Motion of a particle generated by chemical gradients. Part 2. Electrolytes
TL;DR: In this article, a matched asymptotic expansion of a small parameter L/a, where a is the particle radius and L is the length scale characteristic of the physical interaction between solute and particle surface, was used to obtain an expression for particle velocity.
Abstract: When a particle is placed in a fluid in which there is a non-uniform concentration of solute, it will move toward higher or lower concentration depending on whether the solute is attracted to or repelled from the particle surface. A quantitative understanding of this phenomenon requires that the equations representing conservation of mass and momentum within the fluid in the vicinity of the particle are solved. This is accomplished using a method of matched asymptotic expansions in a small parameter L/a, where a is the particle radius and L is the length scale characteristic of the physical interaction between solute and particle surface. This analysis yields an expression for particle velocity, valid in the limit L/a → 0, that agrees with the expression obtained by previous researchers. The result is cast into a more useful algebraic form by relating various integrals involving the solute/particle interaction energy to a measurable thermodynamic property, the Gibbs surface excess of solute Γ. An important result is that the correction for finite L/a is actually O(Γ/C∞ a), where C∞ is the bulk concentration of solute, and could be O(1) even when L/a is orders of magnitude smaller.
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TL;DR: The existence of a slip velocity at solid/fluid interfaces opens a class of flow problems not generally recognized by the fluid-dynamics community as mentioned in this paper, and the existence of slip velocities at solid and fluid interfaces has been studied in the literature.
Abstract: In a historical context the interface between two phases has played only a minor role in the physics of fluid dynamics. It is of course true that boundary conditions at interfaces, usually imposed as continuity of ve locity and stress, determine the velocity field of a given flow; however, this is a more or less passive use of the interface that allows one to ignore the structure of the transition between two phases. When an interface has been assigned a more active role in flow processes, it generally has been assumed that one parameter, the interfacial (surface) tension, accounts for all mech anical phenomena (Young et al. 1 959, Levich & Krylov 1969). In these studies, kinematic effects of the interface were not considered, and the "no-slip" condition on the velocity at interfaces was retained. The basic message of this article is that the interface is a region of small but finite thickness, and that dynamical processes occurring within this region lead not only to interfacial stresses but also to an apparent "slip velocity" that, on a macroscopic length scale, appears to be a violation of the no-slip condition. The existence of a slip velocity at solid/fluid interfaces opens a class of flow problems not generally recognized by the fluid-dynamics community. Three previous articles in this series deal with flow caused by interactions between interfaces and external fields such as electrical potential, tem perature, and solute concentration. Melcher & Taylor ( 1969) and Levich & Krylov (1969) consider fluid/fluid interfaces where stresses produced at the interface by the external field dictate the flow. Saville ( 1977), on the other hand, discusses the action of an electric field on a charged solid/fluid interface and reviews the currently accepted model for electrophoretic
1,343 citations
TL;DR: The response of a model microelectrochemical system to a time-dependent applied voltage is analyzed, including electrochemistry, colloidal science, and microfluidics, including surface conduction, multicomponent electrolytes, and Faradaic processes.
Abstract: The response of a model microelectrochemical system to a time-dependent applied voltage is analyzed. The article begins with a fresh historical review including electrochemistry, colloidal science, and microfluidics. The model problem consists of a symmetric binary electrolyte between parallel-plate blocking electrodes, which suddenly apply a voltage. Compact Stern layers on the electrodes are also taken into account. The Nernst-Planck-Poisson equations are first linearized and solved by Laplace transforms for small voltages, and numerical solutions are obtained for large voltages. The "weakly nonlinear" limit of thin double layers is then analyzed by matched asymptotic expansions in the small parameter epsilon= lambdaD/L, where lambdaD is the screening length and L the electrode separation. At leading order, the system initially behaves like an RC circuit with a response time of lambdaDL/D (not lambdaD2/D), where D is the ionic diffusivity, but nonlinearity violates this common picture and introduces multiple time scales. The charging process slows down, and neutral-salt adsorption by the diffuse part of the double layer couples to bulk diffusion at the time scale, L2/D. In the "strongly nonlinear" regime (controlled by a dimensionless parameter resembling the Dukhin number), this effect produces bulk concentration gradients, and, at very large voltages, transient space charge. The article concludes with an overview of more general situations involving surface conduction, multicomponent electrolytes, and Faradaic processes.
938 citations
TL;DR: In this article, an ion concentration polarization (ICP) was used to separate micro-and nano-sized particles based on their electrophoretic mobilities, which was performed using a strong electric field in the depletion region without the use of internal electrodes.
Abstract: We proposed a novel separation method, which is the first report using ion concentration polarization (ICP) to separate particles continuously. We analyzed the electrical forces that cause the repulsion of particles in the depletion region formed by ICP. Using the electrical repulsion, micro- and nano-sized particles were separated based on their electrophoretic mobilities. Because the separation of particles was performed using a strong electric field in the depletion region without the use of internal electrodes, it offers the advantages of simple, low-cost device fabrication and bubble-free operation compared with conventional continuous electrophoretic separation methods, such as miniaturizing free-flow electrophoresis (μ-FFE). This separation device is expected to be a useful tool for separating various biochemical samples, including cells, proteins, DNAs and even ions.
863 citations
TL;DR: In this paper, it was shown that the "compact layer" and "shear plane" effectively advance into the liquid, due to the crowding of counterions, and that ionic crowding against a blocking surface expands the diffuse double layer and thus decreases its differential capacitance; each trend is enhanced by dielectric saturation.
800 citations
TL;DR: This review provides an introduction to the theory of nanofluidic transport, focusing on the various forces that influence the movement of both solvents and solutes through nanochannels, and reviews the applications of nan offluidic devices in separation science and energy conversion.
Abstract: The evolution from microfluidic to nanofluidic systems has been accompanied by the emergence of new fluid phenomena and the potential for new nanofluidic devices. This review provides an introduction to the theory of nanofluidic transport, focusing on the various forces that influence the movement of both solvents and solutes through nanochannels, and reviews the applications of nanofluidic devices in separation science and energy conversion.
736 citations
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TL;DR: In this article, it was shown that for a colloidal particle of any shape the mobility is independent of the dielectric properties of the particle and the electrostatic boundary conditions on the particle surface.
Abstract: The equations which govern the ion distributions and velocities, the electrostatic potential and the hydrodynamic flow field around a solid colloidal particle in an applied electric field are reexamined. By using the linearity of the equations which determine the electrophoretic mobility, we show that for a colloidal particle of any shape the mobility is independent of the dielectric properties of the particle and the electrostatic boundary conditions on the particle surface. The mobility depends only on the particle size and shape, the properties of the electrolyte solution in which it is suspended, and the charge inside, or electrostatic potential on, the hydrodynamic shear plane in the absence of an applied field or any macroscopic motion.New expressions for the forces acting in the particle are derived and a novel substitution is developed which leads to a significant decoupling of the governing equations. These analytic developments allow for the construction of a rapid, robust numerical scheme for the solution of the governing equations which we have applied to the case of a spherical colloidal particle in a general electrolyte solution. We describe a computer program for the conversion of mobility measurements to zeta potential for a spherical colloidal particle which is far more flexible than the Wiersema graphs which have traditionally been used for the interpretation of mobility data. Furthermore it is free of the high zeta potential convergence difficulties which limited Wiersema's calculations to moderate values of ζ. Some sample computations in typical 1:1 and 2:1 electrolytes are exhibited which illustrate the existence of a maximum in the mobility at high zeta potentials. The physical explanation of this effect is given. The importance of the mobility maximum in testing the validity of the governing equations of electrophoresis and its implications for the colloid chemist's picture of the Stern layer are briefly discussed.
1,563 citations
TL;DR: In this article, it has been demonstrated that small bubbles in pure liquids can be held stationary or driven downwards by means of a sufficiently strong negative temperature gradient in the vertical direction, due to the stresses resulting from the thermal variation of surface tension at the bubble surface.
Abstract: It has been observed experimentally that small bubbles in pure liquids can be held stationary or driven downwards by means of a sufficiently strong negative temperature gradient in the vertical direction. This effect is demonstrated to be due to the stresses resulting from the thermal variation of surface tension at the bubble surface. The flow field within and around the bubble is derived, and an expression for the magnitude of the temperature gradient required to hold the bubble stationary is obtained. This expression is verified experimentally.
894 citations
TL;DR: In this paper, Smoluchowski deduced the expression U = DXζ/4π η (1) for the cataphoretic velocity U ; X is the applied field strength, ζ the potential difference across the double layer, D the dielectric constant and η the viscosity of the medium.
Abstract: § 1. The theory of cataphoresis, and of the complementary phenomenon of electrosmosis, is based on the conception of an “ electrical double layer ” at the interface between the two phases whose relative motion is under consideration.* In the original theory, as propounded by Quincke and Helmholtz, this electrical double layer was regarded as a kind of parallel plate condenser made up of two laminar distributions of electrification, of which one—the so-called “ inner sheet ”—was firmly attached to the rigid phase, while the other—the “ outer sheet ”—resided in the mobile phase ; the separation between the two was considered to be a distance of the order of molecular dimensions. The currently accepted view, initiated by Gouy, differs from that of Helmholtz chiefly in that the outer sheet of the double layer is considered to be a diffuse distribution of electrification—an “ ionic atmosphere ” of the type investigated by Debye and his collaborators in connection with the theory of strong electrolytes. The net electric density in the ionic atmosphere varies continuously from a maximum in the immediate neighbourhood of the fixed inner sheet, to a negligibly small value in the bulk of the liquid, over a distance which is a function of the ionic concentration, and which lies as a rule between molecular dimensions and some thousand micromillimetres. In a deduction which appears to be completely consistent with this more modern view of the double layer, Smoluchowski deduced the expression U = DXζ/4π η (1) for the cataphoretic velocity U ; X is the applied field strength, ζ the potential difference across the double layer, D the dielectric constant and η the viscosity of the medium. The equation is identical with that developed by Helmholtz except for the inclusion of the dielectric constant, but was deduced on a much more general basis, and is claimed by Smoluchowski to be valid for rigid electrically insulating particles of any shape, subject only to the following four restrictions :— 1) That the usual hydrodynamical equations for the motion of a viscous fluid may be assumed to hold both in the bulk of the liquid and within the double layer; (2) That the motion is “stream line motion,” and slow enough for the “inertia terms” in the hydrodynamic equations to be neglected ; (3) That the applied field may be taken as simply superimposed on the field due to the electrical double layer ; and (4) That the thickness of the double layer ( i. e, the distance normal to the interface over which the potential differs appreciably from that in the bulk of the liquid) is small compared with the radius of curvature at any point of the surface.
821 citations
01 Oct 1970
TL;DR: In this paper, the electrophoresis of an arbitrary shape in an unbounded fluid is treated analytically, subject to the restrictions that the local Debye length is much smaller than the local radii of curvature and surface conductivity is negligible.
Abstract: The electrophoresis of an insulating body of arbitrary shape in an unbounded fluid is treated analytically, subject to the restrictions that the local Debye length is much smaller than the local radii of curvature and surface conductivity is negligible. The flow is shown to be irrotational. The Smoluchowski equation is shown to be valid for a particle of any shape, as is frequently assumed.
224 citations
TL;DR: In this article, it is shown that the methods devised by Dukhin for solving the equations for a symmetric two species electrolyte can be simplified, and extended to the case of a general electrolyte.
214 citations