutions

Weak protein interactions are associated with a broad array of biological functions and are often implicated in molecular dysfunction accompanying human disease. In addition, these interactions are a critical determinant in the effective manufacturing, stability, and administration of biotherapeutic proteins. Despite their prominence, much remains unknown about how molecular attributes influence the hydrodynamic and thermodynamic contributions to the overall interaction mechanism. To systematically probe these contributions, we have evaluated self-interaction in a diverse set of proteins that demonstrate a broad range of behaviors from attractive to repulsive. Analysis of the composite trending in the data provides a convenient interconversion among interaction parameters measured from the concentration dependence of the molecular weight, diffusion coefficient, and sedimentation coefficient, as well as insight into the relationship between thermodynamic and hydrodynamic interactions. We find relatively good agreement between our data and a model for interacting hard spheres in the range of weak self-association. In addition, we propose an empirically derived, general scaling relationship applicable across a broad range of self-association and repulsive behaviors.

Study of the Flow Characteristics of Pumped Media in the Confined Morphology of a Ferrofluid Pump With Annular Microscale Constraints

Abstract The driving mechanism of ferrofluid micropumps under the constraints of an annular microscale morphology is not fully understood. The gap between microfabrication technology and the fundamental theory of microfluidics has become a substantial obstacle to the development and application of ferrofluid micropumps. In this study, we first theoretically analyzed the Knudsen numbers of millimeter-scale microfluids using Jacobson's molecular hard sphere model, obtaining the initial conclusion that liquid flow conforms to the continuum hypothesis in geometric morphologies with characteristic dimensions greater than 7 × 10−8 m. Subsequently, using a microscopic lens combined with the particle image velocimetry optical measurement method, the flow patterns in millimeter-scale annular flow channels were captured and we observed wall slip phenomena in which the slip length of the millimeter-scale channel approached the micron level. The slip velocity and flowrate through the cross section of the microscale channel followed a logarithmic function relationship and could be divided into rapid growth, slow growth, and stable stages. As the characteristic scale of the channel was further reduced, the linear relationship between the slip velocity and cross-sectional flowrate in the rapid growth stage was broken, and the nonlinear relationship approximated an exponential function. Finally, a theoretical model for the flow behavior of the driving fluid in a ferrofluid micropump was established using slip boundary conditions. The flow patterns in microscale ring flow under slip conditions conformed to a quadratic function.