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ABSTRACT: The Poisson–Boltzmann equation (P-B) is used as an analytic model in a wide variety of fields in chemistry and physics, because
it describes the charge distribution in a solute. Being highly nonlinear, there are only a few known solutions for simple
boundary geometries and, beyond, iterative numerical schemes are often employed. This study, on the other hand, presents a
systematic perturbation solution of the P-B using a non-dimensional electrokinetic–thermal energy ratio λ which, when it approaches
zero, reduces the P-B to the Debye–Hückel approximation. Perturbation-series solutions are obtained for five basic examples,
and lead to the surprising result that, even when λ is as large as 3 or larger, the perturbation solution is very accurate
with only a few terms included in the series. This is because the perturbation analysis generates very rapidly vanishing coefficients
at higher-order approximations. This result has the important implication that the perturbation method presented in this study
could be applied quite generally for investigating more complicated problems.
KeywordsDebye–Hückel–Electric-charge potential–Electrokinetics–Plate–Sphere–Cylinder–Perturbation series–Poisson–Boltzmann
Journal of Engineering Mathematics 05/2012; 70(4):333-342. · 0.86 Impact Factor
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ABSTRACT: The present study is concerned with unsteady electroosmotic flow (EOF) in a microchannel with the electric charge distribution described by the Poisson-Boltzmann (PB) equation. The nonlinear PB equation is solved by a systematic perturbation with respect to the parameter λ which measures the strength of the wall zeta potential relative to the thermal potential. In the small λ limits (λ<1), we recover the linearized PB equation - the Debye-Hückel approximation. The solutions obtained by using only three terms in the perturbation series are shown to be accurate with errors <1% for λ up to 2. The accurate solution to the PB equation is then used to solve the electrokinetic fluid transport equation for two types of unsteady flow: transient flow driven by a suddenly applied voltage and oscillatory flow driven by a time-harmonic voltage. The solution for the transient flow has important implications on EOF as an effective means for transporting electrolytes in microchannels with various electrokinetic widths. On the other hand, the solution for the oscillatory flow is shown to have important physical implications on EOF in mixing electrolytes in terms of the amplitude and phase of the resulting time-harmonic EOF rate, which depends on the applied frequency and the electrokinetic width of the microchannel as well as on the parameter λ.
Electrophoresis 11/2011; 32(23):3341-7. · 3.30 Impact Factor
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ABSTRACT: The electro-osmotic flow driven by a screen pump, composed of a line array of evenly spaced identical rectangular solid blocks, is investigated under the Debye-Hückel approximation. The geometry of the screen pump is determined by the spacing and aspect ratio of the solid blocks. A constant surface zeta potential is assumed on the block surface. The method of eigenfunction series expansion is applied to solve analytically for the applied electric field, electric charge potential in the fluid, and flow field. Because of the low Reynolds number, Stokes equations are applied for the flow. The analytic result is first confirmed by comparing with the exact solution of the electro-osmotic flow in an infinite channel. Then different geometries of the screen pump and the effect of the electrokinetic width are computed for their influence on the flow rate. Recirculating eddies and reversing flow are found even though the applied electric driving field is unidirectional.
Physical Review E 09/2011; 84(3 Pt 2):036301. · 2.26 Impact Factor
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ABSTRACT: The initial electroosmotic flow through a small pore or microchannel with annular or rectangular cross section is studied under the Debye-Hückel approximation. Analytical series solutions and their asymptotic behavior for small and large non-dimensional electrokinetic widths are found for these two basic cases. The explicit and accurate solutions are particularly useful for examining various geometric/physical effects on the transient time scales and the flow rates for the transient states. The steady flow rate of the smaller channel may be disproportionately smaller than a reference channel if the electric double layer is thick, but will be in close proportion to the area ratio if the electric double layer is thin. A smaller channel compared to a reference channel has a shorter transient time scale, and the transient flow has characters very different from the steady state if the electric double layer is thin. The total transient flow rate of several smaller pores or channels may exceed largely that of a single large pore or channel with the same total cross section on the transient time scale of the smaller channels. The results have important implications on liquid transport in micropores or channels by pulse voltages or more general time-varying voltages.
Electrophoresis 07/2008; 29(14):2970-9. · 3.30 Impact Factor
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ABSTRACT: An efficient Ritz method is developed from the variational principle to solve the Poisson–Boltzmann equation under the Debye–Hückel approximation for studying the EOF in microchannels. The method is applied to the family of superelliptic cross sections which includes the elliptic channel and the rectangular channel as limiting cases. Several accurate tables presented are useful for design of electroosmotic channels, especially rectangular channels with rounded corners. It is shown how the flow rate Q is a sophisticated consequence of the nondimensional electrokinetic width K, the aspect ratio b as well as the superelliptic exponent n.
Electrophoresis 08/2007; 28(18):3296 - 3301. · 3.30 Impact Factor
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ABSTRACT: In this study, we employ the interfacial operator approach to compute surface plasmon modes as well as band structures (including longitudinal modes) for plasmonic crystals in one and two dimensions. In particular, we consider the free-electron model for the metal. It is shown that the localized feature of surface plasmon modes can be resolved near the interface by introducing interfacial variables. For a one-dimensional array of metal, convergence of two branches of surface plasmon modes is studied by varying the filling fraction of the metal. For two-dimensional metallic structures, band flattening, band broadening, and plasmonic band gaps are observed and discussed. The highly degenerate nature and infinite number of surface plasmon modes can be explained by employing the Rayleigh quotient for the TE modes. The cutoff behavior in the TM modes is made clear by considering the energy density of the electromagnetic fields. The transverse electric fields, surface charges, and polarization currents are visualized to help understand various properties of surface plasmon modes. Moreover, the effect of plasma frequency and the transition from dispersive metals to perfect conductors are also explored. Finally, the contribution of Drude damping is considered by perturbation analysis.
Physical Review E 04/2006; 73(3 Pt 2):036605. · 2.26 Impact Factor
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ABSTRACT: In this study, two fast and accurate methods of inverse iteration with multigrid acceleration are developed to compute band structures of photonic crystals of general shape. In particular, we report two-dimensional photonic crystals of silicon air with an optimal full band gap of gap-midgap ratio Deltaomega/omega(mid)=0.2421, which is 30% larger than ever reported in the literature. The crystals consist of a hexagonal array of circular columns, each connected to its nearest neighbors by slender rectangular rods. A systematic study with respect to the geometric parameters of the photonic crystals was made possible with the present method in drawing a three-dimensional band-gap diagram with reasonable computing time.
Physical Review E 09/2003; 68(2 Pt 2):026704. · 2.26 Impact Factor
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ABSTRACT: This study presents a solvable model in renormalization group analysis for the effective eddy viscosity. It is found fruitful to take a simple hypothesis that large-scale eddies are statistically independent of those of smaller scales. A limiting operation of renormalization group analysis yields an inhomogeneous ordinary differential equation for the invariant effective eddy viscosity. The closed-form solution of the equation facilitates derivations of an expression of the Kolmogorov constant C(K) and of the Smagorinsky model for large-eddy simulation of turbulent flow. The Smagorinsky constant C(S) is proportional to C(3/4)(K). In particular, we shall illustrate that the value of C(K) ranges from 1.35 to 2.06, which is in close agreement with the generally accepted experimental values (1.2 approximately 2.2).
Physical Review E 05/2003; 67(4 Pt 2):047301. · 2.26 Impact Factor
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Chien C. Chang
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ABSTRACT: This study presents a solvable model in renormalization group analysis for the effective eddy viscosity. It is found fruitful to take a simple hypothesis that large-scale eddies are statistically independent of those of smaller scales. A limiting operation of renormalization group analysis yields an inhomogeneous ordinary differential equation for the invariant effective eddy viscosity. The closed-form solution of the equation facilitates derivations of an expression of the Kolmogorov constant CK and of the Smagorinsky model for large-eddy simulation of turbulent flow. The Smagorinsky constant CS is proportional to CK3/4. In particular, we shall illustrate that the value of CK ranges from 1.35 to 2.06, which is in close agreement with the generally accepted experimental values (1.2∼2.2).
Phys. Rev. E. 04/2003; 67(4).
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ABSTRACT: It has been difficult to compute the band structures for photonic crystals with metallic components included in the periodic units. The existence of modes of surface plasmon polariton presents the major difficulty not only because of the localized nature of the modes but also of the apparent necessity of handling a nonlinear eigenvalue problem. Here we show that by introducing an interfacial operator within the finite-difference framework, we are able to formulate the problem for computing modes of surface plasmon polariton in the format of standard eigenvalue problems. Results are uncovered by applying the method to periodic structures with corrugated interfaces between metals and dielectric materials, as well as other classes of interfaces.
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ABSTRACT: The electro-osmotic flow through a microchannel with a semicircular cross section is studied under the Debye–Huckel approximation. Analytical series solutions are found for two basic cases. The solutions for the two basic cases considered can be superposed to yield solutions for any combination of constant zeta potentials on the flat or curved wall boundaries. Moreover, in the limit of a thin electric double layer small Debye length compared to the nominal dimension, a method of solution is shown for variable zeta potentials by using the Smoluchowski slip approximation.
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ABSTRACT: This study is aimed at investigation of propagating modes of acoustic waves in periodic solid layers in ideal or viscous fluids. In particular, at the long-wavelength limit, a three-scale homogenization analysis is devel-oped to derive the effective group velocities in analytical forms for the shear-vertical SV waves as well as for the longitudinal-shear-horizontal P-SH waves. It is found that propagating modes, i.e., modes with real group velocities, may be supported even if the fluid phase is viscous. A criterion for the existence of a vanishing effective viscosity is derived based on composite medium constants and the filling ratio of the fluid phase. The critical filling ratios at which an evanescent mode changes to a propagating mode are given for various solid-water systems.
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ABSTRACT: In this study, we continue with a recursive renormalization group (RG) analysis of incompressible turbulence, aiming at investigating various turbulent properties of three-dimensional magneto-hydrodynamics (MHD). In particular, we are able to locate the fixed point (i.e. the invariant effective eddy viscosity) of the RG transformation under the following conditions. (i) The mean magnetic induction is relatively weak compared to the mean flow velocity. (ii) The Alfvén effect holds, that is, the fluctuating velocity and magnetic induction are nearly parallel and approximately equal in magnitude. It is found under these conditions that re-normalization does not incur an increment of the magnetic resistivity, while the coupling effect tends to reduce the invariant effective eddy viscosity. Both the velocity and magnetic energy spectra are shown to follow the Kolmogorov k À5=3 in the inertial subrange; this is consistent with some laboratory measurements and observations in astronomical physics. By assuming further that the velocity and magnetic induction share the same specified form of energy spectrum, we are able to determine the dependence of the (magnetic) Kolmogorov constant C K (C M) and the model constant C S of the Smagorinsky model for large-eddy simulation on some characteristic wavenumbers.
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ABSTRACT: In this study, we propose an interfacial operator approach to compute surface phonon modes for one- and two-dimensional periodic arrays of polar materials in a finite-difference formulation. The key aspect of the approach is to introduce an interfacial variable along the interface between the polar material and the surrounding dielectric material, which represents the local strength of the surface phonon modes along the interface. In this approach, the apparently nonlinear eigenvalue problem can be reformulated as a quadratic eigensystem, and thus further reduced to a standard linear eigenvalue problem. Band structures can be computed directly without the need of examining transmission spectra as in the finite-difference time-domain method, or locating the mode frequency by testing an auxiliary function in other methods. Applying the method to four different types of photonic crystals of polar materials, we are able to uncover several interesting results by studying the effect of dimension, the size (filling ratio) effect, the effects of the transverse optical phonon frequency (ωT), and longitudinal optical phonon frequency (ωL) as well as the effect of shape or geometry of the polar material.
Phys. Rev. B. 73(23).
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ABSTRACT: In this paper, we investigate surface and bulk modes for periodic structures made of negative index materials in one and two dimensions. The negative index material is a composite medium consisting of a network of thin wires and a periodic array of split ring resonators. In different ranges of frequencies, we identify five types of modes: surface plasmon (SP) modes for TE polarization, magnetic surface plasmon (MSP) modes for TM polarization, trapped modes or resonant cavity modes for both TM and TE polarizations, asymmetric surface (AS) modes for TE polarization, and some bulk modes for both polarizations in the range of negative material properties. In particular, we will discuss band flattening and broadening of SP and MSP modes, explain the trapped modes in terms of large positive dielectric constants, examine the properties of AS modes by an interface condition, and use the Rayleigh quotient to account for possibly infinite degeneracy of SP and MSP modes in two dimensions. All of the physical properties are computed by an interfacial operator approach in which we introduce an interfacial variable to measure the local strengths of various surface modes.
Phys. Rev. B. 74(15).
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ABSTRACT: In this study, we investigate the band structures of phononic crystals with particular emphasis on the effects of the mass density ratio and of the contrast of elastic constants. The phononic crystals consist of arrays of different media embedded in a rubber or epoxy. It is shown that the density ratio rather than the contrast of elastic constants is the dominant factor that opens up phononic band gaps. The physical background of this observation is explained by applying the theory of homogenization to investigate the group velocities of the low-frequency bands at the center of symmetry Γ.
Phys. Rev. B. 75(5).
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ABSTRACT: In this study, we investigate the effect of metallic inclusion modeled as perfect conductor on a dielectric photonic crystal (silicon/air) with large full band gap. The dielectric crystal consists of a hexagonal array of circular dielectric columns, each connected to its nearest neighbors by slender rectangular rods. It is found that inclusion of small metallic components inside the circular dielectrics sharply “turns off“ the full band gap of the dielectric photonic crystal. By increasing the radius of metallic inclusion above a threshold value, the full band gap (of the metallodielectric photonic crystal) makes its appearance again and continues to grow in size. On the other hand, metallic inclusion in the air region shows an opposite trend that the full band gap is not turned off, and its size diminishes gradually to zero with increasing the radius of inclusion. These peculiar behaviors can be explained on a unified basis by examining different types of boundary conditions for TM and TE modes, and employing variational arguments based on Rayleigh’s quotients. Moreover, the free-electron model for metallic components is also considered for TM modes. At large plasma frequencies, these modes show very close band structures to those described above for the case of perfect conductors.
Phys. Rev. B. 70(7).
