Quantification of solid pressure in the concentration polarization (CP) layer of colloidal particles and its impact on ultrafiltration

Here we describe the nature and implications of the "concentration polarization" (CP) layer that is formed during ultrafiltration of colloidal particles using a new approach in which the solid pressure, which arises from inter-particle interactions, and the inherent osmotic pressure are separately considered. The approach makes use of the particle transport mass balance between the convective and diffusive fluxes. The particle convection rate is hindered when inter-particle interactions take effect by reducing the particle velocities while the particle diffusion is solely controlled by the Brownian motion. An increase in solid pressure accounts for the reduction of the water potential caused by the relative motions of the particles and the surrounding water. A cell model is adopted to relate the local solid pressure with the local solid fraction and inter-particle interactions. The inter-particle interactions critically determine the form of particle accumulation (i.e. CP or gel/cake) on the membrane. The Shirato-Darcy equation is employed to relate the rate of increase in solid pressure, the relative liquid velocity and the solid fraction. Numerical integration approaches are employed to quantify the properties of the CP layer during both the development as well as the steady state phases (with steady state normally being achieved in a few minutes). The solid fractions are always no higher than those obtained when the inter-particle interactions are not considered. The decrease of the water potential caused by CP formation leads to the increase of both the solid pressure and the osmotic pressure. The dependence of the solid pressure on the solid fraction is usually stronger than that of the osmotic pressure. It is thus apparent that the solid pressure would be expected to dominate water potential reduction for solid fractions above a certain value though the solid pressure will be negligible when the solid fraction is relatively low.     

How to impose stick boundary conditions in coarse-grained hydrodynamics of Brownian colloids and semi-flexible fiber rheology

Long-range hydrodynamics between colloidal particles or fibers is modelled by the fluid particle model. Two methods are considered to impose the fluid boundary conditions at colloidal surfaces. In the first method radial and transverse friction forces between particle and solvent are applied such that the correct friction and torque follows for moving or rotating particles. The force coefficients are calculated analytically and checked by numerical simulation. In the second method a collision rule is used between colloidal particle and solvent particle that imposes the stick boundary conditions exactly. The collision rule comprises a generalisation of the Lowe-Anderson thermostat to radial and transverse velocity differences.     

AC electrokinetic templating of colloidal particle assemblies: effect of electrohydrodynamic flows

The use of spatially nonuniform electric fields for the contact-free colloidal particle assembly into ordered structures of various length scales is a research area of great interest. In the present work, numerical simulations are undertaken in order to advance our understanding of the physical mechanisms that govern this colloidal assembly process and their relation to the electric field characteristics and colloidal system properties. More specifically, the electric-field driven assembly of colloidal silica (d(p) = 0.32 and 2 μm) in DMSO, a near index matching fluid, is studied numerically over a range of voltages and concentration by means of a continuum thermodynamic approach. The equilibrium (u(f) = 0) and nonequilibrium (u(f) ≠ 0) cases were compared to determine whether fluid motion had an effect on the shape and size of assemblies. It was found that the nonequilibrium case was substantially different versus the equilibrium case, in both size and shape of the assembled structure. This dependence was related to the relative magnitudes of the electric-field driven convective motion of particles versus the fluid velocity. Fluid velocity magnitudes on the order of mm/s were predicted for 0.32 μm particles at 1% initial solids content, and the induced fluid velocity was found to be larger at the same voltage/initial volume fraction as the particle size decreased, owing to a larger contribution from entropic forces.     

Numerical calculation of the electrophoretic mobility of concentrated suspensions of soft particles

The electrophoretic mobility of spherical soft particles in concentrated colloidal suspensions is numerically calculated. The particle is modeled as a hard core coated with an ion-penetrable membrane bearing a uniform distribution of fixed charges, while the high particle concentration is taken into account by means of a cell model. The network simulation method used makes it possible to solve the problem without any restrictions on the values of the parameters such as particle concentration, membrane thickness, fixed charge density in the membrane, viscous drag in the membrane, number and valence of ionic species, electrolyte concentration, etc. The theoretical model used is similar to the one presented by Ohshima [H. Ohshima, J. Colloid Interface Sci. 225 (2000) 233], except for the use of the Shilov-Zharkikh, rather than the Levine-Neale, boundary condition for the electric potential, and the inclusion in the force balance equation of an additional term corresponding to the force exerted by the liquid on the core of the moving particle [J.J. López-García, C. Grosse, J. Horno, J. Colloid Interface Sci. 265 (2003) 327]. The obtained results only coincide with existing analytical expressions for low particle concentrations, low particle charge, and when the electrolyte concentration is high, the membrane is thick, and its resistance to the fluid flow is high. This suggests that most interpretations of the electrophoretic mobility of soft particles in concentrated suspensions require numerical calculations.     

Effect of colloidal particle size on adsorbed monodisperse and bidisperse monolayers

Coating hydrogel films or microspheres by an adsorbed colloidal shell is one synthesis method for forming colloidosomes. The colloidal shell allows control of the release rate of encapsulated materials, as well as selective transport. Previous studies found that the packing density of self-assembled, adsorbed colloidal monolayers is independent of the colloidal particle size. In this paper we develop an equilibrium model that correlates the packing density of charged colloidal particles in an adsorbed shell to the particle dimensions in monodisperse and bidisperse systems. In systems where the molar concentration in solution is fixed, the increase in adsorption energy with increasing particle size leads to a monotonic increase in the monolayer packing density with particle radius. However, in systems where the mass fraction of the particles in the adsorbing solutions is fixed, increasing particle size also reduces the molar concentration of particles in solution, thereby reducing the probability of adsorption. The result is a nonmonotonic dependence of the packing density in the adsorbed layer on the particle radius. In bidisperse monolayers composed of two particle sizes, the packing density in the layer increases significantly with size asymmetry. These results may be utilized to design the properties of colloidal shells and coatings to achieve specific properties such as transport rate and selectivity.     

Motion of a colloidal sphere with interfacial self-electrochemical reactions induced by a magnetic field

The motion of a spherical colloidal particle with spontaneous electrochemical reactions occurring on its surface in an ionic solution subjected to an applied magnetic field is analyzed for an arbitrary zeta potential distribution. The thickness of the electric double layer adjacent to the particle surface is assumed to be much less than the particle radius. The solutions of the Laplace equations governing the magnetic scalar potential and electric potential, respectively, lead to the magnetic flux and electric current density distributions in the particle and fluid phases of arbitrary magnetic permeabilities and electric conductivities. The Stokes equations modified with the Lorentz force contribution for the fluid motion are dealt by using a generalized reciprocal theorem, and closed-form formulas for the translational and angular velocities of the colloidal sphere induced by the magnetohydrodynamic effect are obtained. The dipole and quadrupole moments of the zeta potential distribution over the particle surface cause the particle translation and rotation, respectively. The induced velocities of the particle are unexpectedly significant, and their dependence on the characteristics of the particle-fluid system is physically different from that for electromagnetophoretic particles or phoretic swimmers.     

Techniques for measuring surface forces

The forces acting between colloidal particles and between surfaces are of utmost importance for determining the behaviour of dispersed systems and adhesion phenomena. Several techniques are now available for direct measurement of these surface forces. In this review we focus on some of these methods. Two techniques for measuring forces between macroscopic solid surfaces; the interferometric surface force apparatus, known as the SFA, and a novel instrument which is based on a bimorph force sensor, the so-called MASIF, are described in some detail. Forces between a macroscopic surface and a particle can be measured with the atomic force microscope (AFM) using a colloidal probe, or by employing total internal reflection microscipy (TIRM) to monitor the position of a colloidal particle trapped by a laser beam. We also describe two different techniques that can be used for measuring forces between “soft” interfaces, the thin film balance (TFB) for single foam, emulsion and solid/fluid/fluid films, and osmotic stress methods, commonly used for studying interactions in liquid crystalline surfactant phases or in concentrated dispersions. The advantages and limitations of each of these techniques are discussed and typical results are presented.     

Osmotic compressibility of soft colloidal systems

A turbidimetric analysis of particle interaction of model pH-responsive microgel systems consisting of methacrylic acid-ethyl acrylate cross-linked with diallyl phthalate in colloidal suspensions is described. The structure factor at zero scattering angle, S(0), can be determined with good precision for wavelengths greater than 500 nm, and it measures the dispersion's resistance to particle compression. The structure factor of microgels at various cross-linked densities and ionic strengths falls onto a master curve when plotted against the effective volume fraction, phi(eff) = kc, which clearly suggests that particle interaction potential and osmotic compressibility is a function of effective volume fraction. In addition, the deviation of the structure factor, S(0), of our microgel systems with the structure factor of hard spheres, S(PY)(0), exhibits a maximum at phi(eff) approximately 0.2. Beyond this point the osmotic de-swelling force exceeds the osmotic pressure inside the soft particles resulting in particle shrinkage. Good agreement was obtained when the structural properties of our microgel systems obtained from turbidimetric analysis and rheology measurements were compared. Therefore, a simple turbidimetric analysis of these model pH-responsive microgel systems permits a quantitative evaluation of factors governing particle osmotic compressibility.     

Electrodipping force acting on solid particles at a fluid interface

We report experimental results which show that the interfacial deformation around glass particles (radius, 200-300 microm) at an oil-water (or air-water) interface is dominated by an electric force, rather than by gravity. It turns out that this force, called for brevity "electrodipping," is independent of the electrolyte concentration in the water phase. The force is greater for oil-water than for air-water interfaces. Under our experimental conditions, it is due to charges at the particle-oil (instead of particle-water) boundary. The derived theoretical expressions, and the experiment, indicate that this electric force pushes the particles into water. To compute exactly the electric stresses, we solved numerically the electrostatic boundary problem, which reduces to a set of differential equations. Convenient analytical expressions are also derived. Both the experimental and the calculated meniscus profile, which are in excellent agreement, exhibit a logarithmic dependence at long distances. This gives rise to a long-range electric-field-induced capillary attraction between the particles, detected by other authors. Deviation from the logarithmic dependence is observed at short distances from the particle surface due to the electric pressure difference across the meniscus. The latter effect gives rise to an additional short-range contribution to the capillary interaction between two floating particles. The above conclusions are valid for either planar or spherical fluid interfaces, including emulsion drops. The electrodipping force, and the related long-range capillary attraction, can engender two-dimensional aggregation and self-assembly of colloidal particles. These effects could have implications for colloid science and the development of new materials.     

Equation of state of charged colloidal suspensions and its dependence on the thermodynamic route

The thermodynamic properties of highly charged colloidal suspensions in contact with a salt reservoir are investigated in the framework of the renormalized Jellium model (RJM). It is found that the equation of state is very sensitive to the particular thermodynamic route used to obtain it. Specifically, the osmotic pressure calculated within the RJM using the contact value theorem can be very different from the pressure calculated using the Kirkwood-Buff fluctuation relations. On the other hand, Monte Carlo simulations show that both the effective pair potentials and the correlation functions are accurately predicted by the RJM. It is suggested that the lack of self-consistency in the thermodynamics of the RJM is a result of neglected electrostatic correlations between the counterions and coions.     

Phase Transitions of the Liquid–Liquid Type and a Change in the Particle Charge in Colloidal Solutions

The thermodynamic properties are studied for the solutions of charged colloidal particles with ionizable surface groups. The microscopic mechanism of microion binding at surface groups is considered. The free energy of the system in the parameter range where the usual theory of such solutions is inadequate (a range of practical interest) is calculated using the method of the thermodynamic perturbation theory. The first-order phase transition of the liquid–liquid type is shown to be possible; in this phase transition, a phase with a high concentration of colloidal particles that have a higher charge coexists with a phase with a lower concentration of particles that have a lower charge.     

The polydisperse cell model: nonlinear screening and charge renormalization in colloidal mixtures

We propose a model for the calculation of renormalized charges and osmotic properties of mixtures of highly charged colloidal particles. The model is a generalization of the cell model and the notion of charge renormalization as introduced by Alexander et al. [J. Chem. Phys. 80, 5776 (1984)]. The total solution is partitioned into as many different cells as components in the mixture. The radii of these cells are determined self-consistently for a given set of parameters from the solution of the nonlinear Poisson-Boltzmann equation with appropriate boundary conditions. This generalizes Alexanders's model where the (unique) Wigner-Seitz cell radius is solely fixed by the colloid packing fraction. We illustrate the technique by considering a binary mixture of the colloids with the same sign of charge. The present model can be used to calculate thermodynamic properties of highly charged colloidal mixtures at the level of linear theories, while taking the effect of nonlinear screening into account.     

Temperature effects on the structure and dynamics of liquid dimethyl sulfoxide: A molecular dynamics study

The molecular dynamics (MD) simulation technique has been employed to investigate the thermodynamic properties and transport coefficients of the neat liquid dimethyl sulfoxide (DMSO). The fluid has been studied at temperatures in the range 298–353 K and at a pressure equal to 1 atm. The simulations employed a nine-site potential model, which is presented for the first time here, and all the available non-polarizable models. The performance of each model is tested using the same statistical mechanical ensemble and simulation method under the same conditions, revealing its weaknesses and strengths. Thermodynamic properties, microscopic structure and dynamic properties, such as transport coefficients, rotational and single-dipole correlation times have been calculated and compared with available experimental results. Estimations of transport coefficients from various theoretical and empirical models are tested against experimental and MD results. Translational and rotational dynamics suggest the existence of the cage effect and agree with the Stokes–Einstein–Debye relation. The dipole relaxation times calculated are discussed in terms of simple and useful approximations, such as the Glarum–Powles and Fatuzzo–Mason models.     

Fluid-crystal coexistence for proteins and inorganic nanocolloids: Dependence on ionic strength

We investigate theoretically the fluid-crystal coexistence of solutions of globular charged nanoparticles such as proteins and inorganic colloids. The thermodynamic properties of the fluid phase are computed via the optimized Baxter model P. Prinsen and T. Odijk [J. Chem. Phys. 121, 6525 (2004)]. This is done specifically for lysozyme and silicotungstates for which the bare adhesion parameters are evaluated via the experimental second virial coefficients. The electrostatic free energy of the crystal is approximated by supposing the cavities in the interstitial phase between the particles are spherical in form. In the salt-free case a Poisson-Boltzmann equation is solved to calculate the effective charge on a particle and a Donnan approximation is used to derive the chemical potential and osmotic pressure in the presence of salt. The coexistence data of lysozyme and silicotungstates are analyzed within this scheme, especially with regard to the ionic-strength dependence of the chemical potentials. The latter agree within the two phases provided some upward adjustment of the effective charge is allowed for.     

Dispersion of magnetic susceptibility and the microstructure of magnetic fluid

The microstructure of magnetic fluid produced on the basis of kerosene with oleic acid as a stabilizer is studied experimentally. An analytical procedure based on the known dependence of the time of Brownian relaxation of the magnetic moment of the colloidal particle on its size and the expansion of a low-frequency spectrum of dynamic susceptibility into the series of Debye functions is used. Magnetic susceptibility is measured at frequencies from 10 Hz to 100 kHz and temperatures from 225 to 360 K for colloidal solutions with the volume fraction of magnetite from 0.08 to 0.17. The clusters with uncompensated magnetic moments and sizes varying from 50 to 70 nm that are three-or fourfold larger than the mean diameter of a single colloidal particle are found. It is revealed that characteristic sizes of clusters are virtually independent of temperature and concentration of colloidal particles. The contribution of clusters to the equilibrium susceptibility of magnetic fluid grows exponentially with decreasing temperature, being manyfold larger at low temperatures than that of single particles. The obtained temperature dependence of equilibrium susceptibility is compared with that predicted from current theoretical models.     

Reversible coagulation of colloidal suspension in shallow potential wells: Direct numerical simulation

Brownian dynamics computer simulations of aggregation in 2D colloidal suspensions are discussed. The simulations are based on the Langevin equations, pairwise interaction between colloidal particles and take into account Brownian, hydrodynamic and colloidal forces. The chosen mathematical model enables to predict the correct values of diffusion coefficient of freely moving particle, the mean value of kinetic energy for each particle in ensemble of interacting colloidal particles and residence times of colloidal particles inside the potential wells of different depths. The simulations allow monitoring formation and breakage of clusters in a suspension as well as time dependence of the mean cluster size. The article is published in the original.     

Membrane potential in charged porous membranes

For charged porous membranes, the separation efficiency to charged particles and ions is affected by the electrical properties of the membrane surface. Such properties are most commonly quantified in terms of zeta-potential. In this paper, it is shown that the zeta-potential can be calculated numerically from the membrane potential. The membrane potential expression for charged capillary membranes in contact with electrolyte solutions at different concentrations is established by applying the theory of non-equilibrium thermodynamic to the membrane process and considering the space-charge model. This model uses the Nernst–Planck and Navier–Stokes equations for transport through pores, and the non-linear Poisson–Boltzmann equation, which is numerically solved, for the electrostatic condition of the fluid inside pores. The integral expressions of the phenomenological coefficients coupling the differential flow (solute relative to solvent) and the electrical current with the osmotic pressure and the electrical potential gradients are established and calculated numerically. The mobilities of anions and cations are individually specified. The variations of the membrane potential (or the apparent transport number of ions in the membrane pores) are studied as a function of different parameters: zeta-potential, pore radius, mean concentration in the membrane, ratio of external concentrations and type of ions.     

Sedimentation and electrophoresis of a porous floc and a colloidal particle coated with polyelectrolytes

Sedimentation and electrophoresis of porous colloid complex; a colloidal floc and a colloidal particle covered with adsorbed polyelectrolytes are visited to examine the characteristic length of the transport phenomena. In the sedimentation, the overall size of a floc is dominative in the determination of Stokes drag, while the permeability is determined by the largest pore in the floc. This picture is important when break-up of flocs in a turbulent flow is considered. When a colloidal particles is coated with polyelectrolytes, the characteristic length for diffusion is that of the diameter of colloidal particle plus protruding part of polymer chain adsorbed onto the particle. On the other hand, when the porous colloid complex is placed in the electric field, fluid surrounding the complex can easily penetrate into the complex by means of electro-osmosis. The diffusive part of electric double layer located inside of the complex is the source of strong driving force of this osmotic flow. Flow generated in this regime can be treated as a sort of shear driven. The characteristic length scale for transport phenomena is the Debye length or the distance between charged segments. These lengths are much shorter than the case of sedimentation and Brownian diffusion.     

Solvation force induced by short range, exact dissipative particle dynamics effective surfaces on a simple fluid and on polymer brushes

The thermodynamic properties of a simple fluid confined by effective wall forces are calculated using Monte Carlo simulations in the grand canonical ensemble. The solvation force produced by polymer brushes of two different lengths is obtained also. For the particular type of model interactions used, known as the dissipative particle dynamics method, we find that it is possible to obtain an exact, simple expression for the effective force induced by a planar wall composed of identical particles that interact with those in the fluid. We show that despite the short range of all forces in the model, the solvation force can be finite at relatively large distances and therefore does not depend only on the range of the interparticle or solvent-surface forces. As for the polymer brushes, we find that the shape of the solvation force profiles is in fair agreement with scaling and self-consistent field theories. The applications and possible extensions of this work are discussed.