Interparticle Interaction Energy Research Articles

Approximations for the interparticle interaction energy in an exactly solvable two-electron model atom

The capability of different ansatz kernels, denoted as K(r,r{sup '}), in the calculation of the electron-electron interaction energy is investigated here for an exactly solvable two-electron model atom proposed by Moshinsky. The model atom is in the spin-compensated, paramagnetic ground state. The exact expression for the interaction energy in this state, derived by the diagonal of the second-order density matrix, is used as a rigorous background for comparison. It is found that the form of K{sub M}(r,r{sup '})=2rho(r)rho(r{sup '})-gamma{sup p}(r,r{sup '})gamma{sup q}(r{sup '},r), expressed via the rho(r) density distributions and operator powers of the one-body density matrix gamma(r,r{sup '}), results in the exact value for the interparticle interaction energy of the two-electron model atom if and only if p=q=1/2. Approximate forms with p=qnot =1/2 and with pnot =q at (p+q)=1 give deviations from the exact expression.

The dispersion of single-domain BaFe12O19 particles in water

We have studied the dispersion of single-domain BaFe12O19 particles in water and calculated the interparticle interaction energies due to the van der Waals and magnetic dipole-dipole forces. The magnitudes of the attraction energies varied significantly with the particles’ sizes and the separation distances between the particles. The contribution of the van der Waals potential was important only for the closest separation distances (up to 3 nm). The thermal energy exceeded the magnetic dipole-dipole attraction between the particles with radii larger than 20 nm at separation distances of several 100 nm. The electrosteric stabilization of the BaFe12O19 particles was possible only for particles with radii up to 15 nm and thicknesses up to 6 nm. The studied BaFe12O19 powder was a combination of several fractions, with respect to the particles’ sizes. The particles were treated with a surfactant, dodecylbenzylsulfonic acid, using a combination of milling and high-power ultrasound. The zeta-potential of the surfaces of these particles increased in comparison to the raw particles. We analyzed the dispersed particles using thermogravimetry, mass spectrometry, and transmission electron microscopy, and we determined the parameters for the stabilization of the BaFe12O19 powder.

Colloidal dispersions of monodisperse magnetite nanoparticles modified with poly(ethylene glycol)

Monodisperse magnetite nanoparticles modified with poly(ethylene glycol) (PEG) were synthesized using a silane functionalized PEG obtained by reacting 3-aminopropyl triethoxysilane with carboxylic acid-methoxy PEG (mPEG–COOH) using amide reactions. Transmission electron microscopy (TEM), dynamic light scattering (DLS), and zeta potential measurements show the particles are monodisperse ( σ gv ∼ 0.2 ) and stable in water for pH of 3–9 and ionic strengths, up to 0.3 M NaCl. Thermogravimetric analysis coupled with TEM and DLS indicates formation of a dense graft layer on the particle surface. An analysis of the interparticle interaction energy indicates that the particles are stabilized by strong steric repulsions between PEG chains on their surface.

An efficient method for the creation of tunable optical line traps via control of gradient and scattering forces

Interparticle interaction energies and other useful physical characteristics can be extracted from the statistical properties of the motion of particles confined by an optical line trap. In practice, however, the potential energy landscape, U(x), imposed by the line provides an extra, and in general unknown, influence on particle dynamics. We describe a new class of line traps in which both the optical gradient and scattering forces acting on a trapped particle are designed to be linear functions of the line coordinate and in which their magnitude can be counterbalanced to yield a flat U(x). These traps are formed using approximate solutions to general relations concerning non-conservative optical forces that have been the subject of recent investigations [Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, Phys. Rev. Lett. 100, 013602-4 (2008).]. We implement the lines using holographic optical trapping and measure the forces acting on silica microspheres, demonstrating the tunability of the confining potential energy landscape. Furthermore, we show that our approach efficiently directs available laser power to the trap, in contrast to other methods.

Density Dependent Interactions and Structure of Charged Colloidal Dispersions in the Weak Screening Regime

We determine the structure of charge-stabilized colloidal suspensions at low ionic strength over an extended range of particle volume fractions using a combination of light and small angle neutron scattering experiments. The variation of the structure factor with concentration is analyzed within a one-component model of a colloidal suspension. We show that the observed structural behavior corresponds to a nonmonotonic density dependence of the colloid effective charge and the mean interparticle interaction energy. Our findings are corroborated by similar observations from primitive model computer simulations of salt-free colloidal suspensions.

The kinetics of synthesis and mechanism of coagulation of gold nanoparticles in Triton N-42 reverse micelles

The reduction of Au(III) with hydrazine monohydrate in micellar Triton N-42 solutions was shown to be an autocatalytic reaction. Its rate constants were calculated. The growth of a gold nucleus proceeded as a result of surface reduction until the polar micelle nanocavity was completely filled. Calculations according to the Derjaguin-Landau-Verway-Overbeck theory showed that the fate of nanoparticles formed depended on interparticle interaction energy. At a small radius of particles, high surface potential, and fairly thick surfactant surface layer, stable systems were formed. The coagulation zones were calculated depending on the structural parameters of nanoparticles and micelles. If a nanoparticle grew larger than 6.1 nm at a surface potential lower than 10 mV and surface layer thickness ∼1.6 nm, the potential well depth exceeded 3/2 kT in magnitude, and coagulation occurred in the system.

Thermodynamic Analysis of Nanoparticle Size Selective Fractionation Using Gas-Expanded Liquids

A thermodynamic model was developed for the size-selective fractionation of ligand-stabilized nanoparticles by a CO2 gas-expanded liquid precipitation process. The tunable solvent strength of gas-expanded liquids, via CO2 pressurization, results in an effective method to fractionate nanoparticles, based on the size-dependent dispersibility of the particles. Specifically, the thermodynamic model is used to estimate the size of dodecanethiol-capped Ag nanoparticles that can be dispersed at a given CO2 pressure by equating the total interparticle interaction energy to the Boltzmann threshold stabilization energy (−3/2kBT). The ligand−solvent interaction is found to have the greatest impact on the total interaction energy. This model illustrates that the entire length of the ligand is not accessible to the solvent, and three phenomenological model variations were developed to vary the ligand−solvent interaction.

Fe57Mössbauer spectral and muon spin relaxation study of the magnetodynamics of monodispersedγ−Fe2O3nanoparticles

The M\"ossbauer spectra of monodispersed iron oxide nanoparticles with diameters of 4, 7, 9, and $11\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ have been measured between 4.2 and $315\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and fitted within the formalism for stochastic fluctuations of the hyperfine Hamiltonian. In this model, the hyperfine field is assumed to relax between the six $\ifmmode\pm\else\textpm\fi{}x$, $\ifmmode\pm\else\textpm\fi{}y$, and $\ifmmode\pm\else\textpm\fi{}z$ directions in space with a distribution of relaxation rates that is temperature dependent. Muon spin relaxation measurements have been carried out on the $9\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ particles between 4.2 and $295\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. Both techniques reveal three regimes in the magnetic dynamics of these nanoparticles. In the low-temperature regime, between 4.2 and $\ensuremath{\sim}30\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the nanoparticle magnetic moments are blocked and a spin-glass-like state is observed with nearly static hyperfine fields, as is indicated by the well resolved magnetic M\"ossbauer spectra and the slow exponential decay of the muon asymmetry functions. In the high-temperature regime, above $\ensuremath{\sim}125\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the nanoparticle magnetic moments and, hence, the hyperfine fields, relax rapidly and a typical thermally activated superparamagnetic behavior is observed, as is indicated by the M\"ossbauer doublet line shape and the muon asymmetry functions that are unquestionably characteristic of monodispersed nanoparticles. In the intermediate regime between $\ensuremath{\sim}30$ and $125\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the M\"ossbauer spectra are the superposition of broad sextets and doublets and the muon asymmetry functions have been fitted with a sum of two terms, one relaxing term similar to that observed at and above $125\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and one term characteristic of static local fields. Hence, in this intermediate regime, the sample is magnetically inhomogeneous and composed of nanoparticles rapidly and slowly relaxing as a result of interparticle interactions. The magnetic anisotropy constants determined from both the M\"ossbauer spectral and magnetic susceptibility results decrease by a factor $\ensuremath{\sim}4$ with increasing diameter from $4\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}22\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and increase linearly with the percentage of iron(III) ions present at the surface of the nanoparticles. The interparticle interaction energy is estimated to be between 89 and $212\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ from the temperature dependence of the magnetic hyperfine field measured on the $9\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ nanoparticles.

On the fundamentals of kaolin rheology applied to the paper industry

Laminar movement of concentrated kaolin pulps is difficult without the use of dispersing agents. The aim of this work was to contribute to a better understanding of the fundamental aspects of kaolin pulp rheology using colloidal chemistry theory. Zeta potential and rheology were evaluated with and without a dispersant as a function of pH and dispersant concentration. Total interparticle interaction energy was calculated by means of DLVO theory. Finally, rheological data was coordinated with the calculated data from DLVO theory. It was concluded that DLVO theory was qualitatively related to the rheological behaviour of kaolin pulps.

Estimation of Structural Characteristics during Drying of Molded Organic and Organomineral Biogenic Materials

Structuring processes during the drying of molded materials based on peat and sapropel are studied in terms of interparticle interaction energy. The structuring during drying occurs in two steps, which are governed by the ratio between the liquid and solid phases of the system. The change in structural characteristics is caused by a certain type of molecular interaction. The final state of the system is directly affected by the shrinkage rate and anisotropy emerging during molding and drying. It is shown that the fracture activation energy abruptly changes in the transition from one structuring step to the other. The mechanism of this process is considered at various ratios between the concentrations of the organic and mineral matters of disperse materials of biogenic origin.

An approximate method for calculating depletion and structural interactions between colloidal particles

An approximate method is evaluated for calculating the second virial coefficient in a dilute macromolecular solution bounded by two interfaces. The approximation is essentially the superposition of the coefficients calculated independently for each surface. To test this approach, the depletion interaction between two particles in a solution of nonadsorbing, spherical macromolecules is calculated in systems with either hard-wall or long-range electrostatic interactions. In all systems tested, the interparticle interaction energy calculated with the approximation is found to be in good agreement with that calculated using the exact approach (e.g., error less than 2% at the smallest separations). The primary advantage of this approximation is a significant reduction in the computation time required for calculating the depletion interaction, especially in charged systems. The paper also shows that the expressions for predicting the depletion interaction in purely hard-sphere systems can be used in dilute ionic systems, provided the appropriate effective macromolecule size is used. For the attractive depletion interaction, this effective size is determined by the range of the particle–macromolecule interaction (as opposed to the macromolecule–macromolecule interaction).

Aggregation Stability of Aqueous Dispersions of Microcrystalline Cellulose: pH Dependence

The aggregation stability of aqueous dispersions of microcrystalline cellulose (MCC) was studied by the flow ultramicroscopy in a wide range of pH (1–11). The calculations of the molecular and ion-electrostatic components of the interparticle interaction energy, which were performed according to the DLVO theory with and without allowance for the particle conductivity, demonstrated that, in most cases, the loss of the aggregation stability can not be explained without taking into account the concept of additional attraction forces between the MCC particles. It was assumed that such forces could be attributed to the dipole–dipole interactions or hydrogen bonding between hydrated particles.

Viscosity Effect on the Structural Compactness of Latex Flocs Formed under Weak Depletion Attractions

Dilute aqueous dispersions of colloidal polystyrene latex spheres were flocculated by adding a nonadsorbing polymer sample, poly(acrylic acid). The structural compactness of the flocs thus formed was characterized in terms of their mass fractal dimension using the small-angle static light scattering technique. It was found that with low poly(acrylic acid) concentrations and thus weak depletion attraction forces, the dispersion medium viscosity had a marked effect on the floc structure. An increase in the viscosity led to formation of denser flocs. This was revealed in three sets of depletion flocculation experiments: (a) adjusting the background electrolyte concentration at a fixed level of poly(acrylic acid), (b) using water and 30% (w/w) glycerol as the respective solvents, and (c) inducing latex flocculation with two poly(acrylic acids) of different molecular weights at the respective critical polyacid concentrations. Direct force measurements were made with atomic force microscopy to isolate the influence of viscosity on floc structure from that of interparticle interaction energies. We conclude that the formation of denser flocs with increasing medium viscosity can be attributed to the reduced diffusivity of particles in the solution. The latter resulted in an enhanced rate of floc restructuring (through relaxation of attached particles) relative to floc growth.

The structure and strength of depletion force induced particle aggregates

Aggregating fine particulate matter is common practice in many industrial solid-liquid separation processes. Data obtained in this work on dilute aqueous dispersions of model colloidal polystyrene latex spheres indicate that depletion flocculation, which uses non-adsorbing polymer, can yield very compact aggregates. Flocculation of the negatively charged latex particles was induced by the addition of a poly(acrylic acid) at pH 10. The structural compactness of the latex flocs formed in the dilute dispersions was characterised using small-angle static light scattering in terms of mass fractal dimensions. Rheological measurements on the concentrated latex dispersions in the presence of the non-adsorbing polyacid showed Bingham yield stress behaviour. Both the compactness and strength of the latex flocs were found to be significantly dependent upon the level of the polyacid, as well as the concentration of the initial particles. In particular, as the level of the polyacid was raised the floc compactness decreased, whereas its strength increased. They were both seen to level off at high polymer concentrations. Atomic force microscopy measurements were made at varying concentrations of the polyacid to provide a qualitative explanation of the observed floc structural behaviour of the dilute dispersions. By combining the fractal dimension and the Bingham yield stress we were also able to estimate the energy required to separate the flocs into single units in the concentrated dispersions. It was concluded that the interparticle interaction energy is the key to understanding the dependence of both the floc structure and strength on the polymer concentration.

DC and AC magnetization and small-angle neutron scattering studies of strongly interacting particle systems

MnZn-ferrite particles were randomly dispersed in polymer matrix in which the magnetic anisotropy axes in the particles are randomly oriented. DC and AC magnetization and small-angle neutron scattering measurements have been made on the samples. The DC and AC susceptibilities showed spin-glass-like maxima at temperatures that depend on the interparticle interaction. The thermal evolution of remanence I r and coercivity H c was also considerably dependent on the interparticle interaction, i.e., the temperature at which I r and H c tend to zero increased with increasing the interparticle interaction energy. This suggests that a magnetic correlation occurs through the interparticle interaction in the present sample. The temperature dependence of imaginary part of the AC susceptibility χ″ showed a singular cusp at a low temperature which is insensitive to the interparticle interaction. This singularity in χ″ could reflect a blocking phenomena which is attributed to the intraparticle anisotropy. The small-angle magnetic neutron scattering line shape over a temperature range of 6⩽ T⩽260 K could be represented by a function of Lorentzian plus squared Lorentzian which is appropriate for a random anisotropy field system. The data did not show a divergence in the magnetic correlation length, but suggested the formation of a local ferromagnetic cluster composed of the magnetic moments of the adjacent particles below 260 K. The correlation length increases with decreasing temperature. Around 40 K, it shows a maximum and starts to decrease with decreasing temperature. This decrease in the correlation length with decreasing temperature could arise from the random field effect which could originate from the intraparticle anisotropy.

Contribution of Electrostatic Interaction to the Dynamic Stability Coefficient for Coagulation-Flocculation Kinetics of Beta-Iron Oxyhydroxides in Polyelectrolyte Solutions.

This paper describes the colloidal interactions of preformed positively charged iron oxyhydroxide particles: zeta potential, coagulation, and flocculation in the presence of nonionic, cationic and anionic polyelectrolytes. In salt-coagulated systems, the zeta potential shown as a function of the pH and of the KCl salt concentration is the controlling factor in the formation of particle aggregates. In polymer-flocculated systems, a moderately charged anionic polyelectrolyte was used to adsorb onto particle surfaces and then induced the bridging flocculation of oppositely charged iron oxyhydroxide particles, in which maximum flocculation was observed near zero zeta potential. It is pointed out that the relative importance between polymer bridging and charge neutralization mechanism in the coagulation-flocculation processes of colloidal particles is pH dependent. Correlation of the magnitude of zeta potential with the interparticle interaction energy is simulated by using DLVO theory. Moreover, the present study extends the conventional analysis through considering the temporal variation of the fractional surface coverage of colloidal particles by polymers. The theoretical predictions on the stability coefficients are compared with the experimental results.

Increase of van der Waals Attractive Force under Electromagnetic Field

Abstract A formula for the photoincreased van der Waals attractive force is derived, which includes the effects of the frequency of illuminated electromagnetic waves and of the size of particles. The interparticle interaction energy as a function of the frequency of the electromagnetic wave shows a sharp maximum at the resonance frequency. In the small limit of the size, the van der Waals force under on-resonance irradiation conditions approaches that of molecules and in the large limit the force is drastically reduced. The largest interparticle interaction is found at the size of a few tens of nm. This novel effect is found to be typical of nano-systems.

Opacity of Dense, Cold, and Strongly Coupled Plasmas.

The physics of dense $[(1\ensuremath{-}5)\ifmmode\times\else\texttimes\fi{}{10}^{19}{\mathrm{cm}}^{\ensuremath{-}3}]$, cold (1--15 eV), strongly coupled $(\ensuremath{\gamma}\ensuremath{\sim}0.7)$ plasmas is probed with 0.351, 0.527, and 1.054 $\ensuremath{\mu}\mathrm{m}$ opacity measurements in well characterized, laser-heated, aluminum plasmas. Current opacity models are tested, for the first time, in the regime where the probing photon energies are of the same order as the average interparticle interaction energies in the plasma. Predicted enhancements of the opacity at low temperatures are not observed, but overall agreement between experiment and theory is within a factor of 2.

The rheological and colloidal properties of bentonite dispersions in the presence of organic compounds; I, Flow behaviour of sodium-bentonite in water-alcohol

Addition of methanol, ethanol, and n-propanol to aqueous dispersions of Namontmorillonite (2% w/w) reduces the apparent viscosity and the yield value. In the presence of NaCl the flow parameters increase to a maximum at medium alcohol concentration. Thus, the gel strength of Nabentonite dispersions can greatly change with the alcohol content of the dispersion. This is probably caused by the transition of dispersed clay mineral particles (individual silicate layers or lamellae of a few of these layers) to band-type networks with optical homogeneity and weakly attractive particle-particle interaction. When attraction between the particles becomes too strong at high alcohol concentrations, the bands are condensed to short fragments and, eventually, freely moving particles, reducing the viscosity and the yield value. The interparticle interaction energy is calculated for the constant charge model by van Olphen's (1977) method. Agreement with the experimental results is only obtained when the Stern layer adsorption is governed by a specific adsorption energy μ, which increases with decreasing dielectric constant.

Studies of aggregation kinetics of polystyrene latices sterically stabilized by poly(N-isopropylacrylamide).

The aggregation of polystyrene latices sterically stabilized by poly(N-isopropylacrylamide) has been studied using dynamic light scattering. The time dependence of the aggregate growth could be monitored under appropriate experimental conditions. The measured fractal dimension was found to increase with an increase in the aggregation rate and in the salt concentration, as well as the temperature. These unexpected observations do not conform to the predictions of the current models for aggregation of electrostatically stabilized particles. It was also inferred that the restructuring process was different from that reported in the literature. The hydrophobic interaction (or hydration), which is controlled by the salt concentration and temperature, is believed to play a crucial role in the present system in the aggregation process and in the nature of the aggregate structures formed. The results indicate that the fractal dimension increases with an increase in the hydrophobic interaction, i.e., with a decrease in the interparticle interaction energy. This is in good agreement with the predictions of the computer simulations of Shih, Aksay, and Kikuchi [Phys. Rev. A 36, 5015 (1987)].