Modelling of and Mixed Finite Element Methods for Gels in Biomedical Applications

Rognes, Marie E.; Calderer, M. Carme; Micek, Catherine A.

doi:10.1137/090754443

Cited by 11 publications

(16 citation statements)

References 22 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: 80%

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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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confidence: 80%

“…The function φ 1 W elas is the elastic energy per unit volume. An example form of W elas is given by [39]:…”

Section: A Mechanical Modelmentioning

confidence: 99%

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A Dynamic Model of Polyelectrolyte Gels

Mori¹,

Chen

Micek³

et al. 2013

SIAM J. Appl. Math.

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“…Integrating the pressure terms in (2.31) by parts, and using (2.24), it follows that 34) where A stands for the integral of boundary terms 35) in which the indexes {+, -} denote evaluations on the Ω t side and R t side, respectively. It is clear (2.32) yields (2.22) and (2.23).…”

Section: Set Up Of the Modelmentioning

confidence: 99%

“…(5.34) in which ω takes values as (n + 35) and let γ 2 be the second smallest. Then λ 1 ∈ (γ 1 , γ 2 ).…”

Section: Minimum Decay Rate: Uniform Concentrations In (A L)mentioning

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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Effects of permeability and viscosity in linear polymeric gels

Chabaud

Calderer

2015

Math. Meth. Appl. Sci.

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We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body-implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi-static approximation to the dynamics. We focus on the linearized system about stress-free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time-dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non-stress free equilibria and states with residual stress and derive an energy law for the corresponding time-dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement-traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two-dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. the gel is the sum of the elastic stored energy function of the polymer and the Flory-Huggins energy of mixing. The variable fields of the equilibrium problem consist of the volume fractions of the gel components and the deformation gradient tensor of the polymer. These fields are not all independent, but they are related by the equation of balance of mass of the polymer and the saturation condition on the gel. The problem of constrained energy minimization studied by Rognes, Calderer, and Micek in [1] established sufficient conditions on the energy and on the imposed boundary conditions that guarantee existence of a global energy minimizer. The authors also developed, analyzed, and numerically implemented a mixed finite element discretization of the equilibrium linear operator. A challenge of the analysis is the presence of residual stress in the reference configuration of the polymer. The work yielded numerical evaluations of shear stress at the interface between two gels representing bone tissue and the artificial implant. The article [2] also presents a variational formulation of gel equilibrium, from the point of view of the Flory-Rehner theory, which combines Neo-Hookean elasticity with mixing energy and performed numerical simulation. The Flory-Rehner theory is consistent with the Lagrangian formulation of elastic solids and the Eulerian treatment of the solvent fluid.Research in gel mechanics has followed two main modeling approa...

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Modelling of and Mixed Finite Element Methods for Gels in Biomedical Applications

Cited by 11 publications

References 22 publications

A Dynamic Model of Polyelectrolyte Gels

A Dynamic Model of Polyelectrolyte Gels

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

Effects of permeability and viscosity in linear polymeric gels

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