Close Packing Limit Research Articles

Random close packing of semi-flexible polymers in two dimensions: Emergence of local and global order.

Through extensive Monte Carlo simulations, we systematically study the effect of chain stiffness on the packing ability of linear polymers composed of hard spheres in extremely confined monolayers, corresponding effectively to 2D films. First, we explore the limit of random close packing as a function of the equilibrium bending angle and then quantify the local and global order by the degree of crystallinity and the nematic or tetratic orientational order parameter, respectively. A multi-scale wealth of structural behavior is observed, which is inherently absent in the case of athermal individual monomers and is surprisingly richer than its 3D counterpart under bulk conditions. As a general trend, an isotropic to nematic transition is observed at sufficiently high surface coverages, which is followed by the establishment of the tetratic state, which in turn marks the onset of the random close packing. For chains with right-angle bonds, the incompatibility of the imposed bending angle with the neighbor geometry of the triangular crystal leads to a singular intra- and inter-polymer tiling pattern made of squares and triangles with optimal local filling at high surface concentrations. The present study could serve as a first step toward the design of hard colloidal polymers with a tunable structural behavior for 2D applications.

Revised Enskog theory and molecular dynamics simulations of the viscosities and thermal conductivity of the hard-sphere fluid and crystal.

Hard-sphere (HS) shear, longitudinal, cross, and bulk viscosities and the thermal conductivity are obtained by molecular dynamics (MD) simulations, covering the entire density range from the dilute fluid to the solid crystal near close-packing. The transport coefficient data for the HS crystal are largely new and display, unlike for the fluid, a surprisingly simple behavior in that they can be represented well by a simple function of the density compressibility factor. In contrast to the other four transport coefficients (which diverge), the bulk viscosity in the solid is quite small and decreases rapidly with increasing density, tending to zero in the close-packed limit. The so-called cross viscosity exhibits a different behavior to the other viscosities, in being negative over the entire solid range, and changes sign from negative to positive on increasing the density in the fluid phase. The extent to which the viscosity tensor and thermal conductivity of the HS crystal can be represented by revised Enskog theory (RET) is investigated. The RET expressions are sums of an instantaneous (I), a kinetic (K), and a so-called α part. The I part of the transport coefficients evaluated directly by MD are statistically indistinguishable from those of the corresponding kinetic theory (Enskog and RET) expressions. For the K part the integral over the spatial two-particle distribution function at contact was determined and the α part was estimated using the direct correlation function and density functional theory approximations. All three parts were determined in this work which allowed the accuracy of RET for solid systems to be assessed rigorously. It is found that in the case of the thermal conductivity the predictions of RET are in excellent agreement with the MD results. Also, for the shear viscosity the agreement over the entire solid phase is quite good but is considerably worse for the three remaining viscosities in the solid phase.

An alternative general model for the effective longitudinal diffusion in chromatographic beds filled with ordered porous particles

The two-zone moment-analysis method for the determination of the dispersion tensor in hierarchical retentive porous media has been adopted to compute and model the effective longitudinal diffusion Deff, or equivalently the B-term band broadening, in chromatographic beds filled with ordered porous particles. On the one hand, this approach offers accurate numerical results for Deff while keeping computational expenses low. On the other hand, it also gives direct insight for the analytical modelling, readily revealings the two main essential quantities (resp. referred to as the mobile-zone and stationary-zone effective diffusion factors γm and γs) that contribute to Deff. Modelling these two main parameters provided us with two new analytical models for Deff: a general one, valid for diluted and concentrated packings and accurate in the whole range of relevant intra-particle diffusion coefficient Dpz, and an approximate one, reliable for diluted packings and accurate also for concentrated packings with low to intermediate values of Dpz. The large advantage of both models is that they do not need any fitting parameter because all the required information is incorporated into the experimentally accessible geometric obstruction factor in the mobile phase originating from the tortuosity of the through-pore space (limiting case of fully solid particles without any retention). These models hence serve as an alternative to the Effective Medium Theory (EMT) models used so far in the literature. To validate the theory, five ordered geometries have been investigated. The accuracy of the general model proposed has been quantified and found to be comparable with that of the 3rd order approximate Torquato model for four geometries, even for macro-porosities close to the close-packing limit. The case of a 2-d triangular array of ellipsoidal particles with different elongations is also investigated to show the general validity and applicability of the models.

Hydrodynamic permeability of circular-inclusion-doped Brinkman media

Theory and computations are applied to assess the hydrodynamic permeability of cavity- doped hydrogels, central to a variety range of contemporary technological applications. Direct volume-averaging is undertaken in a two-dimensional, Brinkman-hydrodynamic context to test an ensemble-averaging methodology recently proposed for the ion permeability of such media. In two dimensions, the ensemble-averaging integral furnishes a pre-factor $2$ linking the pressure dipole strength of a single inclusion in an unbounded continuum to the effective hydrodynamic permeability of a composite with small inclusion area fraction. The factor is verified by direct computations for dilute simple-square arrays of (cylindrical) inclusions. At area fractions up to the close-packing limit, computations address the hydrodynamic interactions. The theory is shown to accurately predict the effective hydrodynamic permeability of physically relevant composites (Brinkman length of the continuous phase $\ell$ smaller than the inclusion radius $a$ ) for area fractions $\phi \lesssim {\rm \pi}/ 9 \approx 0.3$ . Computations for random ensembles demonstrate that the dilute theory may be extended to higher area fractions by drawing on Rayleigh's self-consistent approximation when the continuous-phase permeability places the continuous-phase flow well into the Darcy regime ( $a / \ell \gtrsim 10$ ). Computations also demonstrate, similarly to Rayleigh theories for scalar diffusion, that microstructural order has a very weak influence on the effective permeability when $\phi \lesssim {\rm \pi}/ 9$ with $\ell / a \lesssim 1$ (Darcy hydrodynamic interactions). Finally, a cursory examination is undertaken of the fluid velocity and its fluctuations arising from shear-viscosity heterogeneity in media with perfectly uniform permeability $\ell ^2$ .

Correct use of excess configurational entropies to study the ideal glass transition in hard-sphere systems with continuous polydispersity

We systematically apply the resolution to the configurational entropy paradox from our previous paper [V. Baranau and U. Tallarek, J. Chem. Phys. 147, 224503 (2017)] to study configurational entropies and the glass transition in polydisperse hard-sphere systems with log-normal particle radius distributions (r) over a wide range of polydispersities δ=⟨Δr2⟩/⟨r⟩=0.1−0.3. The resolution implies the careful use of excess quantities for vibrational and configurational entropies. We obtain the fluid entropy from the fluid equation of state and the vibrational entropy from the glass equation of state; thereby, the configurational entropy becomes their difference. We discovered that the Adam–Gibbs relation is able to fit the asymptotic alpha-relaxation times τα of the hard-sphere systems under study at high volume fractions φ when our excess configurational entropies are supplied. For polydispersity δ = 0.1, the Adam–Gibbs relation is able to fit the data over the entire range of φ studied. Ideal glass transition densities φg obtained in this way are below predictions from the Vogel–Fulcher–Tammann fits. Our results indicate by extrapolation that the glass close packing limit φGCP for monodisperse systems is ∼0.65, consistent with granular matter studies. Our configurational entropies extrapolated to the monodisperse case are found to match Edwards entropies from granular matter studies very well.

Ethyl cellulose nanoparticles as stabilizers for Pickering emulsions

Pickering emulsions stabilized by ethyl cellulose nanoparticles have recently received –great attention for their remarkable stability and numerous industrial applications. De- spite this, the exact stabilization mechanism of such Pickering emulsions is still not fully understood. Both the stabilization of the emulsion by particle adsorption at the inter- face and through network formation in the continuous phase (leading to a yield stress) have been suggested. In this work we study soybean oil-in-water emulsions stabilized by ethyl cellulose nanoparticles and find, by the use of confocal microscopy and interfa- cial tension measurements, that the main stabilization mechanism of this nanoparticle- stabilized emulsions is the adsorption of the particles at the interface, instead of forming a network in the continuous phase. At the same time, oscillatory rheology measurements reveal that the emulsions exhibit a yield stress well below the random close-packing limit for hard spheres, suggesting short-range interactions between the droplets caused by the presence of the particles at the interface. The presence of the particles at the interface in combination with the observed rheological behavior of an attractive emulsion gives a strong indication for a particle-bridged stabilized emulsions.

Long Wavelength Thermal Density Fluctuations in Molecular and Polymer Glass-Forming Liquids: Experimental and Theoretical Analysis under Isobaric Conditions.

We establish via an in-depth analysis of experimental data that the dimensionless compressibility (proportional to the dimensionless amplitude of long wavelength thermal density fluctuations) of one-component normal and supercooled liquids of chemically complex nonpolar and weakly polar molecules and polymers follows extremely well a surprisingly simple and general temperature dependence over an exceptionally wide range of pressures and temperatures. A theoretical basis for this behavior is shown to exist in the venerable van der Waals model and its more modern interpretations. Although associated hydrogen-bonding (and to a lesser degree strongly polar) liquids display modestly more complex behavior, rather simple temperature and pressure dependences are also discovered. A new approach to collapse the temperature- and pressure-dependent dimensionless compressibility data onto a master curve is formulated that differs from the empirical thermodynamic scaling approach. As a practical matter, we also find that the dimensionless compressibility scales well as an inverse power law with temperature with an exponent that is system dependent and decreases with pressure. At very high pressures and low temperatures, the thermal liquid behavior appears to approach (but not reach) a repulsion-dominated random close packing limit. All these findings are relevant to our recent theoretical work on the problem of activated relaxation and vitrification of supercooled molecular and polymeric liquids.

Recent Innovations in Emulsion Science and Technology for Food Applications.

Emulsion technology has been used for decades in the food industry to create a diverse range of products, including homogenized milk, creams, dips, dressings, sauces, desserts, and toppings. Recently, however, there have been important advances in emulsion science that are leading to new approaches to improving food quality and functionality. This article provides an overview of a number of these advanced emulsion technologies, including Pickering emulsions, high internal phase emulsions (HIPEs), nanoemulsions, and multiple emulsions. Pickering emulsions are stabilized by particle-based emulsifiers, which may be synthetic or natural, rather than conventional molecular emulsifiers. HIPEs are emulsions where the concentration of the disperse phase exceeds the close packing limit (usually >74%), which leads to novel textural properties and high resistance to gravitational separation. Nanoemulsions contain very small droplets (typically d < 200 nm), which leads to useful functional attributes, such as high optical clarity, resistance to gravitational separation and aggregation, rapid digestion, and high bioavailability. Multiple emulsions contain droplets that have smaller immiscible droplets inside them, which can be used for reduced-calorie, encapsulation, and delivery purposes. This new generation of advanced emulsions may lead to food and beverage products with improved quality, health, and sustainability.

Effect of interfacial transport on the diffusivity of highly filled polymers

The diffusivity of substances, such as moisture, through polymer composites is often described by an effective macroscopic quantity, even though microscopically the diffusivity might be far from uniform. In this work, we study the theoretical example of a permeable matrix containing equal-sized impermeable spheres. We assume that, due to interface effects, the diffusivity of the matrix in the vicinity of the spheres is higher than its bulk matrix diffusivity. Using numerical simulations of the composite's diffusivity, we show that upon the formation of large clusters of the highly permeable interfaces, i.e. near percolation of the spheres, the diffusivity of the composite rises sharply. For even higher values of the volume fraction of the spheres, up to the close-packing limit, the diffusivity decreases due to the increased tortuosity. This effect is well described by an analytical solution for the composite's diffusivity.

Modeling meso- and microstructure in materials patterned with acoustic focusing

We conduct numerical simulations of acoustic focusing in dense suspensions to map the design space of acoustically patterned materials and understand the relationships between input parameters, structural features, and functional properties. We develop closed-form expressions for acoustic forces on particles, enabling rapid simulation of thousands of particles, and find excellent agreement with experimentally focused patterns over a range of conditions. We map the geometrical and microstructural features of focused particle patterns and their dependence on processing parameters. We find that mesostructural geometrical features (focused line height, width, and profile shape) can be controlled reliably over a broad range by modulating input parameters, and that while microstructural features are less readily modulated via input parameters, they are well-suited for various transport properties in functional materials. Notably, packing density nears the random close packing limit at 0.64, and particle contact density shows anisotropy favoring particle contacts along the focused lines. These results guide process design for controlling the properties of patterned materials, and outline the property ranges accessible via acoustic focusing. Additionally, we discuss the dependence of material functionalities, particularly electrical, thermal, and ionic transport properties, on the meso- and micro-structural features of patterned composite materials in the context of acoustic focusing.

Influence of frictional packing limit on hydrodynamics and performance of gas-solid fluidized beds

The influence of frictional packing limit (FPL) on prediction of hydrodynamics and performance of fluidized bed reactors was studied. Dense gas-solid flows in non-reactive (under isothermal cold and at elevated temperatures) and reactive atmospheres (fluidized bed gasifier) were simulated using Eulerian-Eulerian methodology considering a range of values for FPL. Simulations under cold flow conditions were conducted to establish a range of FPL values that provides physically realistic predictions. It is noticed that bed pressure drop increases with increasing value of FPL when superficial gas velocity (U) is less than or equal to the minimum fluidization velocity. For larger values of U, predicted pressure drop is unaffected by the choice of value of FPL. However, in these cases, the distribution of particles, their velocities and bubbling behavior are significantly affected by FPL. Effect of FPL at elevated temperatures is similar to the one observed at cold flow conditions. It is further noticed that FPL not only affects the predictions on bed hydrodynamics but also has profound influence on reactive flow characteristics such as bed temperature and product gas composition. Sensitivity analysis under cold flow conditions could reveal better predictions when the ratio of FPL to close packing limit is chosen between 0.9 and 0.97.

Coexistence of liquid-like emulsion and solid-like emulsion glass beyond the close-packing limit

The unjamming-jamming transition of a concentrated oil-in-water emulsion is generally characterized by a critical volume fraction of the dispersed phase. However, in this work, we demonstrate that both liquid-like emulsions and solid-like glasses can coexist in a certain range of volume fraction. At a given volume fraction, jammed glasses can be obtained from their emulsions directly by applying mechanical agitation. This agitation-induced ergodic-nonergodic transition is attributed to the reduction of the droplet size. The decrease of the droplet size leads to a significant growth of lubrication repulsions between neighboring particles. This coexistent behavior indicates that size does matter in concentrated dispersions of soft particles. An application example showed that the elasticity of solid-like glass can give the medium supporting ability while the liquid-like emulsion cannot.

Singular behavior of the stresses in the limit of random close packing in collisional, simple shearing flows of frictionless spheres

As random close packing is approached in a granular medium of frictionless spheres, the pressure, shear stress, and second normal stress become singular with exponents 5/2, 5/2, and 7/4, as predicted by a kinetic theory and confirmed by prior discrete element modeling simulations.

Relaxation times, jamming densities, and ideal glass transition densities for hard spheres in a wide range of polydispersities

We computationally study the behavior of asymptotic alpha-relaxation times τα as well as jamming densities for equilibrated frictionless polydisperse hard spheres in wide ranges of particle volume fractions φ. Log-normal particle radii distribution (r) with polydispersities δ=⟨Δr2⟩/⟨r⟩=0.1−0.3 in steps of 0.05 is used. We discover that τα(φ) can be fitted well with the Vogel–Fulcher–Tammann (VFT) form. Through the VFT fits, we estimate positions of the ideal glass transition densities φg. For each equilibrated configuration, we calculate equilibrium kinetic pressure Z. Equilibrium pressures can be well described by the Boublík–Mansoori–Carnahan–Starling–Leland fluid equation of state. For each equilibrated configuration, a jammed particle configuration, which is the closest one in the configuration space, is determined. We measure jamming densities φEJ of these configurations and present plots φEJ(φ) for all polydispersities. We demonstrate that the lines τα(φ), φEJ(φ), and Z(φ), as well as values φg, depend significantly on δ. These results show that φg is, in general, distinct from the random close packing limit (φEJ at φ = 0). We plan to use these data in the future to estimate glass equations of state and the configurational entropy for these hard-sphere systems.

Looking for a compact semi empirical equation of state for hard spheres and the possibility of a glassy transition

The equation of state of a hard sphere fluid at high density should exhibit a simple pole at the random close packing limit. Here we show that trying to obtain a compact semi-empirical equation of state simultaneously compatible with that asymptotic behaviour and with the known virial coefficients raises an analytical difficulty which can be solved if a glassy transition occurs in the disordered metastable phase at a density intermediate between the freezing point and the random close packing limit. The comparison of the estimated value for the transition point with earlier estimations suggests a more subtle situation.

Swelling of multi-responsive spherical polyelectrolyte brushes across a wide range of grafting densities

Multi-stimulus responsive weak polyelectrolyte brushes were grafted by surface-initiated atom transfer radical polymerization from spherical silica nanoparticles across a wide range of grafting densities approaching the limit of close packing of grafting sites: a regime not previously explored in the brush swelling literature. Brushes consisted of poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) homopolymers or poly(2-(dimethylamino)ethyl methacrylate)-r-poly(4-metharyloyloxyazobenzene) random copolymers P (DMAEMA-r-MOAB) and were subjected to pH-, temperature-, and photoisomerization-induced swelling and deswelling. Homopolymer brushes were prepared at grafting densities ranging from 0.12 to 1.4 nm−2 and number average degrees of polymerization ranging from 300 to 1200. Two copolymer brushes with 8 mol% and 13 mol% of the nonionizable, photoresponsive MOAB units were compared with homopolymer brushes prepared at similar grafting densities and degrees of polymerization. Dynamic light scattering was used to measure the hydrodynamic radius of grafted nanoparticles, and thus brush thickness, in water. Brush swelling in response to increased DMAEMA protonation followed a single scaling behavior across the full range of degrees of polymerization and grafting densities. Incorporating azobenzene-containing MOAB comonomers introduced a modest photoresponsive control of brush swelling and decreased the brush thickness relative to the comparable PDMAEMA homopolymer brush, without affecting the scaling with respect to protonation. PDMAEMA brushes exhibited a pH-dependent lower critical solution temperature behavior, resulting in a thermal collapse of up to 50% upon heating. This work demonstrates that scaling behaviors previously tested at low grafting densities hold for extremely high grafting densities and that they are unaffected by moderate nonionizable comonomer content.

Lubrication model of suspension flow in a hydraulic fracture with frictional rheology for shear-induced migration and jamming.

We developed a model for suspension flow in a hydraulic fracture, taking into account frictional rheology to capture the effects of shear-induced particle migration, jamming and transition to close packing. One of the key issues with the existing slurry rheology models is that each of them diverges near the close packing limit, which is typically resolved in numerical simulations via a pragmatic (and mostly unjustified) regularization. Another drawback of the family of existing models for proppant transport in fractures is the assumption of a uniform cross-flow concentration profile, which neglects the effects of shear-induced migration. We developed a self-consistent model for slurry flow with a constitutive relation for suspension rheology, which is applicable in the entire range of particle volume concentration, from dilute suspension through dense suspension to the close packing limit. In addition, we investigated the influence of various constitutive relations for the suspension rheology on the final model for the slurry flow. The selected model for slurry flow was implemented into a two-dimensional lubrication model of proppant transport in a fracture (based on the two-continua approach), and illustrative simulations were conducted in comparison with the family of existing suspension rheology models (having a singularity). Validation against laboratory experiments is discussed.

Theory of tracer diffusion in concentrated hard-sphere suspensions

A phenomenological theory of diffusion and cross-diffusion of tracer particles in concentrated hard-sphere suspensions is developed. Expressions for the diffusion coefficients as functions of the host particle volume fraction are obtained up to the close-packing limit. In concentrated systems the tracer diffusivity decreases because of the reduced pore space available for diffusion. The tracer diffusivity can be modelled by a Stokes–Einstein equation with an effective viscosity that depends on the pore size. Tracer diffusion and segregation during sedimentation cease at a critical trapping volume fraction corresponding to a tracer glass transition. The tracer cross-diffusion coefficient, however, increases near the glass transition and diverges in the close-packed limit.

Rheological properties of concentrated slurries of harvested, incubated and ruptured Nannochloropsis sp. cells

Biorefining of microalgae biomass requires processing of high-solids (> 10%) slurries. To date there is little knowledge of how processes for weakening and rupturing microalgae cells affect the rheological properties of these materials. To fill this gap in the literature, the rheological properties of concentrated slurries of marine microalgae Nannochloropsis sp. were investigated as a function of processing and solids concentration (12, 20 and 24% w/w). Freshly harvested, incubated (autolysed), and high-pressure homogenised (HPH) slurries were found to be shear thinning up to a shear rate of approximately 200 s− 1. Viscosity increases were far more prominent for partially processed versus unprocessed algal pastes at the higher concentrations. Slurry viscosity as a function of cell volume fraction could not be fitted to the Krieger-Dougherty model due to a network structure resulting from extracellular polymeric substances (EPS) and the intracellular cell components released during incubation and cell rupture. The 24% slurry, which was near the close packing limit, was much more viscous than the less concentrated slurries when comprising whole cells (i.e. harvested and incubated slurries). Cell rupture by HPH completely altered the characteristics of the slurry, increasing the viscosity of even the less concentrated slurries, and producing irreversible shear thinning behaviour. The magnitude of the increases in viscosities and the irreversible shear thinning behaviour observed in this study, have significant implications for processing and optimising the solids concentration of algal slurries.

Density-functional theory study of the body-centered-cubic and cI16 hard-sphere crystals.

The properties of the body-centered-cubic (bcc) solid phase of hard spheres are challenging to compute because of its lack of mechanical and thermodynamic stability, yet this structure remains of interest for theoretical and practical reasons. Density-functional theory (DFT) studies of the bcc hard-sphere solid, using the most accurate functionals from fundamental measure theory, have yielded results with unphysical behaviors in structural and thermodynamic properties. We recently reported [Warshavsky et al., J. Chem. Phys. 148, 024502 (2018)] a Monte Carlo (MC) simulation study of hard spheres initiated in a bcc structure. We observed that such systems, even under constant-volume and single-occupancy-cell constraints, rapidly evolved into either a crystalline state with the cI16 structure or one of a few amorphous states. With these observations in mind, we revisited the DFT calculations of the bcc hard-sphere system by allowing for a bcc-to-cI16 structural transformation. Specifically, the free energy functional was minimized with respect to a density profile having two scalar parameters: the traditional alpha parameter characterizing the width of the Gaussian density distribution around each lattice site and a geometric parameter characterizing the bcc-to-cI16 structural transition. The numerical solutions were physically reasonable across the entire density range. At all densities above ρbσ3 = 1.0, a cI16 structure had lower free energy than the corresponding perfect bcc structure. The degree of lattice distortion from bcc to cI16 increased with density up to the close-packing limit. Finally, the predicted values of the structural and thermodynamic properties were in excellent agreement with those extracted from our previous MC simulations.