Spherical Macroparticles Research Articles

Interaction potential of two spherical macroparticles at constant surface potentials

Electrostatic interaction between two spherical macroparticles at constant charges of the particles and constant surface potentials is studied. The interaction potential of two spherical macroparticles as a function of surface separation distance L is calculated using capacitance and potential coefficients. At small separation distances, the capacitance coefficients are defined by Russell’s asymptotical formulae; at large distances, the potential coefficients are calculated by approximate formulae obtained in this paper and, at intermediate distances, the capacitance and potential coefficients are found by smoothly joining the asymptotical and approximate formulae. A comparison of the data obtained in this paper with those of the previous papers is performed. It has been established that this technique provides the rather high precision in the interaction potential calculation for all interparticle distances.

Effect of the size of charged spherical macroparticles on their electrostatic interaction in an equilibrium plasma

The effect of the size of two charged spherical macroparticles on their electrostatic interaction in an equilibrium plasma is analyzed within the linearized Poisson–Botzmann model. It is established that, under the interaction of two charged dielectric macroparticles in an equilibrium plasma, the forces acting on each particle turn out to be generally unequal. The forces become equal only in the case of conducting macroparticles or in the case of dielectric macroparticles of the same size and charge. They also turn out to be equal when the surface potentials of the macroparticles remain constant under the variation of interparticle distances. Formulas are proposed that allow one to calculate the interaction force with a high degree of accuracy under the condition that the radii of macroparticles are much less than the screening length, which is usually satisfied in experiments with dusty plasmas.

Screening and interaction of conductive grains charged by drift-diffusion currents in plasma

The problem of interaction between two ideally conductive spherical charged macroparticles is solved numerically for strongly collisional weakly ionized plasma. The potential and charge density spatial distributions as well as the interaction force are studied and compared with the analytical solution. The electron and ion fluxes that arise due to the absorption of plasma particles by the grain are described using the drift-diffusion approach. The results are obtained for the stationary state, in which the net current through the grain surface is zero.

Ion Drag Force on a Charged Macroparticle in Collisionless Plasma

The problem of calculating the ion drag force acting on a charged macroparticle in collisionless flowing plasma is studied by using an approach based on the direct numerical solution of the Vlasov kinetic equations for plasma components. A uniform plasma flow past a spherical macroparticle is considered. The computations are carried out for different particle sizes and different flow velocities. On the basis of the obtained results the effect of particle size on the ion drag force is analyzed. It is shown that when the particle size is much less than the Debye length in plasma, the ion drag force can be calculated with good accuracy by means of the conventional binary collision approach. A modified version of the binary collision approach is proposed to calculate the ion drag force in the case where the particle size becomes comparable to the Debye length in plasma. It is shown that there is a reasonable agreement between the results obtained using the numerical solution of the kinetic equations and that obtained by the modified binary collision approach.

Interaction of a dielectric macroparticle with a point charge in plasma

The electrostatic interaction of a charged spherical dielectric macroparticle with a point charge in a plasma in the presence of an external uniform electric field is considered. The electrostatic force and the torque acting on the macroparticle have been determined, and the form of the interaction potential has been established for a nonuniform distribution of free charge on the macroparticle surface. A simple (for calculations) expression for the interaction potential that describes well the exact potential at all interparticle distances is proposed. The angular velocity of the spinning of dust particles caused by a nonuniform distribution of free charge over their surface has been estimated.

Entropy effects in self-assembling mechanisms: Also a view from the information theory

At first sight, formation of self-assembled structures such as micelles, bilayers, DNA–cationic bilayers complexes, and the enzyme–substrate complexation in biological catalytic reactions look as totally different phenomena. However, common in all of them is the effective attractive interactions among the self-assembled molecular components that are required to occur. In general, the direct interactions among the components of an assembled structure are not sufficiently attractive to justify self-assembling by itself. Furthermore, in some cases these direct interactions are repulsive as in the case of like charged particles. In this paper, by means of theory and simulation, we consider the effective interactions of the following simple systems: 1) two like-charged parallel rods immersed in an electrolyte solution, 2) two like-charged parallel rods confined between two parallel plates, and 3) a sphere with an open cavity and a spherical macroparticle, both immersed in a hard sphere fluid. In the first two systems we show that effective attractive interaction among the components of a self-assembled structure may only occur when they are in high electrolyte volume fractions. In the lock–key system (3rd case), it is demonstrated that the perfect match between the spherical cavity and macroparticle is the most favorable condition for their self-assembly to occur. All these are examples of attraction driven by the need of the systems to optimize the available volume. That is, self-assembling is driven by the maximum entropy principle, or maximum missing information, as in information theory.

The Numerical Solution of the Nonlinear Poisson-Boltzmann Equation Under the Anisotropic Boundary Condition for Colloidal Plasmas

Based on the model of the Wigner–Seitz cell, the surface potential of the spherical macroparticle (radius a) expands in terms of the monopole (q). A dipole (p) model is assumed for an anisotropic boundary condition of the nonlinear Poisson–Boltzmann equation. Using the finite element method implemented by the FlexPDE software, the potential distribution around the macroparticle is obtained for different ratios p/qa. The calculated results for the potential show that there is an attractive region in the vicinity of the macroparticle when |p/qa|>1.1, and noticeably there is a potential well behind the macroparticle when |p/qa| = 1.1, i.e., there exists both an attractive region and a repulsive region simultaneously. This means that the attractive interaction between macroparticles may arise from the anisotropic distribution of the surrounding plasmas, which well explains some experimental observations.

Effect of the size of macroparticles on their electrostatic interaction in a plasma

The electrostatic interaction of two spherical macroparticles in a plasma has been considered. Primary attention has been focused on investigating the electrostatic interaction at short distances where polarization effects of the surface charge of finite-size macroparticles begin to play a dominant role. The first part of this study is devoted to the interaction of a point charge with a charged conducting sphere in an equilibrium plasma. It has been shown that the presence of a plasma in the system leads to a decrease in the potential barrier when two like-charged macroparticles approach each other and that this decrease proves to be the most significant in the case where the macroparticle radius is comparable to the Debye screening length. The second part of this study is concerned with the interaction of two conducting spheres in the bispherical system of the coordinates under the assumption that the charges of the conducting spheres are constant and under the assumption that the surface potentials of the spheres are constant. The latter case is closer to the physics of electrostatic interaction of two macroparticles in a plasma medium where the electrostatic potential of their surface is determined by the floating potential of the plasma. It has been demonstrated that the interaction potentials in these two cases are substantially different from each other and that, at constant macroparticle charges, the energy of the electrostatic field is an interaction potential, but, in the case of macroparticles with constant surface potentials, which are independent of the interparticle distance, the energy of the electrostatic field is not an interaction potential. In the latter case, account must be taken of the work done by external sources on the macroparticle potentials to maintain them constant. The form of the interaction potential has been established in this case from the analysis of the interaction force in terms of the Maxwell tension tensor. In the third part of this study, the interaction of two macroparticles has been considered in the spherical system of coordinates and analytical expressions for the interaction potentials have been derived for both the case of constant macroparticle charges and the case of constant surface potentials of the macroparticles.

The charge and potential of dust particle in plasma under conditions of diffusion charging

The electric probe theory is used to determine the charge and potential of a spherical dust macroparticle in weakly ionized plasma under conditions where the radius of dust particle is many times the free path of ions and electrons. The stationary negative charge and potential are calculated as functions of the ratio between the coefficients of diffusion of ions and electrons for different values of relative fraction of dust, of the dimensionless Debye radius, and of the temperature ratio of positive ions and electrons.

Entropy driven key-lock assembly

The effective interaction between a sphere with an open cavity (lock) and a spherical macroparticle (key), both immersed in a hard sphere fluid, is studied by means of Monte Carlo simulations. As a result, a two-dimensional map of the key-lock effective interaction potential is constructed, which leads to the proposal of a self-assembling mechanism: There exists trajectories through which the key-lock pair could assemble avoiding trespassing potential barriers. Hence, solely the entropic contribution can induce their self-assembling even in the absence of attractive forces. This study points out the solvent contribution within the underlying mechanisms of substrate-protein assemblydisassembly processes, which are important steps of the enzyme catalysis and protein mediated transport.

The electrical double layer for a fully asymmetric electrolyte around a spherical colloid: An integral equation study

The hypernetted chain/mean spherical approximation (HNC/MSA) integral equation for a totally asymmetric primitive model electrolyte around a spherical macroparticle is obtained and solved numerically in the case of size-asymmetric systems. The ensuing radial distribution functions show a very good agreement when compared to our Monte Carlo and molecular-dynamics simulations for spherical geometry and with respect to previous anisotropic reference HNC calculations in the planar limit. We report an analysis of the potential versus charge relationship, radial distribution functions, mean electrostatic potential, and cumulative reduced charge for representative examples of 1:1 and 2:2 salts with a size-asymmetry ratio of 2. Our results are collated with those of the modified Gouy-Chapman (MGC) and unequal radius modified Gouy-Chapman (URMGC) theories and with those of HNC/MSA in the restricted primitive model (RPM) to assess the importance of size-asymmetry effects. One of the most striking characteristics found is that, contrary to the general belief, away from the point of zero charge the properties of an asymmetric electrical double layer (EDL) are not those corresponding to a symmetric electrolyte with the size and charge of the counterion, i.e., counterions do not always dominate. This behavior suggests the existence of a new phenomenology in the EDL that genuinely belongs to a more realistic size-asymmetric model where steric correlations are taken into account consistently. Such novel features cannot be described by traditional mean-field theories such as MGC, URMGC, or even by enhanced formalisms, such as HNC/MSA, if they are based on the RPM.

A new correlation effect in the Helmholtz and surface potentials of the electrical double layer.

The restricted primitive model of an electrical double layer around a spherical macroparticle is studied by using integral equation theories and Monte Carlo simulations. The resulting theoretical curves for the Helmholtz and surface potentials versus the macroparticle charge show an unexpected positive curvature when the ionic size of uni- and divalent electrolyte species is increased. This is a novel effect that is confirmed here by computer experiments. An explanation of this phenomenon is advanced in terms of the adsorption and layering of the electrolytic species and of the compactness of the diffuse double layer. It is claimed that the interplay between electrostatic and ionic size correlation effects, absent in the classical Poisson-Boltzmann view, is responsible for this singularity.

Interaction of foreign macroparticles in a cholesteric liquid crystal.

We calculate the interaction energy between spherical macroparticles immersed in a cholesteric liquid crystal due to the elastic deformation of the director field. We assume weak anchoring on the surface of the macroparticles and obtain the expression of the interaction energy that is valid for particle radius and interparticle distance sufficiently smaller than the cholesteric pitch. The resultant form of the interaction energy is more complex than that in a nematic liquid crystal. One of the characteristics is its dependence on the particle position as well as the interparticle distance, which arises from the intrinsic structure of a cholesteric liquid crystal, i.e., the absence of translational symmetry due to helical periodicity and local nematic ordering whose orientation depends on the position.

Frequency response of adsorption of a gas onto bidisperse pore-structured solid with modulation of inlet molar flow-rate

We present a theoretical analysis of the frequency response of a continuous-flow adsorber for adsorption of a gas onto bidisperse pore-structured solid, forced by the periodic modulation of the inlet flow-rate. Spherical macroparticle and spherical microparticle geometries are considered. A local adsorption equilibrium of the gas on the exterior surface of microparticle is assumed. It is confirmed that the in-phase characteristic function of the frequency response of a flow adsorber is independent of the overflow parameter, but that the out-of-phase characteristic function is a strong function of the overflow in the lower frequency region. The overall characteristic functions in the microparticle diffusion regime have information of both microparticle diffusion and macropore diffusion. Hence, in this regime we can extract the macropore diffusion parameter as well as the microparticle diffusion parameter from the experimental data of the overall characteristic functions. In the microparticle diffusion regime the capacity parameter of microparticle affects also the characteristic functions.

Amorphous very high surface area silica macrostructures

A procedure for the preparation of amorphous silica spherical macroparticles using strongly basic anion exchange gel resins as shape directing macro-templates is presented. Resin beads were hydroth ...

Aqueous mixtures of spherical and rodlike (flexible) macroparticles: structure and dynamics of latex spheres in saltfree solutions

This paper resumes light scattering investigations of saltfree aqueous solutions of two component mixtures of charged spheres by extending those measurements to systems in which one component is replaced by essentially stiff rodlike particles. In a second step of investigations these were replaced by linear flexible particles. Fd-virus particles (length l=883 nm) or macromolecules of NaPSS of four different contour lengths have been used as representatives. Mostly the concentration of latex spheres was fixed at 0.02 Vol%. The concentration of the other component was varied over a wide range. Concerning the scattering intensity, the contribution of the latex spheres dominates, in particular in the systems containing NaPSS particles. This simplifies the interpretation of data considerably. A rearrangement of the spheres is observed, depending on the shape of the other sort of particles. These conclusions can be drawn from the shift of measured static structure factor SM(q) with concentration c. A power law is found for the q-value of the maximum. The exponent depends on the properties of the second component. For the lower molecular weight (MW) samples of NaPSS below a critical concentration, the exponent is smaller than 1/3, decreasing the more the smaller the MW of the samples is. A tentative explanation in terms of charge number of NaPSS particles is given. The short time dynamics has been explored too. From the data a “dynamically determined structure factor” can be derived, that can be compared with the measured static structure factor. Good (fd) and fair (NaPSS) agreement is obtained respectively. Only at small wavenumbers below the maximum of SM(q) deviations occur which increase with concentration; they are consistent with hydrodynamic interaction.

Interaction between macroparticles in aqueous electrolytes

We report results of theoretical calculations for the interaction between two isolated, structureless spherical macroparticles immersed in aqueous electrolytes comprising waterlike molecules (hard spheres embedded with point dipoles and tetrahedral quadrupoles), 1:1 cations and anions (the diameter of cations is equal to that of anions). The reference hypernetted-chain (RHNC) theory with hard-sphere bridge functions is employed in the calculations. The fluid structure and the potential of mean force scaled by dS/(πdM) (dS and dM denote the solvent and macroparticle diameters, respectively) converge to limiting behaviors with increasing dM as long as the surface charge density of the macroparticle is kept constant. The qualitative aspects of the conclusions are not altered even when dM is set at 10dS. The interaction between neutral macroparticles in pure water is characterized by strong, short-range attraction. However, a relatively minor, short-range repulsive component is added to the interaction when ions are included in water. On the other hand, the presence of apolar particles in water at a trace concentration leads to considerable enhancement of the attraction. Effects of ionic sizes on the interaction between charged macroparticles are substantially large and opposite to those which would be observed using the primitive model. When the size of counterions is sufficiently large and the ionic concentration is sufficiently high, there is a regime where the interaction between highly like-charged macroparticles is strongly attractive.

Analysis of the frequency response method for sorption kinetics in bidispersed structured sorbents

A model for the sorption of gases onto bidispersed structured sorbent material in a batch system, subjected to small periodic perturbations of the system volume is developed. The model takes account of external film mass transfer resistance, coupled macropore and micropore resistance, and the surface barrier. The general model is linearised, and analytical solutions for the characteristic functions for slab, cylindrical and spherical macroparticle and microparticle geometries are determined, for both the general model and degenerate models which are applicable when one or more of the mass transfer resistances are negligible. The characteristic functions are examined to determine the effects of macroparticle and microparticle geometry and the rate controlling mechanisms on the frequency response spectrum. The limitations of the batch frequency response technique for investigation with sorbate—sorbent systems exhibiting H2- or L4-type isotherms are briefly considered.

Electrolyte friction on charged spherical macroparticles: Beyond the Debye–Hückel limit

The contribution to the friction coefficient of a charged spherical polyion diffusing in an ionic solution due to its interactions with the electrolyte ions is calculated, in the absence of hydrodynamic interactions, starting from a formal expression for ζel previously derived by Medina-Noyola and Vizcarra-Rendón. It is shown that such a formal expression can be given a particularly simple form in terms of the equilibrium distribution neqi(r) of the small ions around the tracer, which allows us to study effects ignored in Debye–Hückel level theories. We consider, for example, the effect of a finite size of the small ions and nonlinear contributions to ζel on the polyion charge. We show that the correct Debye–Hückel limit differs from Schurr’s result for ζel, and the source of the disagreement is clearly exposed. The results derived here for ζel are not restricted in the number of species of electrolyte ions or in their charges or mobilities.

Mass-diffusion and self-diffusion properties in systems of strongly charged spherical particles

Using the Fokker–Planck equation as the basic description, various collective and single-particle properties of highly charged spherical macroparticles in solution are derived. The dynamic structure factor S(k,t) for finite k is found to be non-exponential in t due to the viscoelastic properties of the liquid of interacting macroions. By using a mode-coupling procedure the dynamical properties can be completely reduced to the static structure factor S(k), which is calculated according to the method developed by Hansen and Hayter. At short times the system exhibits elastic behaviour and the corresponding high-frequency elastic moduli have been calculated. The bulk modulus is found to increase with c2, where c is the concentration. With regard to the single-particle properties, the velocity autocorrelation function is calculated and from it are obtained the mean-square displacement and the self-diffusion coefficient Ds as a function of concentration and of the coupling strength between macroions.