Penetrable Spheres Research Articles

A Time-Dependent Hierarchical Model for Elastic and Inelastic Scattering Data Analysis of Aerogels and Similar Soft Materials.

Soft nanomaterials like aerogels are subject to thermal fluctuations, so that their structure randomly fluctuates with time. Neutron elastic and inelastic scattering experiments provide unique structural and dynamic information on such systems with nanometer and nanosecond resolution. The data, however, come in the form of space- and time-correlation functions, and models are required to convert them into time-dependent structures. We present here a general time-dependent stochastic model of hierarchical structures, with scale-invariant fractals as a particular case, which enables one to jointly analyze elastic and inelastic scattering data. In order to describe thermal fluctuations, the model builds on time-dependent generalisations of the Boolean model of penetrable spheres, whereby each sphere is allowed to move either ballistically or diffusively. Analytical expressions are obtained for the correlation functions, which can be used for data fitting. The model is then used to jointly analyze previously published small-angle neutron scattering (SANS) and neutron spin-echo (NSE) data measured on silica aerogels. In addition to structural differences, the approach provides insight into the different scale-dependent mobility of the aggregates that make up the aerogels, in relation with their different connectivities.

Dynamic Measure of Hyperuniformity and Nonhyperuniformity in Heterogeneous Media via the Diffusion Spreadability

The recently developed concept of spreadability, $\mathcal{S}(t)$, provides a direct link between time-dependent diffusive transport and the microstructure of two-phase media across length scales. We explicitly compute $\mathcal{S}(t)$ for well-known two-dimensional and three-dimensional model structures, including nonhyperuniform fully penetrable spheres and equilibrium hard spheres, as well as hyperuniform sphere packings derived from perfect glasses, uniformly randomized lattices (URL), disordered stealthy point processes and Bravais lattices. We further confirm that the small-, intermediate- and long-time behaviors of $\mathcal{S}(t)$ sensitively capture the small-, intermediate- and large-scale characteristics of the models. In instances in which the spectral density $\tilde{\chi}_{_V}(\mathbf{k})$ has a power-law form $B|\mathbf{k}|^\alpha$ in the limit $|\mathbf{k}|\rightarrow 0$, the long-time spreadability provides a simple means to extract the value of the coefficients $\alpha$ and $B$ that is robust against noise in $\tilde{\chi}_{_V}(\mathbf{k})$ at small wavenumbers. Interestingly, the excess spreadability $\mathcal{S}(\infty)-\mathcal{S}(t)$ for URL packings has nearly exponential decay at small to intermediate $t$, but transforms to a power-law decay at large $t$. Our study of the aforementioned models enables us to devise an algorithm that accurately extracts large-scale behaviors from diffusion data alone. Lessons learned from such analyses of our models are used to determine accurately the large-scale structural characteristics of a sample Fontainebleau sandstone, which we show is nonhyperuniform. Our study demonstrates the practical applicability of the diffusion spreadability to extract crucial microstructural information from real data across length scales and provides a basis for the inverse design of materials with desirable time-dependent diffusion properties.

Effects of polymer nonideality on depletion-induced phase behaviour of colloidal disks and rods

Colloidal dispersions composed of either platelets or rods exhibit liquid crystalline phase behaviour that is strongly influenced by the addition of nonadsorbing polymers. In this work we examined how polymer segment–segment interactions affect this phase behaviour as compared to using either penetrable hard spheres (PHS) or ideal (‘ghost’) chains as depletants. We find that the simplified polymer description predicts the same phase diagram topologies as the more involved polymer descriptions. Therefore the PHS description is still adequate for qualitative predictions. For sufficiently large polymer sizes we find however that the precise polymer description significantly alters the locations of the phase coexistence regions. Especially the stability region of isotropic–isotropic coexistence is affected by the polymer interactions. To illustrate the quantitative effects some examples are presented.

Multiphase Coexistences in Rod-Polymer Mixtures.

Using recently derived analytical equations of state for hard rod dispersions, we predict the phase behavior of athermal rod–polymer mixtures with free volume theory. The rods are modeled as hard spherocylinders, while the nonadsorbing polymer chains are described as penetrable hard spheres. It is demonstrated that all of the different types of phase states that are stable for pure colloidal rod dispersions can coexist with any combination of these phases if polymers are added, depending on the concentrations, rod aspect ratio, and polymer–rod size ratio. This includes novel two-, three-, and four-phase coexistences and isostructural coexistences between dilute and concentrated phases of the same kind, even for the more ordered (liquid) crystal phases. This work provides insight into the conditions at which particular multiphase coexistences are expected for well-defined model colloidal rod–polymer mixtures. We provide a quantitative map detailing the various types of isostructural coexistences, which confirms an early qualitative hypothesis by Bolhuis et al. (J. Chem. Phys.107, 19971551).

Influence of solvent quality on depletion potentials in colloid-polymer mixtures.

As first explained by the classic Asakura-Oosawa (AO) model, effective attractive forces between colloidal particles induced by depletion of nonadsorbing polymers can drive demixing of colloid-polymer mixtures into colloid-rich and colloid-poor phases, with practical relevance for purification of water, stability of foods and pharmaceuticals, and macromolecular crowding in biological cells. By idealizing polymer coils as effective penetrable spheres, the AO model qualitatively captures the influence of polymer depletion on thermodynamic phase behavior of colloidal suspensions. In previous work, we extended the AO model to incorporate aspherical polymer conformations and showed that fluctuating shapes of random-walk coils can significantly modify depletion potentials [W. K. Lim and A. R. Denton, Soft Matter 12, 2247 (2016); J. Chem. Phys. 144, 024904 (2016)]. We further demonstrated that the shapes of polymers in crowded environments sensitively depend on solvent quality [W. J. Davis and A. R. Denton, J. Chem. Phys. 149, 124901 (2018)]. Here, we apply Monte Carlo simulation to analyze the influence of solvent quality on depletion potentials in mixtures of hard-sphere colloids and nonadsorbing polymer coils, modeled as ellipsoids whose principal radii fluctuate according to random-walk statistics. We consider both self-avoiding and non-self-avoiding random walks, corresponding to polymers in good and theta solvents, respectively. Our simulation results demonstrate that depletion of polymers of equal molecular weight induces much stronger attraction between colloids in good solvents than in theta solvents and confirm that depletion interactions are significantly influenced by aspherical polymer conformations.

ПОРОМАСШТАБНОЕ МОДЕЛИРОВАНИЕ ПОТОКА ЖИДКОСТИ В ПРОНИЦАЕМЫХ СФЕРАХ С ИСПОЛЬЗОВАНИЕМ ПРОЕКЦИОННОГО МЕТОДА ДЛЯ НЕСЖИМАЕМЫХ УРАВНЕНИЙ НАВЬЕ-СТОКСА

The direct numerical simulation (DNS) is an effective and useful tool in the two-phase fluid flow studying. The projection method on the staggered grid was applied in this paper to solve the incompressible Navier-Stokes equations in irregular domains at the pore-scale level (irregular boundary is presented by its level-set function). The permeability of porous medium which was constructed by the random positioning of penetrable spheres of equal radii were numerically calculated and validated by comparing with theoretical estimations of permeability based on the numerical solution of the lattice-Boltzmann equation in irregular domains in previous works. All numerical calculations were performed using PARIS simulator.

Equilibrium properties of penetrable soft spheres

Using the Weeks-Chandler-Andersen separation scheme, we explored equilibrium properties of fully penetrable soft spheres with an attractive tail. When the radial distribution function of the reference system is computed by means of the integral equation theory using the hyper-netted-chain closure, this separation scheme predicts pressure-density isotherms, vapour-liquid phase coexistence, and surface tension in good agreement with the results from Monte Carlo simulations even though the attractive portion of the interparticle potential has a significant effect on the radial distribution function. Despite its simplicity, the model potential we studied, with a certain parameter value, exhibits a non-monotonic temperature dependence of the liquid phase density over a wide range of pressure.

Perfect sampling for Gibbs point processes using partial rejection sampling

We present a perfect sampling algorithm for Gibbs point processes, based on the partial rejection sampling of Guo, Jerrum and Liu (In STOC’17 – Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (2017) 342–355 ACM). Our particular focus is on pairwise interaction processes, penetrable spheres mixture models and area-interaction processes, with a finite interaction range. For an interaction range $2r$ of the target process, the proposed algorithm can generate a perfect sample with $O(\log(1/r))$ expected running time complexity, provided that the intensity of the points is not too high and $\Theta(1/r^{d})$ parallel processor units are available.

Continuum percolation expressed in terms of density distributions.

We present an approach to derive the connectivity properties of pairwise interacting n-body systems in thermal equilibrium. We formulate an integral equation that relates the pair connectedness to the distribution of nearest neighbors. For one-dimensional systems with nearest-neighbor interactions, the nearest-neighbor distribution is in turn related to the pair-correlation function g through a simple integral equation. As a consequence, for those systems, we arrive at an integral equation relating g to the pair connectedness, which is readily solved even analytically if g is specified analytically. We demonstrate the procedure for a variety of pair potentials including fully penetrable spheres as well as impenetrable spheres, the only two systems for which analytical results for the pair connectedness exist. However, the approach is not limited to nearest-neighbor interactions in one dimension. Hence, we also outline the treatment of external fields and long-range interactions and we illustrate how the formalism can applied to higher-dimensional systems using the three-dimensional ideal gas as an example.

Isostructural solid–solid coexistence of colloid–polymer mixtures

Hard spheres mixed with penetrable hard spheres display an isostructural solid–solid transition. This phase transition is fully driven by the entropy gain of the depletants without invoking explicit pair potentials between the colloidal particles. The solid–solid phase coexistence terminates at a critical endpoint with a size ratio q≈0.09. Colloid–polymer mixtures are an excellent candidate for the experimental realization of this intricate solid–solid transition, first predicted by Bolhuis and Frenkel for hard spheres with short range pair attractions [PRL 72, 2211–2214 (1994)].

Ring polymers are much stronger depleting agents than linear ones

ABSTRACTWe investigate the conformations and shapes of circular polymers close to planar, hard walls as well as the ensuing ring–wall and polymer-induced wall–wall interactions in ring polymer solutions. We derive, by means of Monte Carlo simulations, the effective interaction potential between the centres of mass of flexible, unknotted ring polymers and a hard wall for different polymerisation degrees N. Adopting the coarse-grained description of ring polymers as ultrasoft, penetrable spheres, mean-field density functional theory is then employed in order to examine ring polymer solutions under confinement. We demonstrate that, below the semi-dilute regime, ring polymers structure close to walls much more strongly than their linear counterparts, their density profiles featuring pronounced oscillations. Moreover, the polymer-induced depletion potential between the two walls exhibits an oscillatory profile reminiscent of hard-sphere systems, with oscillations that intensify upon increasing polymer concentration. The obtained form of the depletion interaction is shown to be qualitatively different in comparison to the case of linear polymer solutions at the corresponding densities.

Upper Bounds on the Permeability of Random Porous Media

Upper bounds on the permeability of random porous media are presented, which improve significantly on existing bounds. The derived bounds rely on a variational formulation of the upscaling problem from a viscous flow at the pore scale, described by Stokes equation, to a Darcy formulation at the macroscopic scale. A systematic strategy to derive upper bounds based on trial force fields is proposed. Earlier results based on uniform void or interface force fields are presented within this unified framework, together with a new proposal of surface force field and a combination of them. The obtained bounds feature detailed statistical information on the pore morphology, including two- and three-point correlation functions of the pore phase, the solid–fluid interface and its local orientation. The required spatial correlation functions are explicitly derived for the Boolean model of spheres, in which the solid phase is modelled as the union of penetrable spheres. Existing and new bounds are evaluated for this model and compared to full-field simulations on representative volume elements. For the first time, bounds allow to retrieve the correct order of magnitude of permeability for a wide range of porosity and even improve on some estimates. However, none of the bounds reproduces the non-analytic behaviour of the permeability–porosity curve at low solid concentration.

Diffusiophoresis in nonadsorbing polymer solutions: The Asakura-Oosawa model and stratification in drying films.

A colloidal particle placed in an inhomogeneous solution of smaller nonadsorbing polymers will move towards regions of lower polymer concentration in order to reduce the free energy of the interface between the surface of the particle and the solution. This phenomenon is known as diffusiophoresis. Treating the polymer as penetrable hard spheres, as in the Asakura-Oosawa model, a simple analytic expression for the diffusiophoretic drift velocity can be obtained. In the context of drying films we show that diffusiophoresis by this mechanism can lead to stratification under easily accessible experimental conditions. By stratification we mean spontaneous formation of a layer of polymer on top of a layer of the colloid. Transposed to the case of binary colloidal mixtures, this offers an explanation for the stratification observed recently in these systems [A. Fortini etal., Phys. Rev. Lett. 116, 118301 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.118301]. Our results emphasize the importance of treating solvent dynamics explicitly in these problems and caution against the neglect of hydrodynamic interactions or the use of implicit solvent models in which the absence of solvent backflow results in an unbalanced osmotic force that gives rise to large but unphysical effects.

The Practical Use of the Model of Penetrable Spheres.

Two examples are given of the use of the molecular model of penetrable spheres to throw light on problems in liquid state physics.

Structure and phase behaviors of confined two penetrable soft spheres

We study the phase behaviors of two penetrable soft spheres, whose interactions include the soft repulsion and attraction, in a hard spherical pore. The exact partition function, one-body density, and equation of state for the confined two penetrable soft spheres have been calculated using the Fourier transform method. The phase diagrams have been determined from the negative compressibility of the van der Waals type, which imitates the phase transition of many particle system. The addition of the soft repulsion and attraction beyond the soft-core potential gives rise to the van der Waals instability. The soft attraction beyond the soft-core potential significantly enhances the van der Waals instability, whereas the soft repulsion reduces the van der Waals instability. For two hard spheres and hard square-well spheres, the van der Waals instability is not observed. However, the addition of a short-range soft repulsion beyond the hard-core gives rise to the van der Waals instability.

Continuum percolation of polydisperse hyperspheres in infinite dimensions.

We analyze the critical connectivity of systems of penetrable d-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers, and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are treelike for d→∞ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density η(c)→2(-d), independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality z(c) is smaller than unity, while z(c)→1 only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit, the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a log-normal distribution is no longer universal, and that it can be smaller than 2(-d) for d→∞.

Statistical properties of two interacting soft spheres in a hard spherical pore

The structure and thermodynamics of two soft spheres, whose interactions include the soft repulsion and attraction, confined in a hard spherical pore have been considered. The exact partition function, one-body density, and equation of state have been obtained using the Fourier transform method. The pore-size and potential dependence of the structural and thermodynamic properties have been investigated. The two penetrable square-well spheres shows a negative compressibility of the van der Waals type, which imitates the phase transition of many particle system. The van der Waals instability for two penetrable square-well spheres has been studied.

Segregation of penetrable soft spheres under gravity: Mean-field approach

We study the segregation of the additive and non-additive soft-sphere mixtures, which have the purely repulsive interactions, on the relative range of temperature, the cross-interaction between unlike species, the buoyant mass, and the layer numbers. The results show that the most segregation is found at m2/m1=1 and a low temperature, where mi is the mass of species i. The crossover point corresponding to the mixing phase is shifted toward a large mass ratio with increasing the temperature. For the non-additive soft sphere mixture, the crossover point is shifted toward a small mass ratio with increasing the cross-interaction between unlike species. The increase of gravitational strength drives the larger particles sinking to the bottom. The pressure at height z has been calculated from the force balance on a slab of fluids adjacent to the wall. The pressure at the bottom decreases with decreasing the temperature, whereas the pressure at a high altitude increases because of the large particles at the top. For the mixing phase, the large pressure discontinuity is found at z=σ1 because of the density difference between two particles. Through this work, we show that the center of mass of the species, <zi>, is a proper factor for studying the particle segregation.

Third-order thermo-mechanical properties for packs of Platonic solids using statistical micromechanics.

Obtaining an accurate higher order statistical description of heterogeneous materials and using this information to predict effective material behaviour with high fidelity has remained an outstanding problem for many years. In a recent letter, Gillman & Matouš (2014 Phys. Lett. A 378, 3070-3073. ()) accurately evaluated the three-point microstructural parameter that arises in third-order theories and predicted with high accuracy the effective thermal conductivity of highly packed material systems. Expanding this work here, we predict for the first time effective thermo-mechanical properties of granular Platonic solid packs using third-order statistical micromechanics. Systems of impenetrable and penetrable spheres are considered to verify adaptive methods for computing n-point probability functions directly from three-dimensional microstructures, and excellent agreement is shown with simulation. Moreover, a significant shape effect is discovered for the effective thermal conductivity of highly packed composites, whereas a moderate shape effect is exhibited for the elastic constants.

Tomography based numerical simulation of the demagnetizing field in soft magnetic composites

The magneto-static behaviour of soft magnetic composites (SMCs) is investigated using tomography based direct numerical simulation. The microgeometry crucially affects the magnetic properties of the composite since a geometry dependent demagnetizing field is established inside the composite, which lowers the magnetic permeability. We determine the magnetic field information inside the SMC using direct numerical simulation of the magnetic field based on high resolution micro-computed tomography data of the SMC's microstructure as well as artificially generated data made of statistically homogeneous systems of identical fully penetrable spheres and prolate spheroids. Quasi-static electromagnetic behaviour and linear material response are assumed. The 3D magnetostatic Maxwell equations are solved using Whitney finite elements. Simulations show that clustering and percolation behaviour determine the demagnetizing factor of SMCs rather than the particle shape. The demagnetizing factor correlates with the slope of a 2-point probability function at its origin, which is related to the specific surface area of the SMC. Comparison with experimental results indicates that the relatively low permeability of SMCs cannot be explained by demagnetizing effects alone and suggests that the permeability of SMC particles has to be orders of magnitude smaller than the bulk permeability of the particle material.