Electrophoresis of concentrically and eccentrically positioned cylindrical particles in a long tube

We study analytically and numerically the electrophoretic motion of cylindrical particles translating slowly in long tubes filled with an electrolyte solution and subjected to axial electric fields. Both thin and thick double layers are considered. Of particular interest is the case when the particle's and tube's radii are of the same order of magnitude. The model accounts for the flow induced by the particle's motion (the particle acts as a leaky piston) and the electroosmotic flow in the tube. The electrophoretic velocity of the particle and the forces and torques acting on it are determined as functions of the particle's radius, length, and position, the particle's and tube's zeta potentials, the tube's length, and the externally imposed pressures. When the particle is positioned off center, it rotates and its trajectory traces an oscillatory path.     

Electrophoresis of a toroid along the axis of a cylindrical pore

The electrophoresis of a toroid (doughnut-shaped entity) along the axis of a long cylindrical pore is analyzed under the conditions of low surface potential and weak applied electric field. The system under consideration is capable of modeling the electrophoretic behavior of various types of biocolloid such as bacterial DNA, plasmid DNA, and anabaenopsis, in a confined space. The influences of the key parameters of the problem, including the sizes of a toroid, the radius of a pore, and the thickness of the double layer, on the electrophoretic mobility of a toroid are discussed. We show that the electrophoretic behavior of a toroid under typical conditions can be different from that of an integrated entity. For instance, although the presence of the pore wall has the effect of retarding the movement of a particle, it becomes advantageous if a toroid is sufficiently close to the boundary. Several interesting behaviors are also observed, for example, the mobility of a toroid when the boundary effect is significant can be larger than that when it is insignificant.     

Electrophoresis of a charge-regulated spheroid along the axis of an uncharged cylindrical pore

Boundary effects can have a profound influence on the electrophoretic behavior of a charged entity, in particular, when the entity is nonspherical and its surface conditions are dependent upon the nearby environment. In this study, the electrophoresis of a spheroid along the axis of an uncharged cylindrical pore is analyzed for the case where the electrical potential is low and the applied electric field is weak. We consider the case where the surface of a particle contains dissociable acidic and basic functional groups, which simulate biological colloids and entities covered by an artificial membrane. This leads to a mixed-type boundary value problem, which extends the conventional constant-surface-potential and constant-surface-charge-density models to a more general case. The effects of the particle aspect ratio, the relative magnitudes of particle and pore, the thickness of the double layer surrounding a particle, and the pH of the liquid phase on the electrophoretic mobility of a particle are investigated. Several interesting results are observed; for example, if the volume of a particle is fixed, its mobility may have a local maximum as the relative magnitudes of its two axes vary.     

Electrophoresis of a spherical particle along the axis of a cylindrical pore filled with a Carreau fluid

The electrophoresis of a spherical particle along the axis of a cylindrical pore filled with a Carreau fluid is investigated theoretically. In addition to the boundary effect due to the presence of the pore, the influences of the thickness of double layer surrounding a particle and the properties of the fluid on the electrophoretic behavior of the particle are also examined. We show that, in general, the presence of the pore has the effect of retarding the movement of a particle. On the other hand, the shear-thinning nature of the liquid phase is advantageous to its movement. For both Newtonian and Carreau fluids, the mobility of a particle increases monotonically with the decrease in the thickness of double layer, but the mobility is more sensitive to the variation of the thickness of double layer in the latter. The mobility of a particle in a Carreau fluid is larger than that in the corresponding Newtonian fluid, and the difference between the two increases with the decrease in double-layer thickness; in addition, depending upon the values of the parameters assumed, the percentage difference can be in the order of 50%.     

Electrophoresis of a spherical particle along the axis of a cylindrical pore: effect of electroosmotic flow

The electrophoresis of a spherical particle along the axis of a cylindrical pore is investigated under conditions of low surface potential and thick double layer. In particular, the effect of electroosmotic flow is taken into account. The results of numerical simulation reveal that if both particle and pore are positively charged, the variation of the mobility of a particle may have a local minimum as the thickness of the double layer varies, which is not reported in the literature. This is mainly due to the charge induced on the particle surface, which arises from the presence of the charged boundary. Depending upon the level of the surface potential of the pore, the presence of the local minima may lead to a reversal in the direction of particle movement as the thickness of the double layer surrounding it varies: if the surface potential is either too low or too high, reversal does not occur; if it has a medium level, reversal occurs twice. This interesting observation can play a role in electrophoresis measurements. Previous analysis predicts that reversal always occurs once, regardless of the level of the surface potential of the pore.     

Electrophoresis of an ellipsoid along the axis of a cylindrical pore: effect of a charged boundary

The influence of a charged boundary on the electrophoretic behavior of a particle is investigated by considering the electrophoresis of a nonconducting ellipsoid along the axis of a cylindrical pore at the level of the linear Poisson-Boltzmann equation ignoring the polarization effect. The problem considered simulates the electrophoresis conducted in a narrow space such as capillary electrophoresis and electrophoresis through a porous medium. Here, because the effect of electroosmotic flow can be important the electrophoretic behavior is much more complicated than that for the case where a boundary is uncharged. The influences of the thickness of double layer, the aspect ratio of an ellipsoid, the relative radius of a pore, and the charge conditions on the ellipsoid and pore surfaces on the mobility of the ellipsoid are discussed. Several interesting but nonintuitive electrophoretic behaviors are observed.     

Electrophoresis of a sphere along the axis of a cylindrical pore: effects of double-layer polarization and electroosmotic flow

The electrophoresis of a rigid sphere along the axis of a cylindrical pore is investigated theoretically. Previous analysis is extended to the case where the effects of double-layer polarization and electroosmotic flow can be significant. The influences of the surface potential, the thickness of the double layer, and the relative size of a pore on the electrophoretic behavior of a sphere are discussed. Some interesting results are observed. For example, if both a sphere and a pore are positively charged, then the mobility of the sphere has a local minimum as the thickness of its double layer varies. Depending upon the level of the surface potential of a sphere and the degree of significance of the boundary effect, the mobility of the sphere may change its sign twice as the thickness of its double layer varies. This result can play a significant role in electrophoresis measurements.     

Boundary effect on electrophoresis: finite cylinder in a cylindrical pore

The boundary effect on electrophoresis is investigated by considering a finite cylindrical particle moving along the axis of a long cylindrical pore under conditions of low surface potential and weak applied electric field. The influence of the thickness of the double layer, the aspect ratio of a particle, the ratio particle radius/pore radius, and the charged conditions of the surfaces of the particle and pore on the electrophoretic behavior of a particle are investigated. We show that the effect of the aspect ratio of a particle on its electrophoretic behavior for the case where the particle is charged and the pore is uncharged is larger than that for the case where the particle is uncharged and the pore is charged. Also, depending on the parameters chosen, increasing the aspect ratio of a particle can either promote or hinder its movement, which is not reported in previous studies, and can play a role in electrophoresis measurements. Because both the electric and the flow fields in the gap between the particle and the pore are mediated by those near the top and the end of the particle, the end effect is large when the double layer is thick.     

Electrophoresis of two identical rigid spheres in a charged cylindrical pore

The electrophoresis of two identical spheres moving along the axis of a long cylindrical pore under the conditions of low surface potential and weak applied electric field is investigated. The geometry considered allows us to examine simultaneously the effects of boundary and the presence of a nearby entity on the behavior of a particle. The influences of the separation distance between two spheres, the thickness of a double layer, the ratio (radius of sphere/radius of pore), and the charged conditions on the surfaces of the spheres and the pore on the mobility of a particle are investigated. Several interesting results that are not reported in the literature are observed. For instance, although for the case of two positively charged spheres in an uncharged pore the qualitative behavior of a sphere depends largely on its size relative to that of a pore and the thickness of the double layer, this might not be the case when two uncharged spheres are in a positively charged pore. In addition, in the latter, the mobility of a sphere increases with the increases in the separation distance between two spheres, and this effect is pronounced when the ratio (radius of sphere/radius of pore) takes a medium value or the thickness of the double layer is either sufficiently thin or sufficiently thick.     

Drag on two nonuniformly structured flocs moving along the axis of a cylindrical tube

The boundary effect on the drag on two identical, nonuniformly structured flocs moving along the axis of a cylindrical tube filled with a Newtonian fluid is investigated at a small to medium larger Reynolds number. A two-layer model is adopted to simulate various possible structures of a floc, and the flow field inside is described by Darcy–Brinkman model. The results of numerical simulation reveal that a convective flow is present in the rear region of a floc when Reynolds number is on the order of 40. The presence of the tube wall and/or the porous structure of a floc has the effect of reducing that convective flow. For a fixed level of the volume-average permeability of a floc, the influence of the tube wall on the drag depends upon floc structure; the influence on a nonuniformly structured floc is more significant than that on a uniformly structured floc. The more nonuniform the floc structure, the more appreciable the deviation of the drag coefficient–Reynolds number curve from a Stokes’-law-like relation becomes. The smaller the volume-average permeability of a floc and/or the smaller the separation distance between the two flocs, the greater is the deviation, but the presence of the tube wall has the effect of reducing that deviation.     

Stability considerations of the capillary condensation within a cylindrical pore

A proof of the stability of the unduloid configuration within a cylindrical pore is given. The proof involves calculus of variation techniques and in particular the theory of the problem of Bolza is used. It is shown that the unduloid configuration is stable to small disturbances provided dV/dκ < 0, where V represents the volume of the configuration and κ is the curvature of the configuration. If this condition is not satisfied the unduloid configuration is unstable.     

Cusp at the three-fluid contact line in a cylindrical pore

In a cylindrical pore of arbitrary wettability, we analyse the existence of a three-fluid contact line connecting the fluid-fluid interfaces between two bulk phases and the third phase contained in a cusp near the pore wall. This analysis is supported by the very similar, but simpler, analysis for a constriction between parallel plates. From the force balance at the contact line and the equations for the interface curvatures we derive expressions for the cusp height and for the capillary entry pressure related to piston-like displacement between the two bulk phases. The latter is independent of the existence of a cusp and its phase pressure. Based on some realistic assumptions, of which the most important is that a cusp grows continuously from the onset when its phase pressure is increased, we analyse under which conditions a cusp can exist, and, when it exists, what its behaviour is as a function of the cusp phase pressure. We find a simple criterion involving (two ratios of) the three interfacial tensions and two of the three contact angles, which determines whether the three-fluid contact line and, consequently, a cusp exists. The range of contact angles, as well as the size of the cusp increases, when the cusp phase is close to spreading. Not only cusps of the wetting phase can occur, but also of the intermediate-wetting phase. Numerical examples are presented to illustrate the range of behaviour of the cusps.     

The shape of a flexible polymer in a cylindrical pore

We calculate the mean end-to-end distance R of a self-avoiding polymer encapsulated in an infinitely long cylinder with radius D. A self-consistent perturbation theory is used to calculate R as a function of D for impenetrable hard walls and soft walls. In both cases, R obeys the predicted scaling behavior in the limit of large and small D. The crossover from the three-dimensional behavior (D --> infinity) to the fully stretched one-dimensional case (D --> 0) is nonmonotonic. The minimum value of R is found at D approximately 0.46R(F), where R(F) is the Flory radius of R at D --> infinity. The results for soft walls map onto the hard wall case with a larger cylinder radius.     

Finite-size effects in the microscopic structure of a hard-sphere fluid in a narrow cylindrical pore

We examine the microscopic structure of a hard-sphere fluid confined to a small cylindrical pore by means of Monte Carlo simulation. In order to analyze finite-size effects, the simulations are carried out in the framework of different statistical mechanics ensembles. We find that the size effects are specially relevant in the canonical ensemble where noticeable differences are found with the results in the grand canonical ensemble (GCE) and the isothermal isobaric ensemble (IIE) which, in most situations, remain very close to the infinite system results. A customary series expansion in terms of fluctuations of either the number of particles (GCE) or the inverse volume (IIE) allows us to connect with the results of the canonical ensemble.     

Monte Carlo simulations of conformations of chain molecules in a cylindrical pore

Off-lattice Monte Carlo simulations are employed to study the behavior of chain molecules confined in a long cylindrical pore under the condition of hard-body interaction. The emphasis is placed on the chain and bead distributions as well as the location dependence of the chain conformation and anisotropy. The simulation results show that the chains very near the pore surface tend to wrap around the surface in various configurations. This behavior is qualitatively similar to that of the chains near but outside a cylindrical rod. Moreover, the bead concentration near the pore surface increases with increasing surface curvature.     

Electrophoretic motion of a nanorod along the axis of a nanopore under a salt gradient

Wood-based activated carbon was modified by deposition of silver using Tollens method. Adsorbents with various contents of silver were used to study NO(2) and NO (the product of NO(2) reduction by carbon) retention. The surface of the initial and exhausted materials was characterized using adsorption of nitrogen, XRD, SEM/EDX, FTIR and TA. The results indicated that with an increasing content of silver on the surface the capacities to retain NO(2) and NO increase until the plateau is reached. The performance depends on the dispersion of nanoparticles and their chemistry. Highly dispersed small silver metal particles promote formation of chelates with NO(2) and/or with NO. An excess of Tollens reagent results in formation of larger silver crystals and silver oxide nanoparticles. If sufficiently dispersed, they also enhance the retention of NO(2) via formation of nitrates deposited in the pore system. The surface of the carbon matrix is also active in NO(2) retention, providing the small pores and edges of graphene layers, where the reductions of NO(2)/oxidation of carbon take place.     

Structure and rheological behavior of highly charged colloidal particles in a cylindrical pore I. Effect of pore size

In this work we performed nonequilibrium Brownian dynamics (NEBD) computer simulations of highly charged colloidal particles in diluted suspension under a parabolic flow in cylindrical pores. The influence of charged and neutral cylindrical pores on the structure and rheology of suspensions is analyzed. A shear-induced disorder-order-disorder-like transition was monitored for low shear rates and small pore diameters. We calculate the concentration profiles, axial distribution functions, and axial-angular pair correlation functions to determine the structural properties at steady state for a constant shear flow for different pore sizes and flow strengths. Similar behavior has been observed in a planar narrow channel in the case of charged interacting colloidal particles (M.A. Valdez, O. Manero, J. Colloid Interface Sci. 190 (1997) 81). The mobility of the particles in the radial direction decreases rapidly with the flow and becomes practically frozen. The flow exhibits non-Newtonian shear thinning behavior due to interparticle interactions and particle-wall interaction; the apparent viscosity is lower as the pore diameter decreases, giving rise to an apparent slip in the colloidal suspension. The calculated slip velocity was higher than that obtained in a rectangular slit under shear flow.     

Graphoepitaxial assembly of cylinder forming block copolymers in cylindrical holes

The graphoepitaxial assembly of cylinder‐forming block copolymers assembled into holes is investigated through theoretically informed coarse grained Monte Carlo simulations (TICG MC). The aim is to identify conditions leading to assembly of cylinders that span the entire thickness of the holes, thereby enabling applications in lithography. Three hole geometries are considered, including cylinders, elliptical cylinders, and capsule‐shaped holes. Four distinct morphologies of cylinder forming poly(styrene‐b‐methyl methacrylate) (PS‐b‐PMMA) block copolymers are observed in cylinders and elliptical holes, including cylinders, spheres, partial cylinders, and wall‐bound cylinders. Additional morphologies are observed in capsule‐shaped holes. PMMA cylinders that extend through the entire hole are found with PMMA‐wetting surfaces; a weak wetting condition is needed on the bottom of the hole and a strong wetting condition is necessary on the sides of the hole. Simulated are also used to explore the morphologies that arise when holes are overfilled, or when PMMA homopolymers are added in blends with copolymers. We find that overfilling can alter considerably the morphological behavior of copolymers in cylinders and, for blends; we find that when the homopolymer concentration is >10%, the range of conditions for formation of PMMA cylinders that extend through the entire hole is increased. In general, results from simulations (TICG) are shown to be comparable to those of self‐consistent (SCFT) calculations, except for conditions where fluctuations become important. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 430–441