Hard-sphere Repulsion Research Articles

Dewatering and consolidation of clay slurries

Dewatering, which is the process of separating (colloidal) suspended particles from a solvent (usually water), is used in many engineering applications (sanitary engineering, dredging engineering…). Key questions associated with dewatering in the context of the reuse of dredged sediment are (1) what is the process kinetics, (2) how can these processes be optimized and (3) can the dewatered sludge be reused and for which application? Dewatering and consolidation are functions of the suspended particles’ size and type, and their solvent-mediated interaction. In this presentation, some examples will be given about the dewatering of suspensions and slurries as found in engineering applications [1]. The presentation will focus on the behaviour of mineral clay suspensions (kaolinite, montmorillonite, illite…) composed of particles of different particle sizes [2-5]. We will show that, depending on the particle size distribution and solvent properties, the system is either undergoing a slow sedimentation dominated by thermodynamic forces or a rapid sedimentation dominated by gravity. The sedimentation is followed in time using NMR and inferential image analysis, and the particles are characterized by size, density and electrokinetic charge. We show that the time evolution of the sedimentation behaviour can be modelled using an advection-diffusion equation. The advective term is a function of gravity, whereas the diffusion term represents either a hard-sphere repulsion or an effective stress, depending on whether thermodynamic forces dominate the system [6-8]. We show that after solving this advection-diffusion equation numerically, the results based on the theory of Gibson does match the experimental data collected from NMR in the phase of slow kinetics. For the early stages of settling and consolidation, where the system kinetics are fast, we show that the data are contaminated by artifacts due to the limited time of signal acquisition imposed by the NMR method. We show that is possible to overcome those limitations by either sacrificing some information related to particle’s sizes using the NMR or by combining the NMR method with excess pore pressure measurements along the height of the settling columns.

When Ions Defy Electrostatics: The Case of Superchaotropic Nanoion Adsorption

Nanometer-sized anions, like polyoxometalates and borate clusters, bind to nonionic hydrated matter driven by the chaotropic effect, which arises from the favorable dehydration of the ions. Herein, we evaluate the adsorption and activity coefficient of the superchaotropic Keggin polyoxometalate SiW12O404- (SiW) on nonionic surfactant (C8E4) micelles by modeling small-angle X-ray and neutron-scattering spectra. Neither hard sphere nor electrostatic repulsion models reproduce the experimental activity coefficient of adsorbed SiW ions on the micelles. However, the activity and binding of SiW on the micelles is well-described by a Langmuir adsorption isotherm. These results imply that adsorbed SiW ions are non-interacting and "create" around themselves adsorption sites on the micelle. The temperature dependence of the adsorption constant showed that the SiW adsorption is enthalpically driven and entropically unfavorable, in line with the typical chaotropic thermochemical signature. The adsorption enthalpy can be split into an electrostatic term and a water-recovery term to evaluate and qualitatively predict the superchaotropicity of a nanoion.

Effective Hard-Sphere Repulsions between Oleate-Capped Colloidal Metal Oxide Nanocrystals.

Nanocrystal interactions in solvent influence colloidal stability and dictate self-assembly outcomes. Small-angle X-ray scattering is used to study how dilute oleate-capped In2O3 nanocrystals with 7-19 nm core diameters interact when dispersed in a series of nonpolar solvents. Osmotic second virial coefficient analysis finds toluene-dispersed nanocrystals in this size range interact like effective hard spheres with diameters comprising the inorganic core and a ligand-solvent corona with a core-size independent thickness. Hard-sphere-like structure factors are similarly observed for nanocrystals with a 9.7 nm core diameter dispersed in all the solvents investigated. Nanocrystal hydrodynamic diameters from dynamic light scattering measurements correlate with thermodynamic diameters obtained from the osmotic second virial coefficient analysis for all samples. The ability to prepare nanoscale building blocks of different sizes, and dispersed in a variety of solvents, with effective hard-sphere repulsions provides a foundation for assembly, where secondary linking or depletant molecules can be deliberately added to customize interactions to form superstructures such as gel networks or superlattices.

Solutions of Poisson-Nernst Planck Equations with Ion Interaction

This paper develops a mixed Finite Element Method (mFEM) based on both classical rectangular elements (with equal nodal points for all degrees of freedom) and Taylor-hood elements to solve Poisson-Nernst-Planck (PNP) equations with steric effects. The Nernst-Planck (NP) equation for ion fluxes is modified to incorporate finite-size effects in terms of hard-sphere repulsion. The resultant modified NP and Poisson equation representing electrostatic potential is then coupled to form a system of the equation that describes a realistic transport phenomenon in an ion channel. Consequently, we apply mFEM based on both Taylor-hood and classical rectangular elements to discretize the 2D steady system of equations with iterative linearization for the nonlinear components. The numerical scheme is first validated using a 2D analytical solution for PNP equations, the steric components varied and the effects on concentration analyzed and compared against PNP and modified PNP for two monovalent ion species. Classical rectangular elements presented a better comparable approximate solution than Taylor-hood.

Surface electrostatic force in presence of dimer counter-ion

Combining classical density functional theory with modified interfacial statistical associating fluid theory, we investigate the surface electrostatic force (SEF) between two negatively and similarly charged planar surfaces immersed in electrolyte solution consisting of monomer anion and dimer cation. The dimer cation is comprised of two hard spheres (HSs) tangentially bonded together and freely rotating on the top of each other; one of the two HS sites is positively charged and the other is neutral. New findings about novel properties of the SEF caused by the dimer counter-ion are summarized below. (i) In the presence of only univalent counter-ion, the SEF always becomes more and more repulsive with increasing of the dimer counter-ion neutral site size, whether the bulk electrolyte solution is dense or not, and the surface charge strength is small, moderate, or large; repulsion strength of the SEF always reduces with the bulk concentration, and at equal surface distance rapidly increases with the surface charge strength. (ii) In the presence of bivalent counter-ion of monomer or dimer, rather complex SEF behavior emerges. Specifically, (a) at low surface charge strength, the dimer counter-ion with small neutral site size induces one obvious like-charge attraction (LCA) at lower bulk concentrations; whereas at higher bulk concentrations, a repulsive SEF emerges instead, whose strength monotonously decreases with the neutral site size. (b) For high surface charge strengths, the SEF curves exhibit oscillatory structure with the LCA feature; moreover, the dimer counter-ion always induces larger LCA position and strength than the monomer counter-ion does; the LCA strength decreases with the counter-ion neutral site size, and the LCA position always reduces with increase of the surface charge strength. (c) Dependence of the LCA strength on the bulk concentration is conditional on the surface charge strengths considered; to speak specifically, for very high surface charge strength, the LCA strength reduces with the bulk concentration, whereas for not so high value, the opposite is true. The above observations can be analyzed from fluctuation and balance of the three factors: osmotic pressure difference between inside and outside the plates; hard sphere repulsion in congested space, and “hydrogen bond” formation.

Creating a regular array of metal-complexing molecules on an insulator surface at room temperature

Controlling self-assembled nanostructures on bulk insulators at room temperature is crucial towards the fabrication of future molecular devices, e.g., in the field of nanoelectronics, catalysis and sensor applications. However, at temperatures realistic for operation anchoring individual molecules on electrically insulating support surfaces remains a big challenge. Here, we present the formation of an ordered array of single anchored molecules, dimolybdenum tetraacetate, on the (10.4) plane of calcite (CaCO3). Based on our combined study of atomic force microscopy measurements and density functional theory calculations, we show that the molecules neither diffuse nor rotate at room temperature. The strong anchoring is explained by electrostatic interaction of an ideally size-matched molecule. Especially at high coverage, a hard-sphere repulsion of the molecules and the confinement at the calcite surface drives the molecules to form locally ordered arrays, which is conceptually different from attractive linkers as used in metal-organic frameworks. Our work demonstrates that tailoring the molecule-surface interaction opens up the possibility for anchoring individual metal-complexing molecules into ordered arrays.

Numerical Approach for Determining Impact of Steric Effects in Biological Ion Channel

Flow through biological ion channel is understandably complex to support numerous and vital processes that promote life. To account for the biological evolution, mathematical modelling that incorporates electrostatic interaction of ions and effects due to size exclusion has been studied, conceivably with element of difficulty and inaccuracy. In this paper the Nernst-Planck(NP) equation for ion fluxes that uses Lennard Jonnes(LJ) potential to incorporate finite size effects in terms of hard sphere repulsion is examined. To minimize emerging numerical intricacy, the LJ potential is modified by a band limit function with a cut-off length to eliminate troublesome high frequencies in the integral function. This process is achieved through Fourier transform to simplify and hence render the mPNP equation solvable with precision. The resultant modified NP and Poisson equation representing electrostatic potential are then coupled to form system of equation which describes a realistic transport phenomena in ion channel. Consequently, to discretize the 2D steady system of equations, mixed finite element approach based on Taylor hood eight node square referenced elements is adopted. In the method, Galerkin weighted residual approach help obtain sparse matrix and finally Picard Method applied to the nonlinear terms in the algebraic equations to linearize them and improve rate of convergence. Iterative solution for the system of equations then obtained and concentration profiles of ion species under varied steric effects for mPNP are computed and analysed

Integral equation theory of thermodynamics, pair structure, and growing static length scale in metastable hard sphere and Weeks-Chandler-Andersen fluids.

We employ the Ornstein-Zernike integral equation theory with the Percus-Yevick (PY) and modified-Verlet (MV) closures to study the equilibrium structural and thermodynamic properties of metastable monodisperse hard sphere and continuous repulsion Weeks-Chandler-Andersen (WCA) fluids under density and temperature conditions where the system is strongly overcompressed or supercooled, respectively. The theoretical results are compared to crystal-avoiding simulations of these dense monodisperse model one-component fluids. The equation of state (EOS) and dimensionless compressibility are computed using both the virial and compressibility routes. For hard spheres, the MV-based virial route EOS and dimensionless compressibility are in very good agreement with simulation for all packing fractions, much better than the PY analogs. The corresponding MV-based predictions for the static structure factor are also very good. The amplitude of density fluctuations on the local cage scale and in the long wavelength limit, and three technically different measures of the density correlation length, are studied with both closures. All five properties grow in a roughly exponential manner with density in the metastable regime up to packing fractions of 58% with no sign of saturation. The MV-based results are in good agreement with our crystal-avoiding simulations. Interestingly, the density dependences of long and short wavelength quantities are closely related. The MV-based theory is also quite accurate for the thermodynamics and structure of supercooled monodisperse WCA fluids. Overall our findings are also relevant as critical input to microscopic theories that relate the equilibrium pair correlation function or static structure factor to dynamical constraints, barriers, and activated relaxation in glass-forming liquids.

Comparative Modeling of Ions and Solvent Properties in Ca-Na Montmorillonite by Atomistic Simulations and Fluid Density Functional Theory

AbstractMolecular dynamics (MD) simulations provide an accurate description of the mineral–fluid interface from the perspective of the atomistic level taking into account all atom interactions. This simulation approach is computationally expensive if applied to large molecular systems. Classical Fluid Density Functional Theory (f-DFT) delivers structural and thermodynamic information at comparatively small computational costs. Numerous applications of f-DFT for electrolytes neglect an explicit consideration of solvent. In this work, an unrestricted three-component model (3CM) of f-DFT was applied, which incorporates Lennard-Jones (LJ) attractions for the description of the short-range interactions of fluid–fluid and fluid–wall rather than the hard sphere repulsions, named DFT/LJ-3CM. The DFT/LJ-3CM model considers ions as charged LJ particles and treats solvent molecules as neutral LJ particles. To validate the performance of the DFT/LJ-3CM, the f-DFT calculations were compared with atomistic simulations for montmorillonite (Mnt) with various hydrated states in electrolyte solutions. This benchmarking was used to assess critically the advantages and limitations of the f-DFT model. The calibrated DFT/LJ-3CM model for Na and Ca Mnt was applied to calculate cation selectivity for the ion exchange equilibrium with effective ion radius and swelling behavior of Mnt. The predictions of the DFT/LJ-3CM model were found to be in good agreement with the atomistic simulations and experimental data under a wide range of conditions. At the same time, the DFT calculations were 3–4 orders of magnitude faster than conventional MD simulations. Thus, the DFT/LJ-3CM model can be a computationally effective alternative to atomistic simulation in providing structural and thermodynamic properties of fluid–clay mineral interfaces. The DFT/LJ-3CM model provides a robust approach, which can be used for upscaling in reactive transport simulators and modeling ion migration taking place under more complex thermo-chemo-hydro-mechanical conditions.

Ion-specific clustering of metal-amphiphile complexes in rare earth separations.

The nanoscale structure of a complex fluid can play a major role in the selective adsorption of ions at the nanometric interfaces, which is crucial in industrial and technological applications. Here we study the effect of anions and lanthanide ions on the nanoscale structure of a complex fluid formed by metal-amphiphile complexes, using small angle X-ray scattering. The nano- and mesoscale structures we observed can be directly connected to the preferential transfer of light (La and Nd) or heavy (Er and Lu) lanthanides into the complex fluid from an aqueous solution. While toluene-based complex fluids containing trioctylmethylammonium-nitrate (TOMA-nitrate) always show the same mesoscale hierarchical structure regardless of lanthanide loading and prefer light lanthanides, those containing TOMA-thiocyanate show an evolution of the mesoscale structure as a function of the lanthanide loading and prefer heavy lanthanides. The hierarchical structure indicates the presence of attractive interactions between ion-amphiphile aggregates, causing them to form clusters. A clustering model that accounts for the hard sphere repulsions and short-range attractions between the aggregates has been adapted to model the X-ray scattering results. The new model successfully describes the nanoscale structure and helps in understanding the mechanisms responsible for amphiphile assisted ion transport between immiscible liquids. Accordingly, our results imply different mechanisms of lanthanide transport depending on the anion present in the complex fluid and correspond with anion-dependent trends in rare earth separations.

Charge effects in water solution of a low generation sodium carboxylate terminated dendrimers

Dendrimers, a class of novel nano-sized materials characterized by a highly ramified structure and a relevant number of functional surface groups, represent a versatile platform for a wide range of nanotechnology applications. The number and characteristic of their modifiable surface groups strongly influence the solution properties as well as some relevant processes involved in molecular recognition, signal processing as well as for binding various targeting or guest molecules. Owing to these properties dendrimers have attracted the interest of the researchers in the development of new prototypes implicated in many technological applications. However, despite of their recent extensive studies, their colloidal stability in solution environment have been relatively less investigated. With the aim to furnish useful insight about the inter-dendrimer interaction in solution, a Small Angle X-ray Scattering (SAXS) structural investigation on the sodium carboxylate terminated (-COO-Na) generations G1.5 of poly(amidoamine) dendrimers in water solution has been performed. The presence of a pronounced interference peak in the SAXS spectra in a wide range of concentrations gives evidence of a long range electrostatic interaction which has been ascribed to the ionisation of the surface carboxylate terminal end-groups. The experimental inter-dendrimer structure factor S(q) has been analysed in the framework of liquid integral equation theory. From that, we derived an effective inter-particle interaction composed of a screened Coulombic plus hard-sphere repulsion potential, which allows the estimation of the dendrimer effective surface charge Z eff . The present analysis strongly supports the finding that colloidal stability and interaction of dendrimers in solution environment is strongly influenced by charge effects.

The SAFT for prediction of hydrate formation conditions of gas mixtures in the presence of methane, glycerol, ethylene glycol, and triethylene glycol

The statistical associating fluid theory (SAFT) equation of state is employed to calculate the fugacities of all components in the vapor and liquid phases in equilibrium with the coexisting hydrate phase. Since the SAFT takes into account the interactions of hard-sphere repulsion, hard chain formation, dispersion and association between molecules, it can successfully predict the inhibiting effect of methanol, glycerol, ethylene glycol, and triethylene glycol on the hydrate formation of gas mixtures containing methane, ethane, propane and carbon dioxide. The predictions were in excellent agreement with experimental data. However, the traditional cubic equations of state fail for systems containing water, alcohols and hydrocarbons in general.

Mobile obstacles accelerate and inhibit the bundle formation in two-patch colloidal particle.

Aggregation of protein into bundles is responsible for many neurodegenerative diseases. In this work, we show how two-patch colloidal particles self-assemble into chains and a sudden transition to bundles takes place by tuning the patch size and solvent condition. We study the kinetics of formation of chains, bundles, and networklike structures using patchy Brownian cluster dynamics. We also analyze the ways to inhibit and accelerate the formation of these bundles. We show that in the presence of inert immobile obstacles, the kinetics of formation of bundles slows down. However, in the presence of mobile aggregating particles, which exhibit interspecies hard sphere repulsion and intraspecies attraction, the kinetics of bundle formation accelerates slightly. We also show that if we introduce mobile obstacles, which exhibit interspecies attraction and intraspecies hard sphere repulsion, the kinetics of formation of bundles is inhibited. This is similar to the inhibitory effect of peptide P4 on the formation of insulin fibers. We are providing a model of mobile obstacles undergoing directional interactions to inhibit the formation of bundles.

Molecular theory of a ferromagnetic nematic liquid crystal.

We employ a version of classical density functional theory to study the phase behavior of a simple model liquid crystal in an external field. The uniaxially symmetric molecules have a spherically symmetric core with superimposed orientation-dependent attractions. The interaction between the cores consists of a hard-sphere repulsion plus an isotropic square-well attraction. The anisotropic part of the interaction potential allows for the formation of a uniaxially symmetric nematic phase. The orientation of the molecules couples to an external polar field. The external field is capable of rotating the nematic director n[over ̂] in the x-z plane. The field is also capable of changing the topology of the phase diagram in that it suppresses the phase coexistence between an isotropic liquid and a nematic phase observed in the absence of the field. We study the transition from an unpolar to a polar nematic phase in terms of the orientation-distribution function (odf), nematic and polar order parameters, and components of n[over ̂]. If represented suitably the odf allows us to study orientational changes during the switching process between nonpolar and polar nematic phases. We also give a simple argument that explains why nematic order is lost whereas polar order persists up to the gas-liquid critical point along the coexistence curve. We also discuss the relevance of our theory for future experimental studies.

Sticky, active microrheology: Part 1. Linear-response

Attractive colloidal-scale forces between macromolecules in biological fluids are suspected to play a role in important system dynamics, including association times, spatially heterogeneous viscosity, and anomalous diffusion. Passive and active microrheology provide a natural connection between observable particle motion and viscosity in such systems via generalized Stokes-Einstein and Stokes’ drag law relations. While such models are robust for purely repulsive colloidal-scale interactions, no such theory exists to model the effects of attractive forces. Here we present such a model for the linear-response regime, where a Brownian probe particle is driven gently through a complex fluid by an external force that weakly augments thermal fluctuations. As the probe moves through the bath, hard-sphere repulsion results in an accumulation of particles on its upstream face and a trailing depletion zone, producing particle drag that slows the probe. Linear-response viscosity can be inferred constitutively from this speed reduction. One expects attractive forces to make the suspension more viscous, but surprisingly, weak attractions exerted by upstream particles actively pull the probe forward, giving it a “hypoviscous” environment through which it slides more easily. As attractions grow stronger, particles join to the probe in a long-lasting doublet, extracting particles from the upstream region and depositing them behind the probe. At a critical value of the second virial coefficient common to all potentials we studied, the distorted structure reverses direction, and continued growth of attraction strength causes the probe to drag a cluster of density along, dramatically increasing viscosity. But at this transition, the structure is neutral under the balance of attraction and repulsion, allowing the probe to “cloak” itself and move through the bath undetected and unhindered relative to hard-sphere dispersions. This poses an intriguing mechanism by which proteins or other macromolecules may change their surface chemistry in order to alter the viscosity of the surrounding medium to speed their own motion, or simply to pass undetected through a cell.

Ramachandran maps for side chains in globular proteins.

The Ramachandran plot for backbone ϕ,ψ-angles in a blocked monopeptide has played a central role in understanding protein structure. Curiously, a similar analysis for side chain χ-angles has been comparatively neglected. Instead, efforts have focused on compiling various types of side chain libraries extracted from proteins of known structure. Departing from this trend, the following analysis presents backbone-based maps of side chains in blocked monopeptides. As in the original ϕ,ψ-plot, these maps are derived solely from hard-sphere steric repulsion. Remarkably, the side chain biases exhibit marked similarities to corresponding biases seen in high-resolution protein structures. Consequently, some of the entropic cost for side chain localization in proteins is prepaid prior to the onset of folding events because conformational bias is built into the chain at the covalent level. Furthermore, side chain conformations are seen to experience fewer steric restrictions for backbone conformations in either the α or β basins, those map regions where repetitive ϕ,ψ-angles result in α-helices or strands of β-sheet, respectively. Here, these α and β basins are entropically favored for steric reasons alone; a blocked monopeptide is too short to accommodate the peptide hydrogen bonds that stabilize repetitive secondary structure. Thus, despite differing energetics, α/β-basins are favored for both monopeptides and repetitive secondary structure, underpinning an energetically unfrustrated compatibility between these two levels of protein structure.

Extension of a van der Waals Equation of State to the Solid-liquid-vapor Phases of a Lennard-Jones System

An existing van der Waals (vdW) equation of state (EOS) based on hard-sphere repulsions during intermolecular interactions was extended to a Lennard-Jones (LJ) system. Specifically, an EOS for a perfect solid and liquid v5 (Y. Kataoka and Y. Yamada, J. Comput. Chem. Jpn.,12, 181–189 (2013)) was simplified to satisfy the Rigby conditions for a generalized vdW EOS. The phase diagram of argon was reproduced in pressure-temperature space using this extended EOS, and a phase diagram in temperature-density space was also generated and found to be qualitatively consistent with the LJ system.

A cluster size distribution theory to study the thermodynamics and phase behavior of multi-bonding single site solutes in patchy colloidal mixtures.

We study binary mixtures of multi-bonding single site solute particles in a solvent comprising patchy colloid particles. The particles in the mixture interact by very short-ranged attraction and hard-sphere repulsion. The attractive patch on the solute can bond with multiple solvent particles, whereas the patch on the solvent is restricted to bond only once. From a quasi-chemical analysis of association, in the hard-sphere reference we develop an accurate multi-body correlation information for the distribution of solvent particles over the patch region of the solute. We use this information within Wertheim's multi-density formalism to develop a cluster size distribution theory that is capable of capturing the physics of multi-body association for any geometry of association sites on the solute. We use this general framework to study a mixture containing Janus solutes and one- or two-patch solvent particles over a range of concentration of the solute and association strengths. We find that a mixture of two-patch solvent (with both patches of the same kind) and multi-bonding solutes with different patch geometries can have a vapor-liquid equilibrium, although the pure components themselves cannot phase separate. The liquid state occurs at very low densities, forming a so-called empty liquid. For the relative association strengths studied in this work, we observe that the vapor-liquid coexistence curve broadens as the concentration of the patchy solvent particles in the liquid phase is increased. The pressure-composition phase equilibrium curves show negative azeotropes for these mixtures. We also observe that, for these mixtures, as the size of the patch on the solute particles is decreased, the critical temperature and the critical packing fraction decreases.

Numerical aspects of classical density functional theory for one-dimensional vapor-liquid interfaces

This study compares different algorithms for solving the equations of classical density functional theory for the one-dimensional vapor-liquid interface of real fluids: Picard iteration, Anderson mixing, a restarted quasi Newton method and two versions of the matrix-free inexact Newton method, one using analytic the other using numerically approximated derivatives. All algorithms converge to the same density profiles in all studied test cases. The calculation time of the slowest algorithm (Picard iteration), however, is up to twenty times longer than for the fastest algorithms (Anderson mixing and inexact Newton method with numerically approximated derivatives). Further, the computational speed-up of parallelization and of a convergence criterion based on the value of surface tension instead of the decrease of the norm of the residual are discussed. A Helmholtz energy functional consistent with the perturbed-chain statistical associating fluid theory (PC-SAFT) is applied which includes contributions due to hard-sphere repulsion, chain formation, dispersion as well as association. For the association term, we compare two widely used functionals.

Protein-protein interactions at high concentrations. Isothermal titration calorimetry determination of human serum albumin-lysozyme interaction enthalpy at several pH values

Macromolecular crowding affects many processes in the cell, where solvent mediated short and long range protein-protein electrostatic interactions, as well as protein-solvent interactions, occur at high protein concentrations in a complex mix of ionic species. The predominant view of macromolecular crowding considers only hard-sphere repulsions (solely entropic). However, as shown recently in the literature, the longer-range interactions can be stronger than the excluded volume effects. In this work, we quantify these interactions using isothermal titration calorimetry. We determined the interaction enthalpy between two proteins, lysozyme and human serum albumin, at high concentrations, at several pH values and, for one case, at three temperatures. We demonstrate that the long-range electrostatic interactions, attractive or repulsive, due to the charges on the protein surfaces dominate the interaction between them, and that the strength of these solvent-mediated forces can be tuned by changing the pH.