Class, Andreas G.

Marthaler, Philipp G.

English

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Received: 25 June 2021 Accepted: 20 August 2021DOI: 10.1002/pamm.202100209Integral effects of the Debye layer on a sedimenting particlewith zeta-potential variationsPhilipp G. Marthaler1,∗ and Andreas G. Class1,∗∗1 Institute for Thermal Energy Technology and Safety (ITES), Karlsruhe Institute of Technology (KIT),Herrmann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, GermanyFluid-solid interfaces give rise to electro-hydrodynamic effects at microscale. In a macroscale model, those effects can berepresented by jump conditions. This report focuses on the derivation of such jump conditions for integral parameters, likemass flux, forces and charge fluxes. In particular, the effect of spatial and temporal variations in the ζ-potential on the solid’ssurface are discussed, using the generic problem of a sedimenting particle.© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.The dynamic behavior induced by varying ζ-potentials in electro-hydrodynamic problems is of great interest for bothtechnical applications and life sciences. Examples are lab-on-a-chip-systems or micro-organisms. In technical systems, asdiscussed e.g. by Ramos et al. [9], research focuses on the design of applied surface potentials in order to optimize the in-duced flow pattern. The understanding of micro-organisms is based on the interaction of a fluid electrolyte with charged andflexible bio-membranes. One major topic in this context is the understanding of micro-organism motility as e.g. investigatedby Stone & Samuel [11].Time-dependent behavior originates from variations in the surface charge of the corresponding solid or fluctuations of themacroscale flow field. Consequently, the general problem of interest is an arbitrarily shaped solid subjected to a generic sur-face charge. Following a number of previous publications [4,6,12], we investigate the problem of a small particle sedimentingin an electrolyte liquid. The liquid is assumed to be of constant density and viscosity containing two symmetric species ofions. In particular, both ion species have equal diffusion coefficients and opposite charges. The charged surface of the particleis in direct contact with a liquid electrolyte, i.e. a fluid characterized by its flow and pressure field, and ion concentrationfields. The electro-hydrodynamic behavior arises from a small layer of dimensionless thickness δ close to the solid’s surface.Singular problems can be modeled by a method originally developed for flame surfaces by Class et al. [5] and adapted toelectro-hydrodynamic problems by Marthaler and Class [8]. The method combines a contravariant description of the corre-sponding complex-shaped surface following Aris [1] with an asymptotic approach based on Bender [2] and Hinch [7] to findanalytical solutions for the set of differential equations.We use the generic conservation equation as starting point of our analysis. Based on our previous work [5, 8], we describethe arbitrary surface of our particle with the contravariant metric gij(xj , t). Due to the assumption, that the surface normalcoordinate x1 is orthogonal to both tangential coordinates xα at each point of the surface, the simplifications g11 = 1 and g1αhold. With the volume element√g(xj , t)the conservation of a generic scalar a(xj , t)takes the form∂t (√g a) + ∂xj(√g Jj (a))=√g S (a) , (1)where Jj (a) includes all fluxes which can e.g. be specified as convective, diffusive or electrophoretic. The fluxes are balancedby a source term S (a) and a transient term. By replacing the generic parameter a by the relevant parameters for our exampleproblem, flow velocity vj(xj , t), pressure p(xj , t), electric charges of equal valence and opposite sign c±(xj , t)and electricpotential ϕ(xj , t),we find a system of charge conservation equationsPe ∂t (√g c±) + ∂xj(√g(−gij∂xic± + Pe c±vj ∓ c±gij∂xiϕ))= 0, (2)complemented by momentum and mass conservation and Gauß’s law. The Reynolds number in a typical microfluidic problemis small enough to neglect the transient and convective terms in the momentum equation. Thus, it reduces to a balance ofNewton and Maxwell stresses, and the only time derivative in the governing equations appears in the charge balance.At the solid boundary x1 = 0, no-slip and impermeability conditions hold. Far from the wall x1 → ∞, we have an attenuationcondition to the outer flow field. At the boundaries, the electrical problem is governed byJ1 (c±)∣∣x1=0= 0, ϕ∣∣x1=0= ζ (xα, t) , c±∣∣x1→∞ → 1, ϕ∣∣x1→∞ → ϕ∞. (3)Attenuation conditions for the potential and charges (that take their equilibirum value 1 far from the charged wall) and a no-fluxcondition at the wall are complemented by a Dirichlet-type boundary condition of the potential. The ζ-potential, representing∗ Corresponding author: e-mail philipp.marthaler@kit.edu, phone +49 721 608 25781∗∗ e-mail andreas.class@kit.eduThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distributionin any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.PAMM · Proc. Appl. Math. Mech. 2021;21:1 e202100209. www.gamm-proceedings.com 1 of 2https://doi.org/10.1002/pamm.202100209 © 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.2 of 2 Section 9: Laminar flows and transitionFig. 1: Qualitative jump conditions for normal fluxes on the sphere surface. The red dotted line represents the value zero. Arrows and greenline demonstrate the respective jump conditions at the circumferential position. The reagrding ζ-potential in dimensionless form was chosenas ζ = ζ0 + ζ̃ sin (k θ). The used parameters are ζ0 = 1, ζ̃ = 0.1, wavenumber k = 11, and the polar coordinate θ ∈ [0, 2π]. The indicesin brackets represent the order of the displyed jump, (3) stands for values of order δ3. The particle sediments from top to bottom, the valueθ = 0 is located in the lee position.the excitation term in our problem, is assumed to be known. Introducing temporal or spatial varying ζ-potentials, the system’sresponse behavior can be investigated. After introducing the coordinate stretch x1 = δX , we compute jump conditions for theintegral parameters following the procedures in Marthaler & Class [8] and Class et al. [5]. For the derivation of the generalform of the jump condition√g[J1 (a)]=∫ ∞Xsurface(√g (S (A)− S (a))− ∂t (√g (A− a))− ∂xα (√g (Jα (A)− Jα (a)))) dX (4)we refer to the mentioned publications. The overarching objective of the procedure is the replacement of detailed physicsinside the layer by jump conditions. Therefore, the jump condition contains differences of the parameters of the continuousmodel A(xj , t)and the parameters of the jumping model a(xj , t). Spatial variations of the ζ-potential along the particle sur-face are computed with the above relation (4). Examples for the results, considering a stationary case, are displayed in Fig. 1.In particular, the hydrodynamic effects give rise to corrections to the forces acting on the surrounding fluid. The normal stressdistribution indicates an extra force acting on the fluid on the floward side of the sedimenting particle and a negative correctionon the lee side.Temporal variations in the ζ-potential are distinguished by their fluctuation frequency. Slow time scales of order t ∼ δ−1or slower let the transient term in the Nernst-Planck equations vanish. This leads to a quasi-stationary system with the well-known Gouy-Chapman solution. Fast time scales of at leastt ∼ δ0τ(0) yield the time-dependent basic system∂X(−√g(0)∂XΦ(0))= Q(0), ∂τ(0)(√g(0)C±(0))+ ∂X(−√g(0)∂XC±(0) ∓√g(0)C±(0)∂XΦ(0))= 0, (5)which has to be solved numerically. Developing a numerical code for this problem will increase the understanding for manytechnical or bio-mechanical problems in the context of electro-hydrodynamics. Yet, the latter requires an extension of themodel, incoporating the interaction between soft tissue and the electrolyte. That model extension defines a second roadmapfor future research.Acknowledgements We acknowledge the Framatome Professional School for support. Open access funding enabled and organized byProjekt DEAL.References[1] R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics (Dover Publ., New York, 1989).[2] C. M. Bender, S. A. Orszag, Advanced Mathemetical Methods for Scientists and Engineers I (Springer, New York, 1999)[3] S. G. Bike and D. C. Prieve, J. Colloid Interface Sci. 136, 95 (1990)[4] F. Booth, J. Chem. Phys. 22, 1956 (1954)[5] A. G. Class, B. J. Matkowsky and A. Y. Klimenko, J.F.M. 491, pp. 11–49 (2003).[6] R. G. Cox, J.F.M. 168, pp. 169–194 (1986).[7] E. J. Hinch, Perturbation Methods (Cambridge Univ. Press, Cambridge, 2002).[8] P. G. Marthaler, A. G. Class, PAMM 20, e202000044 (2021).[9] A. Ramos, A. González, A. Castellanos, N. G. Green and H. Morgan, Phys. Rev. E 67, 056302 (2003)[10] M. Smoluchowski, Handbuch der Elektrizität und des Magnetismus II (Barth, Leipzig, 1921)[11] H. A. Stone and A. D. T. Samuel, Phys. Rev. Lett. 77, 4102 (1996)[12] E. Yariv, O. Schnitzer and I. Frankel, J.F.M. 685, pp. 306–334 (2011).© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH. www.gamm-proceedings.com

Integral effects of the Debye layer on a sedimenting particle with zeta‐potential variations

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Integral effects of the Debye layer on a sedimenting particle with zeta‐potential variations

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