Mathematical models of continuous flow electrophoresis: Electrophoresis technology

Two aspects of continuous flow electrophoresis were studied: (1) the structure of the flow field in continuous flow devices; and (2) the electrokinetic properties of suspended particles relevant to electrophoretic separations. Mathematical models were developed to describe flow structure and stability, with particular emphasis on effects due to buoyancy. To describe the fractionation of an arbitrary particulate sample by continuous flow electrophoresis, a general mathematical model was constructed. In this model, chamber dimensions, field strength, buffer composition, and other design variables can be altered at will to study their effects on resolution and throughput. All these mathematical models were implemented on a digital computer and the codes are available for general use. Experimental and theoretical work with particulate samples probed how particle mobility is related to buffer composition. It was found that ions on the surface of small particles are mobile, contrary to the widely accepted view. This influences particle mobility and suspension conductivity. A novel technique was used to measure the mobility of particles in concentrated suspensions

Electrokinetic Phenomena and Anomalous Conduction

The effects of immobilizing a thin layer of water adjacent to the surface of a colloidal particle are calculated using the Dynamic Stern Layer model introduced by Zukoski and Saville.1 Supposing water is immobilized by, for example, unreacted monomer or polymer chains dangling from the particle surface allows one to divide the diffuse layer into two regions, an outer region where transport is by convection, electromigration and diffusion and a region near the surface where counterions move only by electromigration and diffusion. The present calculations account for all the relevant processes in a mathematically rigorous fashion. Even a thin layer has dramatic effects in diminishing particle mobility and increasing the dielectric response

Method for electrohydrodynamically assembling patterned colloidal structures

A method apparatus is provided for electrophoretically depositing particles onto an electrode, and electrohydrodynamically assembling the particles into crystalline structures. Specifically, the present method and apparatus creates a current flowing through a solution to cause identically charged electrophoretically deposited colloidal particles to attract each other over very large distances (<5 particle diameters) on the surface of electrodes to form two-dimensional colloidal crystals. The attractive force can be created with both DC and AC fields and can modulated by adjusting either the field strength or frequency of the current. Modulating this lateral attraction between the particles causes the reversible formation of two-dimensional fluid and crystalline colloidal states on the electrode surface. Further manipulation allows for the formation of two or three-dimensional colloidal crystals, as well as more complex designed structures. Once the required structures are formed, these three-dimension colloidal crystals can be permanently frozen or glued by controlled coagulation induced by to the applied field to form a stable crystalline structure

Apparatus for electrohydrodynamically assembling patterned colloidal structures

A method apparatus is provided for electrophoretically depositing particles onto an electrode, and electrohydrodynamically assembling the particles into crystalline structures. Specifically, the present method and apparatus creates a current flowing through a solution to cause identically charged electrophoretically deposited colloidal particles to attract each other over very large distances (<5 particle diameters) on the surface of electrodes to form two-dimensional colloidal crystals. The attractive force can be created with both DC and AC fields and can modulated by adjusting either the field strength or frequency of the current. Modulating this lateral attraction between the particles causes the reversible formation of two-dimensional fluid and crystalline colloidal states on the electrode surface. Further manipulation allows for the formation of two or three-dimensional colloidal crystals, as well as more complex designed structures. Once the required structures are formed, these three-dimension colloidal crystals can be permanently frozen or glued by controlled coagulation induced by to the applied field to form a stable crystalline structure

Modeling the electrophoretic deposition of colloidal particles

This letter presents the results of numerical simulations of the buildup of a layer of colloidal particles on an electrode. In a low-frequency electric field, particles suspended in a low-conductivity liquid migrate to one electrode and then to the other. During each cycle, deposits are formed and dissipated. The current-voltage characteristics of the process reflect properties of the suspension and the deposited layer. Using a flux corrected transport (FCT) algorithm, the transport equation for the particle phase is solved simultaneously with equations describing the electric field. The model reproduces the main features of the current-voltage relation.Ministerio de Ciencia y Tecnología BFM2000-105

Electrohydrodynamically patterned colloidal crystals

A method for assembling patterned crystalline arrays of colloidal particles using ultraviolet illumination of an optically-sensitive semiconducting anode while using the anode to apply an electronic field to the colloidal particles. The ultraviolet illumination increases current density, and consequently, the flow of the colloidal particles. As a result, colloidal particles can be caused to migrate from non-illuminated areas of the anode to illuminated areas of the anode. Selective illumination of the anode can also be used to permanently affix colloidal crystals to illuminated areas of the anode while not affixing them to non-illuminated areas of the anode

Electrohydrodynamic Printing and Manufacturing

An stable electrohydrodynamic filament is obtained by causing a straight electrohydrodynamic filament formed from a liquid to emerge from a Taylor cone, the filament having a diameter of from 10 nm to 100.mu.m. Such filaments are useful in electrohydrodynamic printing and manufacturing techniques and their application in liquid drop/particle and fiber production, colloidal deployment and assembly, and composite materials processing

Anisotropic Adsorption of Molecular Assemblies on Crystalline Surfaces

Orientational order of surfactant micelles and proteins on crystalline templates has been observed but, given that the template unit cell is significantly smaller than the characteristic size of the adsorbate, this order cannot be attributed to lattice epitaxy. We interpret the template-directed orientation of rodlike molecular assemblies as arising from anisotropic van der Waals interactions between the assembly and crystalline surfaces where the anisotropic van der Waals interaction is calculated using the Lifshitz methodology. Provided the assembly is sufficiently large, substrate anisotropy provides a torque that overcomes rotational Brownian motion near the surface. The probability of a particular orientation is computed by solving a Smoluchowski equation that describes the balance between van der Waals and Brownian torques. Torque aligns both micelles and protein fibrils; the interaction energy is minimized when the assembly lies perpendicular to a symmetry axis of a crystalline substrate. Theoretical predictions agree with experiments for both hemi-cylindrical micelles and protein fibrils adsorbed on graphite. Introduction Surfactants in aqueous solution form micelles due, in part, to the limited solubility of their hydrocarbon tails. Spherical, cylindrical, bilayer, and bicontinuous structures occur, depending on the characteristics of the molecules and their concentrations. Orientational relationships appear insensitive to the composition of the adsorbate. For example, Aksay et al. In a recent study, First, we develop a statistical model based on the balance between van der Waals and Brownian torques for calculation of the probability of a particular orientation. The model is then applied to two different molecular assemblies: hemi-cylindrical micelles and cylindrical protein fibrils. The calculations confirm that the torque experienced by a molecular aggregate is sufficient to overcome the Brownian forces so that an assembly aligns with respect to a symmetry axis of the graphite. Detailed treatments of the methodologies are given in the appendixes. Finally, we discuss the characteristics of molecular assembly such as its dynamic character, waviness, and interactions between assemblies. The formulation of the van der Waals interaction between a rod and an anisotropic surface is set out in the appendixes for cylindrical and hemi-cylindrical rods following the treatment by Parsegian and Weiss. 22 The detailed calculation of the dielectric properties of the graphite is also set out there, along with a discussion of the dielectric functions employed for the micelles and the protein fibrils. A Probabilistic Description of Orientational Ordering. Anisotropic interactions between a cylindrical or hemi-cylindrical rod and a surface mimic those between a molecular assembly and an inorganic surface. The orientational ordering is pictured as follows. In solution, each molecule or assembly undergoes Brownian motion while being attracted to the surface by ubiquitous van der Waals forces. On the surface, rodlike structures form where the van der Waals interaction is anisotropic and induces a torque that alters the probability of a particular orientation. Near a surface, certain orientations are favored, and random rotational motion is suppressed. However, rotary motion parallel to the surface continues, and raft-like aggregates (cf. To describe the rotational motion parallel to the surface, we use a Smoluchowski equation As described in the appendixes, the van der Waals interaction energy between a cylindrical or hemi-cylindrical rod of length, l, and radius, a, and an anisotropic substrate is where A H aniso (θ) denotes a Hamaker constant with θ being the angle between the long axis of the rod and one of the symmetry axes of a substrate, as indicated in Anisotropic Adsorption of Molecular Assemblies J. Phys. Chem. B C of eq 2 and the probability distribution function can be found from solutions of where A′ is obtained from differentiation of the anisotropic Hamaker constant with respect to p x , and scaled on kT. The right-hand side of eq 4 is proportional to the torque experienced by a rod at a certain separation R from the surface. From eq 4, the probability function ψ follows as where C is an integration constant. Upon normalization, the probability density function f(p x ) is with  ) A′a 2 l/6R 3 . Clearly the probability density becomes uniform, i.e., f(p x ) ) 1, for isotropic substrates where  ) 0. For  &gt; 0, the probability is maximum for p x ) 0 (where rods are aligned perpendicular to the symmetry axis) and a minimum at p x ) 1. Using the maximum and minimum probabilities, the tendency for the alignment of molecular assembly can be evaluated as the relative probability ratio: where P parallel and P perpendicular denote probabilities for parallel and perpendicular configurations of the rod, respectively. Anisotropic Adsorption of Micelles and Protein Fibrils. Next we look at two applications of the theory, first for micelles and then for protein fibrils. For hemi-cylindrical micelles on HOPG in an aqueous solution, we use the formulation for a hemi-cylindrical rod, eq A.24. The series representation for the Hamaker function A H aniso given in the appendix was summed into ultraviolet region (∼10 17 rad/s); the wave vector integration employed a 16-point Gauss-Laguerre quadrature method. 24 The data summarized in the appendix enable calculation of the dimensionless torque parameter A′ for a hemi-cylindrical rod interacting with graphite across a thin film of water. When the rod is long and close to the surface, the relevant quantity  can be sizable even though A′ is small, e.g., 1.991 × 10 -5 . 7 To analyze the orientation of protein fibrils, we use the formulation for a cylindrical rod, eq A.18 with the Hamaker function summed as before. Taking the radius of a fibril as half of the length of an individual protein strand (i.e., a ) 1.75 nm) and A′ ) 9.4614 × 10 -5 , we calculate the relative probability ratio between the graphite surface and protein fibril for different fibril lengths and separations. Additional evidence for anisotropic adsorption is furnished by calculating the probability as a function of orientation at a particular separation. 1 dp x exp ( - Figure 5. Relative probability relation for hemi-cylindrical micelles (a ) 2.5 nm) of different lengths adsorbed on graphite as a function of separation between a hemi-cylindrical micelle and the surface. A large relative probability represents a strong tendency for alignment perpendicular to the symmetry axis of the graphite. Discussion The theory presented in the previous sections improves our understanding of the anisotropic adsorption for both micelles and protein fibrils but does not cover some characteristics of molecular assembly: (i) the dynamic nature, (ii) micelle flexibility or waviness, and (iii) interactions between assemblies. In this section, we discuss each and its implication for the theory. Dynamic Character of Molecular Assembly. In the present theory, we neglect the dynamics of the micelle disintegration and reassembly shown in However, to gain a complete understanding of the anisotropic adsorption, we should employ a temporal change of length of the molecular assembly due to continuous disintegration and reformation of molecular assembly. The temporal change of length might be treated as a &quot;reaction&quot; of molecular assembly whose rate constant is related to the disintegration and reformation energies of the molecular assembly. Flexibility or Waviness of Molecular Assembly. The theory presented above describes molecular assemblies as rigid rods. Molecular assemblies are, however, inherently &quot;flexible&quot; or &quot;wavy&quot; owing to an existence of a finite persistence length. A persistence length of rodlike molecular assembly can be understood as a length scale where elastic and thermal energies are in balance. The persistence length is typically O(10) nm for cylindrical micelle Conclusions We have set out a detailed theory for the anisotropic adsorption of molecular assemblies on crystalline surfaces. A particular orientation is favored because rotational motion is influenced by a van der Waals torque; orientation arises from the interaction between an anisotropic object (a rod) and an anisotropic substrate. Although the intrinsic torque parameter is small, the combination of a relatively large rod volume and close separation provides the necessary torque. The theory presented provides an explicit relationship between the energy and the orientational order, involving simple geometry and physical properties. The theory also captures the essential features of the anisotropic adsorption of a molecular assembly and provides a basic formalism to which further detailed considerations can be incorporated, such as the dynamic character of molecular assemblies, the waviness of molecular assemblies, the interaction between molecular assemblies. Acknowledgment. We are grateful for discussions with H. C. Schniepp

A boundary-layer analysis of heat and mass transfer in free convection around horizontal cylindrical bodies

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