Electroviscous Forces Research Articles

Capillary Orientation and Morphology Dictated Oscillatory Electro-magneto-imbibition of Viscoelastic Electrolytes.

We explore the intricate dynamics of imbibition by a viscoelastic electrolyte within an arbitrarily oriented, nonuniform microcapillary, while under the stimulus of external electromagnetic fields and internal electroviscous forces stemming from streaming potential. The microcapillary walls are envisaged to be tapered relative to each other, with the entire system inclined with respect to the horizontal plane. The rheological behavior of the electrolyte is characterized using the Phan-Thien-Tanner (PTT) model. To manipulate the imbibition dynamics, external transverse magnetic and electric fields are imposed. Incorporating all contributing forces, we obtain semianalytical formulations for the velocity and shear stress distributions. We identify distinct stages during the imbibition process: (i) initial, (ii) filling, (iii) oscillatory, and (iv) stagnation stages. Moreover, we scrutinize the impact of four pivotal parameters, namely, the Weissenberg number (Wi), the Hartmann number (Ha), the transverse electric to viscous force (EVF) number (S), and the electric Reynolds number (Ree), on the imbibition dynamics across different inclinations and taper angles. We also delineate the parameter space for these four parameters, identifying where the onset of oscillations occurs. Finally, through scaling analysis, we establish the existence of four distinct regimes corresponding to the aforementioned stages: (i) the linear regime, (ii) the Washburn regime, (iii) the oscillatory regime, and (iv) the equilibrium regime. Our findings are anticipated to enhance the understanding of capillary imbibition under such complex flow conditions and contribute significantly to the advancement of capillarity-driven microfluidic devices.

Improving ion rejection by optimizing the structure and charge distribution of the Janus membrane

Due to the asymmetry of charge distribution and geometric structure, the Janus membrane can simultaneously intercept counter-ions and co-ions. This work constructed four Janus membrane configurations using numerical simulation methods. Due to the different potential within the membrane and the selectivity level, there is a significant deviation in the interception performance. The results found that the change in the height of the Janus membrane affects the distribution of anion and cation concentration inside the nanochannel, and the asymmetry of the nanochannel height is more conducive to interception than a uniform nanochannel. The effect of the nanochannel length of the first section on the rejection rate is higher than that of the second section. Considering the design of the membrane, we recommend that the pore length of the first section be set to about 50 nm. Since the flow potential balances the electroviscous force, the optimization of the flow rate for ion transport is often opposite to the interception effect. The Janus membrane has asymmetric pore size and charge distribution characteristics and has excellent retention ability for divalent salts under low-pressure intensity and high-concentration conditions.

Micro-imbibition electro-magneto-hydrodynamics of viscoelastic fluids with interactive streaming potential

• Imbibition electro-magneto-hydrodynamics of ionic viscoelastic fluid explored • Capillarity of a sPTT fluid with induced streaming potential solved semi-analytically • Roles of Hartmann number, Electroviscous number, and Deborah number investigated • The critical Hartmann number decreases with increase in Deborah number • Capillarity with rheology independent ( x ¯ ∼ t ¯ ), and later, rheology modulated regime ( x ¯ ∼ t ¯ ). Conjugated surface tension and electro-magneto-hydrodynamics driven micro-capillary imbibition of an ionic viscoelastic fluid, in presence of interacting streaming potential, has been investigated in this article. The rheology of the fluid is modelled by appealing to the linearized simplified Phan-Thien Tanner (sPTT) model. By taking into account the electrolytic nature of the fluid, additional electroviscous forces due to induced streaming potential is also considered in the model. Transverse electric and magnetic fields are applied externally to the fluid to investigate the possibility of achieving volumetric flow gain and/or fluidic control using body force manipulation. Considering all the participating mechanisms, we deduce a semi-analytical formalism and present a comprehensive picture on the dependence of various parameters on capillary imbibition dynamics. We explore the roles played by the governing non-dimensional parameters, viz. Hartmann number, Electroviscous number, and Deborah number. For low Hartmann number regimes, there is evidence of augmented imbibition, while at higher values the flow is suppressed. We also show that the critical Hartmann number decreases with increase in Deborah number, allowing for such viscoelastic fluids to achieve more efficient imbibition velocity with less strength of applied fields. The flow regime is noted to consist of two parts – a rheology independent regime ( x ¯ ∼ t ¯ ), and later, a rheology modulated regime ( x ¯ ∼ t ¯ ). These findings are expected to provide a significant understanding of the imbibition hydrodynamics of such complex fluids, and at the same time, provide means for flow augmentation and control in microfluidic devices.

Electroviscous effect for a confined nanosphere in solution.

A charged colloidal particle suspended in an electrolyte experiences electroviscous stresses arising from motion-driven electrohydrodynamic phenomena. Under certain conditions, the additional contribution from electroviscous drag forces to the total drag experienced by the moving particle can lead to measurable deviations of particle diffusion coefficients from values predicted by the well known Stokes-Einstein relation that describes diffusive behavior of small particles in an unbounded charge-free fluid. In this study, we investigate the role of electroviscous stresses on nanoparticle diffusion in confined geometries using both simulations and experiment. We compare our experimental measurements with the results of a numerically solved continuum model based on the Poisson-Nernst-Planck-Stokes system of equations and find good agreement between experiment and theory. Depending on the radius of the counterion species in solution and the degree of confinement, we find that the viscous drag on polystyrene nanoparticles can be augmented by approximately 10-25% compared to the values predicted by pure hydrodynamic models in the absence of free charge in the fluid. This enhancement corresponds approximately to a 5-10% increase compared to the electroviscous contribution for a charged particle in an unbounded fluid. Contrary to recent reports in the experimental literature, we find neither experimental nor theoretical evidence of an anomalously large enhancement of electroviscous forces on a confined charged nanoparticle in solution.

A new liquid transport model considering complex influencing factors for nano- to micro-sized circular tubes and porous media

A new liquid transport model in wetted nano- to microsized circular tubes is proposed using basic dynamical analyses that comprehensively consider the Lifshitz–van der Waals force (LWF), the electroviscous force, the weak liquid compressibility, and the Bingham-plastic behavior. The model predicts that the average velocity is initially zero and increases nonlinearly with a concave shape before increasing linearly with the pressure gradient (ΔP/L) and is validated using the experimental data. The threshold pressure gradient (TPG) and the lower limit of the movable-fluid radius (Rm) are calculated based on the proposed model, which are mainly determined by the yield stresses from the Bingham plastic behavior and are also affected by the compressibility and LWF. Considering the microstructural complexity of real porous media, the average velocity model is also applicable for tight porous media with a capillary equivalent radius from the permeability. The calculated average velocity is non-Darcy with TPG. The TPG decreases as the permeability increases, and the Rm decreases with the pressure gradient in the low range and remains constant at the higher ranges, which is primarily between 10 and 30 nm. All these results for porous media are compared with the experimental data of core seepage and show good agreement in general. The proposed model has a clear parametric representation compared with previous nonlinear models. It explains the underlying reasons for the nonlinear, low-velocity flow mechanism in nano- to microsized tubes and pores and provides theoretical guidance for liquid transport in porous media and oil recovery from tight oil reservoirs.

Microscale effects of Bingham-plastic liquid behavior considering electroviscous effects in nano- or microsized circular tubes

Although microscale phenomena are ubiquitous in fluid flow through nano- or microsized channels and pores, the mechanisms remain unclear. To clarify these mechanisms, we investigate herein Bingham-plastic liquids with electroviscous effects (EVEs) in nano- and microsized circular tubes. The constitutive equation and electroviscous forces are introduced into the governing equations, and approximate analytical solutions are obtained. Velocity reduction results from the combined effects of the Bingham characteristics and EVEs. The Bingham behavior reduces the strength of the EVE electric field, and EVEs increase the width of the flow core. The dimensionless average velocity decreases as the tube radius decreases and goes to zero upon approaching the lower limit of the movable fluid radius (Rm). As the pressure gradient decreases, the average velocity first decreases linearly, then decreases nonlinearly in a concave shape, and finally approaches zero as the pressure reaches the threshold pressure gradient (TPG). The Bingham plastic behavior causes both the Rm and the nonlinear flow with TPG, and Rm is still caused by the van der Waals forces under liquid compressibility more obviously. The EVE parameters only affect the degree of nonlinearity when the liquid exhibits Bingham-plastic behavior. These results are consistent with experimentally observed de-ionized water flowing in microscale silica tubes. We infer that the Newtonian fluid displays the Bingham-plastic behavior in nano- or microsized channels in what we call “microscale effects.” These results elucidate the mechanism that leads to nonlinear or low-speed non-Darcy flow in nano- or microsized channels and pores from the liquid characteristic and liquid-solid interaction.

Streaming-potential phenomena in the thin-Debye-layer limit. Part 3. Shear-induced electroviscous repulsion

We employ the moderate-Péclet-number macroscale model developed in part 2 of this sequence (Schnitzer et al., J. Fluid Mech., vol. 704, 2012, pp. 109–136) towards the calculation of electroviscous forces on charged solid particles engendered by an imposed relative motion between these particles and the electrolyte solution in which they are suspended. In particular, we are interested in the kinematic irreversibility of these forces, stemming from the diffusio-osmotic slip which accompanies the salt-concentration polarisation induced by that imposed motion. We illustrate the electroviscous irreversibility using two prototypic problems, one involving side-by-side sedimentation of two spherical particles, and the other involving a force-free spherical particle suspended in the vicinity of a planar wall and exposed to a simple shear flow. We focus on the pertinent limit of near-contact configurations, where use of lubrication approximations provides closed-form expressions for the leading-order lateral repulsion. In this approximation scheme, the need to solve the advection–diffusion equation governing the salt-concentration polarisation is circumvented.

Lift forces on colloidal particles in combined electroosmotic and Poiseuille flow.

Colloidal particles suspended in aqueous electrolyte solutions flowing through microchannels are subject to lift forces that repel the particles from the wall due to the voltage and pressure gradients commonly used to drive flows in microfluidic devices. There are very few studies that have considered particles subject to both an electric field and a pressure gradient, however. Evanescent-wave particle tracking velocimetry was therefore used to investigate the near-wall dynamics of a dilute suspension of 245 nm radius polystyrene particles in a monovalent electrolyte solution in Poiseuille and combined electroosmotic (EO) and Poiseuille flow through 30-μm-deep fused-silica channels. The lift force observed in Poiseuille flow, which is estimated from the near-wall particle distribution, appears to be proportional to the shear rate, a scaling consistent with hydrodynamic lift forces previously reported in field-flow fractionation studies. The estimates of the lift force observed in combined flow suggest that the force magnitude exceeds the sum of the lift forces observed in EO flow at the same electric field or in Poiseuille flow at the same shear rate. Moreover, the force magnitude appears to be proportional to the electric field magnitude and have a power law dependence on the shear rate with an exponent between 0.4 and 0.5. This unexpected scaling suggests that the repulsive lift force observed in combined electroosmotic and Poiseuille flow is a new phenomenon, distinct from previously reported electroviscous, hydrodynamic lift, or dielectrophoretic-like forces, and warrants further study.

Streaming-potential phenomena in the thin-Debye-layer limit. Part 2. Moderate Péclet numbers

AbstractMacroscale description of streaming-potential phenomena in the thin-double-layer limit, and in particular the associated electro-viscous forces, has been a matter of long-standing controversy. In part 1 of this work (Yariv, Schnitzer & Frankel,J. Fluid Mech., vol. 685, 2011, pp. 306–334) we identified that the product of the Hartmann ($\lambda $) and Péclet ($\mathit{Pe}$) numbers is$O({\delta }^{\ensuremath{-} 2} )$,$\delta $being the dimensionless Debye thickness. This scaling relationship defines a one-family class of limit processes appropriate to the consistent analysis of this singular problem. In that earlier contribution we focused on the generic problems associated with moderate$\lambda $and large$\mathit{Pe}$, where the streaming-potential magnitude is comparable to the thermal voltage. Here we consider the companion generic limit of moderate Péclet numbers and large Hartmann numbers, deriving the appropriate macroscale model wherein the Debye-layer physics is represented by effective boundary conditions. Since the induced electric field is asymptotically smaller, calculation of these conditions requires higher asymptotic orders in analysing the Debye-scale transport. Nonetheless, the leading-order electro-viscous forces are of the same$O({\delta }^{2} )$relative magnitude as those previously obtained in the large-$\mathit{Pe}$limit. The structure of these forces is different, however, first because the small Maxwell stresses do not contribute at leading order, and second because salt polarization results in a dominant diffuso-osmotic slip. Since the salt distribution is governed by an advection–diffusion equation, this slip gives rise to electro-viscous forces which are nonlinear in the driving flow. The resulting scheme is illustrated by the calculation of the electro-viscous excess drag in the prototype problem of a translating sphere.

Streaming-potential phenomena in the thin-Debye-layer limit. Part 1. General theory

AbstractElectrokinetic streaming-potential phenomena are driven by imposed relative motion between liquid electrolytes and charged solids. Owing to non-uniform convective ‘surface’ current within the Debye layer Ohmic currents from the electro-neutral bulk are required to ensure charge conservation thereby inducing a bulk electric field. This, in turn, results in electro-viscous drag enhancement. The appropriate modelling of these phenomena in the limit of thin Debye layers $\delta \ensuremath{\rightarrow} 0$ ($\delta $ denoting the dimensionless Debye thickness) has been a matter of ongoing controversy apparently settled by Cox’s seminal analysis (J. Fluid Mech., vol. 338, 1997, p. 1). This analysis predicts electro-viscous forces that scale as ${\delta }^{4} $ resulting from the perturbation of the original Stokes flow with the Maxwell-stress contribution only appearing at higher orders. Using scaling analysis we clarify the distinction between the normalizations pertinent to field- and motion-driven electrokinetic phenomena, respectively. In the latter class we demonstrate that the product of the Hartmann & Péclet numbers is $O({\delta }^{\ensuremath{-} 2} )$ contrary to Cox (1997) where both parameters are assumed $O(1)$. We focus on the case where motion-induced fields are comparable to the thermal scale and accordingly present a singular-perturbation analysis for the limit where the Hartmann number is $O(1)$ and the Péclet number is $O({\delta }^{\ensuremath{-} 2} )$. Electric-current matching between the Debye layer and the electro-neutral bulk provides an inhomogeneous Neumann condition governing the electric field in the latter. This field, in turn, results in a velocity perturbation generated by a Smoluchowski-type slip condition. Owing to the dominant convection, the present analysis yields an asymptotic structure considerably simpler than that of Cox (1997): the electro-viscous effect now already appears at $O({\delta }^{2} )$ and is contributed by both Maxwell and viscous stresses. The present paradigm is illustrated for the prototypic problem of a sphere sedimenting in an unbounded fluid domain with the resulting drag correction differing from that calculated by Cox (1997). Independently of current matching, salt-flux matching between the Debye layer and the bulk domain needs also to be satisfied. This subtle point has apparently gone unnoticed in the literature, perhaps because it is trivially satisfied in field-driven problems. In the present limit this requirement seems incompatible with the uniform salt distribution in the convection-dominated bulk domain. This paradox is resolved by identifying the dual singularity associated with the limit $\delta \ensuremath{\rightarrow} 0$ in motion-driven problems resulting in a diffusive layer of $O({\delta }^{2/ 3} )$ thickness beyond the familiar $O(\delta )$-wide Debye layer.

Non-Newtonian flow in microporous structures under the electroviscous effect

Single phase non-Newtonian microporous flow combined with the electroviscous effect is investigated in the pore-scale under conditions of various rheological properties and electrokinetic parameters. The lattice Boltzmann method is employed to solve both the electric potential field and flow velocity field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends on both the fluid rheological behavior and pore surface area ratio significantly. For the shear thinning fluid with power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electrovicous effect plays a more important role compared to the Newtonian fluid and shear thickening fluid. The high pore surface area ratio in the porous structure increases the electroviscous force and the induced flow resistance becomes important even to the flow of Newtonian and shear thickening fluids.

Tangential electroviscous drag on a sphere surrounded by a thin double layer near a wall for arbitrary particle–wall separations

When a charged particle moves along a charged wall in a polar fluid, it experiences an electroviscous lift force normal to the surface and an electroviscous drag, superimposed on the viscous drag, parallel to the surface. Here a theoretical analysis is presented to determine the electroviscous drag on a charged spherical particle surrounded by a thin electrical double layer near a charged plane wall, when the particle translates parallel to the wall without rotation, in a symmetric electrolyte solution at rest. The electroviscous (electro-hydrodynamic) forces, arising from the coupling between the electrical and hydrodynamic equations, are determined as a solution of three partial differential equations, for electroviscous ion concentration (perturbed ion clouds), electroviscous potential (perturbed electric potential) and electroviscous or electro-hydrodynamic flow field (perturbed flow field). The problem was previously solved for small gap widths and low Peclet numbers in the inner region around the gap between the sphere and the wall, using lubrication theory. Here the restriction on the particle–wall distances is removed, and an analytical and numerical solution is obtained valid for the whole domain of interest. For large sphere–wall separations the solution approaches that for the electroviscous drag on an isolated sphere in an unbounded fluid. For small particle–wall distances it differs from that obtained by the use of lubrication theory, showing that lubrication theory is inadequate for electroviscous problems. The analytical results are in complete agreement with the full numerical calculations. For small particle–wall distances a model is given which provides both physical insight and an easy way to calculate the force with high precision.

Electroviscous effect on non-Newtonian fluid flow in microchannels

Understanding non-Newtonian flow in microchannels is of both fundamental and practical significance for various microfluidic devices. A numerical study of non-Newtonian flow in microchannels combined with electroviscous effect has been conducted. The electric potential in the electroviscous force term is calculated by solving a lattice Boltzmann equation. And another lattice Boltzmann equation without derivations of the velocity when calculating the shear is employed to obtain flow field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends significantly on the fluid rheological behavior. For the shear thinning fluid of the power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electroviscous effect plays a more important role. And its effect on the flow increases as the ratio of the Debye length to the channel height increases and the exponent n decreases. While the shear thickening fluid of n > 1 is less affected by the electroviscous force, it can be neglected in practical applications.

Understanding and Modulating Opalescence and Viscosity in a Monoclonal Antibody Formulation

Opalescence and high viscosities can pose challenges for high concentration formulation of antibodies. Both phenomena result from protein–protein intermolecular interactions that can be modulated with solution ionic strength. We studied a therapeutic monoclonal antibody (mAb) that exhibits high viscosity in solutions at low ionic strength (∼20cP at 90mg/mL and 23°C) and significant opalescence at isotonic ionic strength (approximately 100 nephelometric turbidity units at 90mg/mL and 23°C). The intermolecular interactions responsible for these effects were characterized using membrane osmometry, static light scattering, and zeta potential measurements. The net protein–protein interactions were repulsive at low ionic strength (∼4mM) and attractive at isotonic ionic strengths. The high viscosities are attributed to electroviscous forces at low ionic strength and the significant opalescence at isotonic ionic strength is correlated with attractive antibody interactions. Furthermore, there appears to be a connection to critical phenomena and it is suggested that the extent of opalescence is dependent on the proximity to the critical point. We demonstrate that by balancing the repulsive and attractive forces via intermediate ionic strengths and by increasing the mAb concentration above the apparent critical concentration both opalescence and viscosity can be simultaneously minimized. © 2009 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 99:82–93, 2010

Streaming potential and electroviscous effects in periodical pressure-driven microchannel flow

An analytical solution for pressure-driven periodical electrokinetic flows in a two-dimensional uniform microchannel is presented based on the Poisson–Boltzmann equation for electrical double layer and the Navier–Stokes equations for incompressible viscous fluid. The analytical results indicate that the periodical streaming potential strongly depends on the periodical Reynolds number (Re=ωh2∕ν) which is a function of the frequency, the channel size, and the kinetic viscosity of fluids. For Re&amp;lt;1, the streaming potential behaves similarly to that of steady flow, whereas it decreases rapidly with Re as Re&amp;gt;1. In addition, the electroviscous force affects greatly both the periodical flow and streaming potential, particularly when the nondimensional electrokinetic diameter κh is small. The electroviscous force has been found to depend on three factors: first, the electroviscous parameter, which is defined as the ratio of the maximum electroviscous force to the pressure gradient; second, the distribution parameter describing the distribution of the electroviscous force over the cross section of the microchannel; third, the coupling coefficient, which is a function of both the periodical Reynolds number and electroviscous parameter, determining both the amplitude attenuation and phase offset of the electroviscous force.

Periodical streaming potential and electro-viscous effects in microchannel flow

This paper presents an analytical solution to periodical streaming potential, flow-induced electric field and velocity of periodical pressure-driven flows in two-dimensional uniform microchannel based on the Poisson-Boltzmann equations for electric double layer and Navier-Stokes equation for liquid flow. Dimensional analysis indicates that electric-viscous force depends on three factors: (1) Electric-viscous number representing a ratio between maximum of electric-viscous force and pressure gradient in a steady state, (2) profile function describing the distribution profile of electro-viscous force in channel section, and (3) coupling coefficient reflecting behavior of amplitude damping and phase offset of electro-viscous force. Analytical results indicate that flow-induced electric field and flow velocity depend on frequency Reynolds number (Re = ωh 2/ν). Flow-induced electric field varies very slowly with Re when Re 1. Electro-viscous effect on flow-induced electric field and flow velocity are very significant when the rate of the channel width to the thickness of electric double layer is small.

Electroviscous sphere–wall interactions

A theoretical analysis is presented to determine the forces of interaction between an electrically charged spherical particle and a charged plane wall when the particle translates parallel to the wall and rotates around its axis in a symmetric electrolyte solution at rest. The electroviscous effects, arising from the coupling between the electrical and hydrodynamic equations, are determined as a solution of three partial differential equations, derived from Cox's general theory [R.G. Cox, J. Fluid Mech. 338 (1997) 1], for electroviscous ion concentration, electroviscous potential and electroviscous flow field. It is a priori assumed that the double layer thickness surrounding each charged surfaces is much smaller than the particle size. Using the matched asymptotic expansion technique, the electroviscous forces experienced by the sphere are explicitly determined analytically for small particle–wall distances, but low and intermediate Peclet numbers.

Modeling and analysis of interfacial electro-kinetic effects on thin film lubrication

At the interface between a solid and a following liquid, charge migration within the electric double layer causes an electro-viscous force to develop which manifests itself as an enhanced fluid viscosity. Based on the Poisson–Boltzmann equation, a new mathematical model of the electro-viscosity is developed. Numerical analysis is carried out taking into account the influence of the electric double layer on thin film lubrication. Analysis results show that the electro-viscosity leads to a significant increase in the film thickness in the thin film regime (below 100 nm).

Electroviscous cylinder–wall interactions

A theoretical analysis is presented to determine the forces of interaction between an electrically charged cylindrical particle and a charged plane boundary wall when the particle translates parallel to the wall and rotates around its axis in a symmetric electrolyte solution at rest. The electroviscous effects, arising from the coupling between the electrical and hydrodynamic equations, are determined as a solution of three partial differential equations, derived from R.G. Cox's general theory [J. Fluid Mech. 338 (1997) 1], for electroviscous ion concentration, electroviscous potential, and electroviscous flow field. It is assumed a priori that the double layer thickness surrounding each charged surface is much smaller than the length scale of the problem. Using the matched asymptotic expansion technique, the electroviscous forces experienced by the cylinder are explicitly determined analytically for small particle–wall distances for low and intermediate Peclet numbers. It is found that the tangential force usually increases the drag above the purely hydrodynamic drag, although for certain conditions the drag can be reduced. Similarly the normal force is usually repulsive, i.e., it is an electrokinetic lift force, but under certain conditions the normal force can be attractive.

The electroviscous force between charged particles: beyond the thin-double-layer approximation

We have investigated the hydrodynamic drag force between charged particles in electrolyte solutions, specifically the electroviscous force that arises from the distortion of the electrical double layers by the flow field. We report an improvement on the thin-double-layer theory (S.G. Bike, D.C. Prieve, J. Colloid Interface Sci. 136 (1990) 95–112), using a more accurate boundary condition for the radial charge current. The differences become important when the double layers start to overlap. We have found that nonlinear hydrodynamic effects are small, whereas nonlinear electric effects can be significant, in some instances leading to qualitatively different behavior. If the ion diffusivities are highly asymmetric, the electroviscous force can be reduced by an order of magnitude when there is an excess of the mobile ions in the double layer. The common supposition that there are substantial differences in the electroviscous force predicted by constant-charge and constant-potential boundary conditions is incorrect; our calculations show that it is an artifact introduced by the Debye–Hückel approximation.