A Dynamic Model of Polyelectrolyte Gels

Mori, Yoichiro; Chen, Haoran; Micek, Catherine A.; Calderer, M. Carme

doi:10.1137/110855296

Cited by 35 publications

(50 citation statements)

References 49 publications

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“…The use of free energy identities as a guiding principle in formulating equations originates in the work of Onsager [46], and this approach has been widely adopted in soft condensed matter physics [9,10,25,15]. The present work is closely related to our recent work in [37,39,41,6], wherein the free energy identity played an essential role in ionic electrodiffusion problems arising in physiology and the material sciences. One practical benefit of the physically consistent formulation of our model is that it treats fast cable (or electrotonic/electrical current) effects and the much slower effects mediated by ion concentration gradients in a single unified framework.…”

Section: Introductionmentioning

confidence: 93%

A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression

Mori

2015

Physica D: Nonlinear Phenomena

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Ionic electrodiffusion and osmotic water flow are central processes in many physiological systems. We formulate a system of partial differential equations that governs ion movement and water flow in biological tissue. A salient feature of this model is that it satisfies a free energy identity, ensuring the thermodynamic consistency of the model. A numerical scheme is developed for the model in one spatial dimension and is applied to a model of cortical spreading depression, a propagating breakdown of ionic and cell volume homeostasis in the brain.

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Section: Introductionmentioning

confidence: 93%

A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression

Mori

2015

Physica D: Nonlinear Phenomena

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“…This would be true for an applied flow where at this outer boundary the pressure balances the applied strain or for a flow where U tends to zero. After integration by parts, equation (18) shows that the sum of the two surface integrals on Γ t and the two volume integrals I Ω S and I Ω P must equal to the volume integral I E , which has been shown to be equal to the rate of change of an elastic free energy E elas 18 :…”

Section: B Boundary Conditionsmentioning

confidence: 99%

“…Cogan and Keener developed a two-phase flow model for cellular cytosol, with a dominant viscous dissipation of a deformable skeleton (that depends on the rate of strain) and no elastic energy (that depends on the strain) 17 . The dynamic coupling between mechanical and electrochemical effects of polyelectrolyte gels has been formulated by Mori et al 18 . Focusing on the variational derivation of the governing equations in the mean-field framework, their dynamical model consists of many parameters that are related to the underlying microscopic physics and transport in the transition region around the domain boundary.…”

Section: Introductionmentioning

confidence: 99%

Slightly deformable Darcy drop in linear flows

2019

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The small-deformation of a poroelastic drop due to an external linear flow field in the suspending medium is investigated. A two-phase flow model is developed to study the small-deformation of such a poroelastic drop under linear flows. Inside the drop a deformable porous network with a bending rigidity and a viscosity is fully immersed in a viscous fluid. When the viscous dissipation of the interior fluid phase is negligible (compared to the friction between the fluid and the skeleton), the two-phase flow is reduced to a poroelastic Darcy fluid with a deformable porous network. At the interface between the poroelastic drop and the exterior Stokesian fluid, a novel set of boundary conditions are derived by the free energy dissipation principle. Both slip and permeability are taken into account and the permeating flow induces dissipation that depends on the elastic stress of the interior solid. Assuming that the porous network is slightly deformed from its original spherical shape due to its large bending rigidity, a small-deformation analysis is conducted. A steady equilibrium is computed for two different linear flows: a uniaxial extensional flow and a planar shear flow. By exploring the interfacial slip, permeability and network elasticity various flow patterns are found at equilibrium of these slightly deformed poroelastic drops. Linear dynamics of the small-amplitude deviation of the poroelastic drop from the spherical shape is governed by a nonlinear eigenvalue problem, and the eigenvalues are determined numerically, and asymptotically in certain limits. These results lay the foundation for studying the rheology of a suspension of poroelastic spherical particles, and give insight to possible flow patterns of a system of selfpropelling swimmers with porous flow (such as intracellular cytosol) inside.

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“…Conclusion. In this paper, we briefly reviewed the model presented in the previous work [30] and applied Onsagers variational principle to rederive the dynamic equations and boundary conditions both in the mechanical case and in the ion electrodiffusive case. Under certain assumptions on the form of mechanical energy, especially convexity, we proved the uniqueness of steady state solution.…”

Section: (561) If We Further Assumementioning

confidence: 99%

“…We derive the equations of gel dynamics using Onsager's variational principle. In [30], we stated the kinematic relations and proposed the dynamic equations, from which the energy relation was derived as a consequence. Here, we start from the kinematic and energy relations, and using the variational approach, to derive the dynamic equations.…”

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confidence: 99%

Analysis and simulation of a model of polyelectrolyte gel in one spatial dimension

Chen¹,

Calderer²,

Mori³

2014

Nonlinearity

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Abstract. We analyze a model of polyelectrolyte gels that was proposed by the authors in previous work. We first demonstrate that the model can be derived using Onsager's variational principle, a general procedure for obtaining equations in soft condensed matter physics. The model is shown to have a unique steady state under the assumption that a suitably defined mechanical energy density satisfies a convexity condition. We then perform a detailed study of the stability of the steady state in the spatially one-dimensional case, obtaining bounds on the relaxation rate. Numerical simulations for the spatially one-dimensional problem are presented, confirming the analytical calculations on stability.1. Introduction. Gels are crosslinked, three dimensional polymer networks that absorb solvent and swell without dissolution [21,22,16,39,2]. In this paper, we study polyelectrolyte gels, in which the polymer network carries charge, and delivers counterions to the solvent. An important feature of many polyelectrolyte gels is that they undergo large and often discontinuous volume changes (called the volume phase transition) in response to various environmental parameters, including pH, temperature and ionic composition [14,37,38]. Some physically interesting characteristics of the volume phase transition include robust hysteresis, coexistence of phases and complicated transient dynamics [12,37]. These volume changes are at the basis of numerous applications of gels to artificial devices [3,31,29,7,13,19] and are thought to underlie certain physiological processes [46,41,37,29]. It is thus of both practical and theoretical interest to develop a dynamic model of polyelectrolyte gels.There have been numerous modeling studies of gels. Studies using purely mechanical models include [36,40] in which static problems are addressed, and [5,44,35,9,10,24,25] in which dynamic problems are the focus. Models of polyelectrolyte gels include the static models of [37,20,34] and the dynamic models of [17,15,18,26,28,43,42,45,27,6,1,20]. In this paper, we initiate a detailed study of a dynamic PDE model of polyelectrolyte gels introduced in [30]. Our model is distinguished from previous models in its careful derivation of the boundary conditions at the gel-fluid interface as well as its satisfaction of a free energy identity, including interfacial terms. We show that our model can be derived from Onsager's variational principle which is a systematic way of deriving equations in soft condensed matter systems [10,33]. The one-dimensional stability calculation is a generalization of the classical work by [40] in which the neutral gel case is studied. We prove the uniqueness of the steady state solution and study the stability as the model being restricted to one spatial dimension. Finally, we have successfully simulated our polyelectrolyte gel model in the one spatial dimension. The simulated equilibrium solution and exponential decay rate both match our analytical calculation.The paper is organized as follows. In section 2, we apply Onsager'...

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A Dynamic Model of Polyelectrolyte Gels

Cited by 35 publications

References 49 publications

A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression

A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression

Slightly deformable Darcy drop in linear flows

Analysis and simulation of a model of polyelectrolyte gel in one spatial dimension

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