Linear Superposition Approximation Research Articles

Choice of DLVO approximation method for quantifying the affinity between latex particles and membranes

The Derjaguin-Landau-Verwey-Overbeek (DLVO) model, as well as the extended model (XDLVO), is popularly employed to quantify the interfacial interactions underlying membrane fouling. However, disagreements between membrane fouling extent and the interaction energies derived from DLVO or XDLVO models have been reported, which suggests gaps in the understanding. This study demonstrates that the DLVO approximation methods for predicting the interfacial foulant-membrane interaction are sensitive to the boundary condition assumptions (e.g., constant charge versus constant potential). In particular, while both the Poisson-Boltzmann (P–B) and linear superposition approximation (LSA) equations can quantify the electrostatics (EL) interaction energy component, the former assumes constant potential, while the latter additionally considers constant charge scenarios. The relative accuracy of these two equations were evaluated here. For dead-end filtration tests, flux decline trends, OCT analysis results and fouling model parameters were obtained. Regarding fouling of pristine PCTE membrane by latex particles of opposite charge signs, both the P–B and LSA equations contribute to predict relative fouling extents correctly. However, for the case of the BPEI-coated membrane, the P–B equation failed whereas the LSA equation gave good agreements. For cross-flow filtration tests in organic solvents, LSA also out-performed the P–B equation in providing more accurate predictions of membrane fouling. The results here highlight the shortcomings in the commonly used P–B equation and are expected to be potentially valuable in the development of better approximations for quantifying interfacial interaction energies.

Electrostatic interaction between two spheroidal particles at large separations

Asymptotic expressions for the electrostatic interaction energy between two weakly charged spheroidal particles (prolate and oblate) with uniform surface potential or surface charge density at large separations in an electrolyte solution are derived on the basis of the linear superposition approximation. The electrostatic interaction between two spheroids is found to be the screened Coulomb interaction between two hypothetical point particles with orientation-dependent charges. Explicit interaction energy equations, which are expressed in terms of spheroidal wave functions, are given for four particular configurations of two similar spheroids, that is, side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates. Comparison is also made with results obtained using Derjaguin’s approximation. Electrostatic interaction between two spheroids • Asymptotic expressions for the electrostatic interaction energy between two spheroidal particles are derived. • Constant surface potential and charge density cases are considered. • Side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates are considered.

Can nanoscale surface charge heterogeneity really explain colloid detachment from primary minima upon reduction of solution ionic strength?

This study theoretically examined colloid detachment from primary minima with ionic strength (IS) reduction on heterogeneous collector surfaces. The chemically and physically heterogeneous collector surfaces were modeled as a planar surface carrying nanoscale patches of different zeta potentials and nanoscale pillars/hemispheroids, respectively. The surface element integration technique was used to calculate interaction energies between colloid and collector surfaces. Two boundary conditions for the double-layer interaction energy were considered, namely constant surface potential (CSP), and linear superposition approximation (LSA). In contrast to prevailing opinions in the literature, our results show that colloids attached on the chemically heterogeneous surface cannot be detached by IS reduction under CSP condition due to an increase of the adhesive force/torque with decreasing IS. Detachment from chemically heterogeneous surfaces by IS reduction can occur under LSA condition only when the flow velocity is very high. In contrast, the presence of nanoscale physical heterogeneity can cause colloid detachment from primary minima by IS reduction under both CSP and LSA conditions at flow velocities commonly used in experimental studies because of a significant reduction in the adhesive forces/torques.

Approximate expressions for the surface charge density/surface potential relationship and double-layer potential distribution for a spherical or cylindrical colloidal particle based on the modified Poisson-Boltzmann equation

Approximate expressions for the surface charge density/surface potential relationship and double-layer potential distribution are derived for a spherical or cylindrical colloidal particle in an electrolyte solution. The obtained expressions are based on an approximate form of the modified Poisson-Boltzmann equation taking into account the ion size effects through the Carnahan-Starling activity coefficients of electrolyte ions. We further derive approximate expression for the effective surface potentials of a spherical or cylindrical particle and for the electrostatic interaction energy between two spherical or cylindrical particles on the basis of the linear superposition approximation.

Numerical and experimental investigations on 342 nm femtosecond laser ablation of K24 superalloy

With the feature of ultrashort pulse width and ultrahigh peak power, femtosecond laser can realize real sense of ‘cold’ machining and it has been widely applied in micro and nano scale manufacturing. K24 is a new cast Ni-based superalloy with high strength, strong resistance to corrosion and excellent heat stability under high temperature. Micro drilling method of this kind of material has becomes a tough issue so far. In this paper, interaction process between 342nm femtosecond laser and K24 superalloy is theoretically described. Laser-dependent absorption coefficient and reflectivity of K24 superalloy are derived applying Planck equations while other physical properties like lattice heat capacity and electronic specific heat coefficient are calculated correlated with solid state physics utilizing linear superposition approximation. Based on these, finite difference method is employed to solve one-dimensional two-temperature model. Through corresponding verification experiments the accuracy of mathematical model is proved. The influence rules of machining parameters of 342nm femtosecond laser beam drilling, such as the feed distance of laser spot, scanning times, scanning velocity and laser average power, on the morphology of micro holes are deeply investigated through orthogonal experiments. Following theoretical analysis provide solid foundations for the future application of femtosecond laser drilling on superalloy.

Floating and Sinking of a Pair of Spheres at a Liquid-Fluid Interface.

Spheres floating at liquid-fluid interfaces cause interfacial deformations such that their weight is balanced by the resultant forces of surface tension and hydrostatic pressure while also satisfying a contact angle condition. Determining the meniscus shape around several floating spheres is a complicated problem because the vertical locations of the spheres and the horizontal projections of the three-phase contact lines are not known a priori. Here, a new computational algorithm is developed to simultaneously satisfy the nonlinear Laplace-Young equation for the meniscus shape, the vertical force balance, and the geometric properties of the spheres. We implement this algorithm to find the shape of the interface around a pair of floating spheres and the horizontal force required to keep them at a fixed center-center separation. Our numerical simulations show that the ability of a pair of spheres to float (rather than sink) depends on their separation. Similar to previous work on horizontal cylinders, sinking may be induced at close range for small spheres that float when isolated. However, we also discover a new and unexpected behavior: at intermediate inter-particle distances, spheres that would sink in isolation can float as a pair. This effect is more pronounced for spheres of radius comparable to the capillary length, suggesting that this effect is a result of hydrostatic pressure, rather than surface tension. An approximate solution confirms these phenomena and shows that the mechanism is indeed the increased supporting force provided by the hydrostatic pressure. Finally, the horizontal force of capillary attraction between the spheres is calculated using the results of the numerical simulations. These results show that the classic linear superposition approximation (due to Nicolson) can lose its validity for relatively heavy particles at close range.

Approximate analytic expressions for the electrostatic interaction energy between two colloidal particles based on the modified Poisson-Boltzmann equation

Simple analytic expressions are derived for the electrostatic interaction energy between two charged colloidal particles in an electrolyte solution. The obtained expressions are based on an approximate form of the modified Poisson-Boltzmann equation taking into account the ion size effects through Carnahan-Starling activity coefficients of electrolyte ions. We derive the electrostatic interaction energy between two parallel plates on the basis of the linear superposition approximation. We further employ Derjaguin’s approximation to derive the corresponding expressions for the electrostatic interaction energy between two spheres, two parallel cylinders, or two crossed cylinders.

In situ contrast calibration to determine the height of individual diffusing nanoparticles in a tunable confinement

We study the behavior of charged spherical Au nanoparticles in a nanofluidic slit as a function of the separation of the symmetrically charged confining surfaces. A dedicated setup called the nano-fluidic confinement apparatus allows us to parallelize the two confining surfaces and to continuously approach them down to direct contact. Interferometric scattering detection is used to measure the particle contrast with 2 ms temporal resolution. We obtain the confinement gap distance from the interference signal of the glass and the oxide-covered silicon wafer surface with nanometer accuracy. We present a three parameter model that describes the optical signal of the particles as a function of particle height and gap distance. The model is verified using nanoparticles immobilized at the glass and the substrate surface. For freely diffusing particles, the envelope of the particle signal as a function of gap distance and the known particle height at tight confinement is used to calibrate the particle signal in situ and obtain all free model parameters. Due to the periodic contrast variation for large gap distances, we obtain a set of possible particle heights for a given contrast value. For a range of small gap distances, this assignment is unique, and the particle height can be measured directly with high accuracy. The high temporal resolution allows us to measure the height occupation probability, which provides a direct link to the free-energy landscape the particles are probing via the Boltzmann distribution. Accordingly by fitting the results to a physical model based on the linear superposition approximation, the physical parameters governing the particle-glass interaction are quantified.

Driven particles at fluid interfaces acting as capillary dipoles

The dynamics of spherical particles driven along an interface between two immiscible fluids is investigated asymptotically. Under the assumptions of a pinned three-phase contact line (TCL) and very different viscosities of the two fluids, a particle assumes a tilted orientation. As it moves, it causes a deformation of the fluid interface which is also computed. The case of two interacting driven particles is studied via the linear superposition approximation. It is shown that the capillary interaction force resulting from the particle motion is dipolar in terms of the azimuthal angle and decays with the fifth power of the inter-particle separation, similar to a capillary quadrupole originating from undulations of the TCL. The dipolar interaction is demonstrated to exceed the quadrupolar interaction at moderate particle velocities.

Simple transform methods of a force curve obtained by surface force apparatus to the density distribution of a liquid near a surface

We propose two simple methods that transform a force curve obtained by a surface force apparatus (SFA) into a density distribution of a liquid near a surface of the SFA probe. The transform methods are derived based on the statistical mechanics of simple liquids, where the liquid is an ensemble of small spheres. The solvent species is limited to only one component and two-body potential between the solvent spheres is arbitrary. However, two-body potential between the SFA probe and the solvent is restricted to rigid potential (i.e., the transform methods are derived within the restriction of the rigid potential). In addition, Kirkwood and linear superposition approximations are applied in order to derive the transform methods. The transform methods are simply tested in both hard-sphere fluid and Lennard-Jones (LJ) fluid with hard core potential. The tests are computationally practiced using a three-dimensional integral equation theory. It is found that the transform method with Kirkwood superposition approximation (transform method 1) generally reproduces the more precise solvation structure than that with linear superposition approximation (transform method 2). In the test of the hard sphere solvent, it is found that the reproducibility becomes better as the number density of the solvent is lower. Furthermore, it is found in the test of the LJ fluid that the reproducibility becomes better as the two-body potential between the SFA probe and the solvent approaches rigid potential. This is because, the transform methods are derived within the model of the rigid potential. It is verified that the transform methods are useful for obtaining of a rough image of the solvation structure. (However, if evaporation or solidification, a phase transition in a local space sandwiched between the two surfaces, occurs while the experiment, the transform methods should not be used.)

An analytical method for determining the interaction energy between multiple identical spherical colloidal zinc oxide nanoparticles

A modification of the linear superposition approximation for the linearized Poisson- Boltzmann equation, describing the interaction of colloidal nanoparticles, is proposed based on the implementation of a phenomenological charge regulation model to allow for the more accurate description of the interaction at small particle separations and multi-particle interactions.

Electrostatic interaction between two spherical soft particles at small separations

An approximate analytic expression is given for the potential energy of the electrostatic interaction between two identical spherical soft particles (ion-penetrable porous spheres) in a symmetrical electrolyte solution with the help of Derjaguin's approximation. This method uses accurate analytic expressions for the corresponding interaction energy between two parallel similar soft plates (ion-penetrable membranes) obtained previously (J. Colloid Interface Sci., 350 (2010) 249). Agreement with numerical data (J. Colloid Interface Sci., 36 (1989) 523) is excellent for arbitrary potentials and small particle separations, where the Debye–Hückel linearization approximation and the linear superposition approximation are not allowed.

Exact Analytical Expressions for the Potential of Electrical Double Layer Interactions for a Sphere−Plate System

Exact, closed-form analytical expressions are presented for evaluating the potential energy of electrical double layer (EDL) interactions between a sphere and an infinite flat plate for three different types of interactions: constant potential, constant charge, and an intermediate case as given by the linear superposition approximation (LSA). By taking advantage of the simpler sphere-plate geometry, simplifying assumptions used in the original Derjaguin approximation (DA) for sphere-sphere interaction are avoided, yielding expressions that are more accurate and applicable over the full range of κa. These analytical expressions are significant improvements over the existing equations in the literature that are valid only for large κa because the new equations facilitate the modeling of EDL interactions between nanoscale particles and surfaces over a wide range of ionic strength.

Exact expressions for colloidal plane–particle interaction forces and energies withapplications to atomic force microscopy

We begin by deriving a general useful theoretical relationship between the plane–particleinteraction forces in solution, and the corresponding plane–plane interaction energies. Thisis the main result of the paper. It provides a simple tool to obtain closed-formparticle–plane forces from knowledge of plane–plane interaction energies. To illustrate thesimplicity of use of this general formalism, we apply it to find particle–plane interactionswithin the Derjaguin–Landau–Verwey–Overbeek (DLVO) framework. Specifically,we obtain analytical expressions for forces and interaction energies in the vander Waals and the electrical double layer cases. The van der Waals expression iscalculated here for benchmarking purposes and is compared with well-establishedexpressions from Hamaker theory. The interactions for the electric double layersituation are computed in two cases: the linear superposition approximation andthe constant surface potential. In both cases, our closed-form expressions werecompared with existent numerical results. We also use the main result of thispaper to generate an analytical force-separation expression based on atomic forcemicroscope experiments for a tip and surface immersed in an aqueous solution, andcompare it with the corresponding numerical results. Finally, based on our mainresult, we generalize the Derjaguin approximation by calculating the next order ofapproximation, thus obtaining a formula valuable for colloidal interaction estimations.

Capillary Forces between Spherical Particles Floating at a Liquid−Liquid Interface

We study the capillary forces acting on sub-millimeter particles (0.02-0.6 mm) trapped at a liquid-liquid interface due to gravity-induced interface deformations. An analytical procedure is developed to solve the linearized capillary (Young-Laplace) equation and calculate the forces for an arbitrary number of particles, allowing also for a background curvature of the interface. The full solution is expressed in a series of Bessel functions with coefficients determined by the contact angle at the particle surface. For sub-millimeter spherical particles, it is shown that the forces calculated using the lowest order term of the full solution (linear superposition approximation; LSA) are accurate to within a few percents. Consequently the many particle capillary force is simply the sum of the isolated pair interactions. To test these theoretical results, we use video microscopy to follow the motion of individual particles and pairs of interacting particles at a liquid-liquid interface with a slight macroscopic background curvature. Particle velocities are determined by the balance of capillary forces and viscous drag. The measured velocities (and thus the capillary forces) are well described by the LSA solution with a single fitting parameter.

Mechanism of Particle Deposition on Silicon Surface during Dilute HF Cleans

The mechanism of particle deposition onto silicon wafer surfaces during dilute HF cleans is discussed. Direct surface force measurement using an atomic force microscope showed that particle redeposition on a silicon surface is due to the dominant van der Waals interaction between the particle and the wafer surface. The addition of surfactants can affect the clean effectiveness of a dilute HF solution by changing the surface interaction forces between a particle and a wafer. We show that there is a simple correlation between particle deposition and the product of the zeta-potentials of the wafer and particle. This correlation can be explained by introducing the linear superposition approximation to the derivation of electrical double-layer interactions. The addition of surfactant will also decrease dispersion attraction, introduce steric repulsion, and eliminate adhesion force, as indicated by the results of surface force measurements. © 2003 The Electrochemical Society. All rights reserved.

Brownian motion in confinement

This research modifies an earlier approach based on the single-wall reflection method to predict the perpendicular and parallel diffusion coefficients of a Brownian sphere in confinement. The modified version provides predictions that match experimental data reported in the literature more accurately than those provided by other available models, including the linear superposition approximation, the coherent superposition approximation, and Oseen's equation.

Effect of Surface Roughness on the Interaction Energy between a Colloidal Sphere and a Flat Plate

A set of analytical equations are developed for calculating the van der Waals and electrostatic interaction energies between a rough spherical particle and a smooth plate. The particle roughness is modeled as hemispherical asperities of fixed radius, in agreement with SEM micrographs of polystyrene latex spheres. The van der Waals energy is calculated using a pairwise summation technique for both nonretarded and retarded cases. The electrostatic energy is calculated using the linear superposition approximation and assuming the asperity-plate and particle-plate interactions are additive. We find that at large separations, the increased electrostatic repulsion produced by the roughness results in a shallower secondary potential well. At closer separations, however, the increased van der Waals attraction produces a dramatically lower repulsive barrier (as much as two orders of magnitude smaller in the example presented). This lower barrier would yield much larger capture rates than predicted by DLVO theory for smooth surfaces, in qualitative agreement with experimental observations.

Solvation Force in Confined Molecular Fluids Using Molecular Dynamics Simulation

The solvation force in water, treated as a molecular fluid, confined between two mica surfaces is calculated using molecular dynamics simulations. In order to maintain a constant chemical potential as the wall separation is varied, the mean density of the confined fluid is calculated using a linear superposition approximation. By calculating the chemical potential using the standard test particle method, the approximation is shown to be valid for distances of separation down to about two molecular diameters. Both the density profiles and the solvation force isotherm are oscillatory with a periodicity of about one molecular diameter. The oscillations in structural force show an approximately exponential decay with separation distance.

Electrostatic Interaction between Two Dissimilar Spheres: Image-Interaction Correction to the Linear Superposition Approximation

The leading term of the electrostatic interaction energy between two colloidal spheres in an electrolyte solution is given by a screened Coulomb interaction due to a simple linear superposition of the unperturbed potentials of the respective spheres. We have previously shown that the next-order correction to the linear superposition approximation corresponds to the interaction between one sphere and its image with respect to the other sphere. The image interaction depends strongly on the type of boundary conditions on the interacting spheres. In the present paper we consider the image-interaction energy for three cases on the basis of the linearized Poisson–Boltzmann equation: (a) the constant surface potential case, (b) the constant surface charge density case, and (c) the mixed case in which one sphere has a constant surface potential and the other has a constant surface charge density. It is shown that the image interaction is always attractive for case (a) but may be either attractive or repulsive for case (b). The change in sign of the interaction force or the appearance of the extremum in the interaction energy may occur when the image-interaction correction term and the leading term are of the opposite sign and the former exceeds the latter.