arXiv subjects

Barbara Wagner

Publications and source records attributed to Barbara Wagner.

At least 19 recordsLinked to original sources

arXiv2026

An asymptotic model of Poisson--Nernst--Planck--Stokes systems in narrow channels

Ion transport through narrow channels is described by the coupled Poisson--Nernst--Planck--Stokes equations (PNPS) on a continuum scale. However, direct numerical simulations in two or three dimensions of boundary value problems for small aspect ratio geometries, a crucial characteristic of nanopores, can quickly become computationally intensive and thus limit the insights into the underlying mechanisms that control electrokinetic phenomena. Taking advantage of the small aspect ratio, we derive a systematic asymptotic reduction of the PNPS system. In contrast to existing one-dimensional reductions, which assume a Debye length much smaller than the channel radius, our analysis identifies a distinguished asymptotic regime in which the Debye length is allowed to be comparable to the channel width. Our approach has a significantly larger range of validity and contains existing approximations such as the Helmholtz--Smoluchowski approximation as limiting cases. The derived asymptotic model extends also to a generalized PNPS system, where finite-size constraints and solvation effects are taken into account and thus applies to other well-known models such as the Bikerman--Freise model. Using our asymptotic model we demonstrate that the ion current can undergo a number of different flow transitions and in particular predict that positively charged ions can be pushed against their electrostatic gradient. Furthermore, we show how finite-size effects can influence the ion current and enhance ion selectivity. Finally, we revisit case studies of protein-based channels from the literature to illustrate the predictive potential of our asymptotic model.

math.AP↗

arXiv2025

The diffusive dynamics and electrochemical regulation of weak polyelectrolytes across liquid interfaces

We propose a framework to study the spatio-temporal evolution of liquid-liquid phase separation of weak polyelectrolytes in ionic solutions. Unlike strong polyelectrolytes, which carry a fixed charge, the charge state of weak polyelectrolytes is modulated by the electrochemical environment through protonation and deprotonation processes. Leveraging numerical simulations and analysis, our work reveals how solution acidity (pH) influences the formation, interactions, and structural properties of phase-separated coacervates. We find that pH gradients can be maintained across coacervate interfaces resulting in a clear distinction in the electro-chemical properties within and outside the coacervate. By regulating the charge state of weak polyelectrolytes, pH gradients interact and modulate the electric double layer forming at coacervate interfaces eventually determining how they interact. Further linear and nonlinear analyses of stationary localised solutions reveal a rich spectrum of behaviours that significantly distinguish weak from strong polyelectrolytes. Overall, our results demonstrate the importance of charge regulation on phase-separating solutions of charge-bearing molecules and the possibility of harnessing charge-regulated mechanisms to control coacervates and shape their stability and spatial organisation.

cond-mat.soft↗

arXiv2024

Shape of Polystyrene Droplets on Soft PDMS: Exploring the Gap Between Theory and Experiment at the Three-Phase Contact Line

The shapes of liquid polystyrene (PS) droplets on viscoelastic polydimethylsiloxane (PDMS) substrates are investigated experimentally using atomic force microscopy for a range of droplet sizes and substrate elasticities. These shapes, which comprise the PS-air, PS-PDMS, and PDMS-air interfaces as well as the three-phase contact line, are compared to theoretical predictions using axisymmetric sharp-interface models derived through energy minimization. We find that the polystyrene droplets are cloaked by a thin layer of uncrosslinked molecules migrating from the PDMS substrate. By incorporating the effects of cloaking into the surface energies in our theoretical model, we show that the global features of the experimental droplet shapes are in excellent quantitative agreement for all droplet sizes and substrate elasticities. However, our comparisons also reveal systematic discrepancies between the experimental results and the theoretical predictions in the vicinity of the three-phase contact line. Moreover, the relative importance of these discrepancies systematically increases for softer substrates and smaller droplets. We demonstrate that global variations in system parameters, such as surface tension and elastic shear moduli, cannot explain these differences but instead point to a locally larger elastocapillary length, whose possible origin is discussed thoroughly.

cond-mat.soft↗

arXiv2024

Interface dynamics in a degenerate Cahn-Hilliard model for viscoelastic phase separation

The formal sharp-interface asymptotics in a degenerate Cahn-Hilliard model for viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable are shown to lead to non-local lower-order counterparts of the classical surface diffusion flow. The diffuse-interface model is a variant of the Zhou-Zhang-E model and has an Onsager gradient-flow structure with a rank-deficient mobility matrix reflecting the ODE character of stress relaxation. In the case of constant coupling, we find that the evolution of the zero level set of the order parameter approximates the so-called intermediate surface diffusion flow. For non-constant coupling functions monotonically connecting the two phases, our asymptotic analysis leads to a new family of third-order evolution laws with associated propagation operators behaving, at leading order, like the square root of the minus Laplace-Beltrami operator. In this case, the normal velocity of the moving sharp interface arises as the Lagrange multiplier in a constrained elliptic equation, which is at the core of our derivation. The constrained elliptic problem can be solved rigorously by a variational argument, and is shown to encode the gradient structure of the effective geometric evolution law. The asymptotics are presented for deep quench, an intermediate free boundary problem based on the double-obstacle potential.

math.AP↗

arXiv2023

Counterion-controlled phase equilibria in a charge-regulated polymer solution

We study phase equilibria in a minimal model of charge-regulated polymer solutions. Our model consists of a single polymer species whose charge state arises from protonation-deprotonation processes in the presence of a dissolved acid, whose anions serve as screening counterions. We explicitly account for variability in the polymers' charge states. Homogeneous equilibria in this model system are characterised by the total concentration of polymers, the concentration of counter-ions and the charge distributions of polymers which can be computed with the help of analytical approximations. We use these analytical results to characterise how parameter values and solution acidity influence equilibrium charge distributions and identify for which regimes uni-modal and multi-modal charge distributions arise. We then study the interplay between charge regulation, solution acidity and phase separation. We find that charge regulation has a significant impact on polymer solubility and allows for non-linear responses to the solution acidity: re-entrant phase behaviour is possible in response to increasing solution acidity. Moreover, we show that phase separation can yield to the coexistence of local environments characterised by different charge distributions and mixture compositions.

cond-mat.soft↗

arXiv2022

The electric double layer at the interface between a polyelectrolyte gel and salt bath

The electric double layer (EDL) that forms at the interface between a polyelectrolyte gel and a salt bath is studied using asymptotic and numerical methods. Specifically, matched asymptotic expansions, based on the smallness of the Debye length relative to the typical gel dimensions, are used to construct solutions of the governing equations and derive electroneutral models with consistent jump conditions across the gel-bath interface. A general approach for solving the equations of incompressible nonlinear elasticity in a curved boundary layer is developed and used to resolve the gel mechanics in the EDL. A critical feature of the model is that it accounts for phase separation within the gel, which gives rise to diffuse interfaces with a characteristic thickness described by the Kuhn length. We show that the solutions of the electroneutral model can only be asymptotically matched to the solutions in the EDL, in general, when the Kuhn length greatly exceeds the Debye length. Conversely, if the Debye length is similar to or larger than the Kuhn length, then the entire gel can self-organise into periodic, electrically charged domains via phase separation. The breakdown of electroneutrality demonstrates that the commonly invoked electroneutral assumption must be used with caution, as it generally only applies when the Debye length is much smaller than the Kuhn length.

cond-mat.soft↗

arXiv2021

Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers

The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.

math.NA↗

arXiv2021

A kinetic model of a polyelectrolyte gel undergoing phase separation

In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a hydrodynamic model for a salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, the nonlinear elasticity of the polyelectrolyte network, and multi-component diffusive transport following a Stefan--Maxwell approach. The time-dependent model describes the evolution of the gel across multiple time and spatial scales and so is able to capture the large-scale solvent flux and the emergence of long-time pattern formation in the system. We explore the model for the case of a constrained gel undergoing uni-axial deformations. Numerical simulations show that rapid changes in the gel volume occur once the volume phase transition sets in, as well as the triggering of spinodal decomposition that leads to strong inhomogeneities in the lateral stresses, potentially leading to experimentally visible patterns.

cond-mat.soft↗

arXiv2021

The dynamics of a collapsing polyelectrolyte gel

We analyse the dynamics of different routes to collapse of a constrained polyelectrolyte gel in contact with an ionic bath. The evolution of the gel is described by a model that incorporates non-linear elasticity, Stefan-Maxwell diffusion and interfacial gradient free energy to account for phase separation of the gel. A bifurcation analysis of the homogeneous equilibrium states reveals three solution branches at low ion concentrations in the bath, giving way to only one above a critical ion concentration. We present numerical solutions that capture both the spatial heterogeneity and the multiple time-scales involved in the process of collapse. These solutions are complemented by two analytical studies. Firstly, a phase-plane analysis that reveals the existence of a depletion front for the transition from the highly swollen to the new collapsed equilibrium state. This depletion front is initiated after the fast ionic diffusion has set the initial condition for this time regime. Secondly, we perform a linear stability analysis about the homogeneous states that show that for a range of ion concentrations in the bath, spinodal decomposition of the swollen state gives rise to localized solvent-rich(poor) and, due to the electro-neutrality condition, ion-poor(rich) phases that coarsen on the route to collapse. This dynamics of a collapsing polyelectrolyte gel has not been described before.

cond-mat.soft↗

arXiv2018

Stability of concentrated suspensions under Couette and Poiseuille flow

The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated and connections to the stability problem for the related classical Bingham-flow problem are discussed.

physics.flu-dyn↗

arXiv2017

Impact of energy dissipation on interface shapes and on rates for dewetting from liquid substrates

We revisit the fundamental problem of liquid-liquid dewetting and perform a detailed comparison of theoretical predictions based on thin-film models with experimental measurements obtained by atomic force microscopy (AFM). Specifically, we consider the dewetting of a liquid polystyrene (PS) layer from a liquid polymethyl methacrylate (PMMA) layer, where the thicknesses and the viscosities of PS and PMMA layers are similar. The excellent agreement of experiment and theory reveals that dewetting rates for such systems follow no universal power law, in contrast to dewetting scenarios on solid substrates. Our new energetic approach allows to assess the physical importance of different contributions to the energy-dissipation mechanism, for which we analyze the local flow fields and the local dissipation rates.

physics.flu-dyn↗

arXiv2017

Thin film models for active gels

In this study we present a free-boundary problem for an active liquid crystal based on the Beris-Edwards theory that uses a tensorial order parameter and includes active contributions to the stress tensor to analyse the rich defect structure observed in applications such as the Adenosinetriphosphate (ATP) driven motion of a thin film of an actin filament network. The small aspect ratio of the film geometry allows for an asymptotic approximation of the free-boundary problem in the limit of weak elasticity of the network and strong active terms. The new thin film model captures the defect dynamcs in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Lesli-Erickson-Parodi theory. Analytic expression are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.

math.AP↗

arXiv2017

Localized Instabilities and Spinodal Decomposition in Driven Systems in the Presence of Elasticity

We study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material ux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material ux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the time-dependent and non-uniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a at moving front. The second instability is related to the Mullins- Sekerka instability in the presence of elasticity and arises at the moving front between the two phases when the ux is reversed. Stability analyses of the full model and the corresponding sharp-interface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries.

cond-mat.mtrl-sci↗

arXiv2017

Models for the two-phase flow of concentrated suspensions

A new two-phase model for concentrated suspensions is derived that incorporates a constitutive law combining the rheology for non-Brownian suspension and granular flow. The resulting model exhibits a yield-stress behavior for the solid phase depending on the collision pressure. This property is investigated for the simple geometry of plane Poiseuille flow, where an unyielded or jammed zone of finite width arises in the center of the channel. For the steady states of this problem, the governing equations are reduced to a boundary value problem for a system of ordinary differential equations and the conditions for existence of solutions with jammed regions are investigated using phase-space methods. For the general time-dependent case a new drift-flux model is derived using matched asymptotic expansions that takes into account the boundary layers at the walls and the interface between the yielded and unyielded region. The drift-flux model is used to numerically study the dynamic behavior of the suspension flow including the appearance and evolution of an unyielded or jammed region.

physics.flu-dyn↗

arXiv2014

Controlled topological transitions in thin film phase separation

In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. By bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model. Using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that this newly created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

physics.flu-dyn↗

arXiv2012

Droplets on liquids and their long way into equilibrium

The morphological paths towards equilibrium droplets during the late stages of the dewetting process of a liquid film from a liquid substrate is investigated experimentally and theoretically. As liquids, short chained polystyrene (PS) and polymethyl-methacrylate (PMMA) are used, which can be considered as Newontian liquids well above their glass transition temperatures. Careful imaging of the PS/air interface of the droplets during equilibration by \emph{in situ} scanning force microscopy and the PS/PMMA interface after removal of the PS droplets reveal a surprisingly deep penetration of the PS droplets into the PMMA layer. Droplets of sufficiently small volumes develop the typical lens shape and were used to extract the ratio of the PS/air and PS/PMMA surface tensions and the contact angles by comparison to theoretical exact equilibrium solutions of the liquid/liquid system. Using these results in our dynamical thin-film model we find that before the droplets reach their equilibrium they undergo several intermediate stages each with a well-defined signature in shape. Moreover, the intermediate droplet shapes are independent of the details of the initial configuration, while the time scale they are reached depend strongly on the droplet volume. This is shown by the numerical solutions of the thin-film model and demonstrated by quantitative comparison to experimental results.

cond-mat.soft↗

arXiv2012

Stationary solutions of liquid two-layer thin film models

We investigate stationary solutions of a thin-film model for liquid two-layer flows in an energetic formulation that is motivated by its gradient flow structure. The goal is to achieve a rigorous understanding of the contact-angle conditions for such two-layer systems. We pursue this by investigating a corresponding energy that favors the upper liquid to dewet from the lower liquid substrate, leaving behind a layer of thickness $h_*$. After proving existence of stationary solutions for the resulting system of thin-film equations we focus on the limit $h_*\to 0$ via matched asymptotic analysis. This yields a corresponding sharp-interface model and a matched asymptotic solution that includes logarithmic switch-back terms. We compare this with results obtained using $Γ$-convergence, where we establish existence and uniqueness of energetic minimizers in that limit.

math.AP↗

arXiv2008

Spinodal dewetting of thin films with large interfacial slip: implications from the dispersion relation

We compare the dispersion relations for spinodally dewetting thin liquid films for increasing magnitude of interfacial slip length in the lubrication limit. While the shape of the dispersion relation, in particular the position of the maximum, are equal for no-slip up to moderate slip lengths, the position of the maximum shifts to much larger wavelengths for large slip lengths. Here, we discuss the implications of this fact for recently developed methods to assess the disjoining pressure in spinodally unstable thin films by measuring the shape of the roughness power spectrum. For PS films on OTS covered Si wafers (with slip length b approx. 1 mum) we predict a 20% shift of the position of the maximum of the power spectrum which should be detectable in experiments.

cond-mat.soft↗