Incipient Dynamics of Swelling of Gels

Zhang, Hang; Calderer, M. Carme

doi:10.1137/070680941

Cited by 14 publications

(28 citation statements)

References 20 publications

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“…We propose a novel class of boundary conditions at the gel-fluid interface. Some previously proposed boundary conditions can be seen as limiting cases of the general class we present here [8,51,39]. We then prove a free energy identity satisfied by this system.…”

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confidence: 79%

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A Dynamic Model of Polyelectrolyte Gels

Mori¹,

Chen

Micek³

et al. 2013

SIAM J. Appl. Math.

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Abstract. We derive a model of the coupled mechanical and electrochemical effects of polyelectrolyte gels. We assume that the gel, which is immersed in a fluid domain, is an immiscible and incompressible mixture of a solid polymeric component and the fluid. As the gel swells and de-swells, the gel-fluid interface can move. Our model consists of a system of partial differential equations for mass and linear momentum balance of the polymer and fluid components of the gel, the NavierStokes equations in the surrounding fluid domain, and the Poisson-Nernst-Planck equations for the ionic concentrations on the whole domain. These are supplemented by a novel and general class of boundary conditions expressing mass and linear momentum balance across the moving gel-fluid interface. Our boundary conditions include the permeability boundary conditions proposed in earlier studies. A salient feature of our model is that it satisfies a free energy dissipation identity, in accordance with the second law of thermodynamics. We also show, using boundary layer analysis, that the well-established Donnan condition for equilibrium arises naturally as a consequence of taking the electroneutral limit in our model.

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mentioning

confidence: 79%

“…As such, this was a small deformation theory. The dynamic theory of electrically neutral gels has since been developed by many authors [8,51,39,11,12,28,29].…”

mentioning

confidence: 99%

A Dynamic Model of Polyelectrolyte Gels

Mori¹,

Chen

Micek³

et al. 2013

SIAM J. Appl. Math.

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“…[56]). Other possible applications may include for example problems involving diffusion inside of elastoplastic solids [34,32,57] or diffuse interfaces [36].…”

Section: Discussionmentioning

confidence: 99%

“…In contrast, in the context of continuum mechanics of mixtures, Lagrangian coordinates for individual components are commonly used (see for example Refs. [31,32,33]). Our approach can be seen as a natural generalization of the classical one-phase Lagrangian dynamics that also uses only one set of Lagrangian coordinates.…”

Section: State Variablesmentioning

confidence: 99%

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

Peshkov

Grmela

Romenski

2014

Continuum Mech. Thermodyn.

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On the example of two-phase continua experiencing stress induced solid-fluid phase transitions we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization.

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“…There are two possibilities: (i) in (5.35), the smallest is a PEI. In this case, 44) and hence the right hand side is positive. (ii) in (5.35), the smallest is the PEP.…”

Section: Proof Of (541) It Is Convenient To Write (534) In the Formmentioning

confidence: 95%

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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Incipient Dynamics of Swelling of Gels

Cited by 14 publications

References 20 publications

A Dynamic Model of Polyelectrolyte Gels

A Dynamic Model of Polyelectrolyte Gels

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

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

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