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A DYNAMIC MODEL OF POLYELECTROLYTE GELS
"... 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-f ..."
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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 Navier-Stokes 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.
MODELLING OF AND MIXED FINITE ELEMENT METHODS FOR GELS IN BIOMEDICAL APPLICATIONS ∗
, 1305
"... Abstract. A set of equilibrium equations for a biphasic polymer gel are considered with the end purpose of studying stress and deformation in confinement problems encountered in connection with biomedical implants. The existence of minimizers for the gel energy is established first. Further, the sma ..."
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Abstract. A set of equilibrium equations for a biphasic polymer gel are considered with the end purpose of studying stress and deformation in confinement problems encountered in connection with biomedical implants. The existence of minimizers for the gel energy is established first. Further, the small-strain equations are derived and related to the linear elasticity equations with parameters dependent on the elasticity of the polymer and the mixing of the polymer and solvent. Two numerical methods are considered, namely a two-field displacement-pressure formulation and a three-field stressdisplacement-rotation formulation with weakly imposed symmetry. The symmetry of the stress tensor is affected by the residual stress induced by the polymer-solvent mixing. A novel variation of the stress-displacement formulation of linear elasticity with weak symmetry is therefore proposed and analyzed. Finally, the numerical methods are used to simulate the stresses arising in a confined gel implant.
ANALYSIS AND SIMULATION OF A MODEL OF POLYELECTROLYTE GEL IN ONE SPATIAL DIMENSION
"... Abstract. We analyze a model of polyelectrolyte gels that was proposed by the authors in previ-ous 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 ..."
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Abstract. We analyze a model of polyelectrolyte gels that was proposed by the authors in previ-ous 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 simula-tions for the spatially one-dimensional problem are presented, confirming the analytical calculations on stability.
EFFECTS OF PERMEABILITY AND VISCOSITY IN LINEAR POLYMERIC GELS
"... Abstract. We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum. 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 pr ..."
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Abstract. We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum. 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 relevant equilibrium solutions, 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. The fact that some equilibrium solutions of interest are not stress free brings additional challenges to the analysis, and, in particular, to the derivation of the energy law of the systems. It also singles out the special role of the rotations in the analysis. From the point of view of applications, we point out that the conditions that guarantee stability of solutions also provide criteria to select material parameters for devices. The boundary conditions that we consider are of two types, first displacement-traction conditions for the governing equation of the polymer component, and secondly permeability conditions for the fluid equation. We present a rigorous study of these conditions in terms of balance laws of the fluid across the interface between the gel and its environment [20], and use it to justify heuristic permeability formulations found in the literature [39], [14]. We also consider the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component. We establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two-dimensional, finite
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, 2008
"... We consider a gel as an immiscible mixture of polymer and solvent, and derive governing equations of the dynamics. They include the balance of mass and linear momentum of the individual components. The model allows to account for nonlinear elasticity, viscoelasticity, transport and diffusion. The to ..."
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We consider a gel as an immiscible mixture of polymer and solvent, and derive governing equations of the dynamics. They include the balance of mass and linear momentum of the individual components. The model allows to account for nonlinear elasticity, viscoelasticity, transport and diffusion. The total free energy of the system combines the elastic contribution of the polymer with the Flory-Huggins energy of mixing. The system is also formulated in terms of the center of mass velocity and the diffusive velocity, involving the total and the relative stresses. This allows for the identification of special regimes, such as the purely diffusive and the transport ones. We also obtain an equation for the rate of change of the total energy yielding decay for special choices of boundary conditions. The energy law motivates the Rayleghian variational approach discussed in the last part of the article. We consider the case of a gel in a one-dimensional strip domain in order to study special features of the dynamics, in particular, the early dynamics. We find that the monotonicity of the extensional stress is a necessary condition to guarantee the propagation
SOLID-FLUID DYNAMICS OF YIELD-STRESS FLUIDS
"... We present a two-phase, solid-fluid, continuum model of yield-stress fluids describing both solid deformations in unyielded regions and liquid flows in yielded regions. Solutions of its governing equations, that are written in the form of local conservation laws of Godunov type, are proven to agree ..."
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We present a two-phase, solid-fluid, continuum model of yield-stress fluids describing both solid deformations in unyielded regions and liquid flows in yielded regions. Solutions of its governing equations, that are written in the form of local conservation laws of Godunov type, are proven to agree with mechanics and thermodynamics. The structure of the governing equations expressing their physical content is required to be preserved also in their discretized versions that arise in three numerical illustrations in which one-dimensional shock-wave type solutions are explored.
1LONG-TIME EXISTENCE OF CLASSICAL SOLUTIONS TO A 1-D SWELLING GEL
"... Abstract. In this paper we derived a model which describes the swelling dynamics of a gel and study the system in one-dimensional geometry with a free boundary. The governing equations are hyperbolic with a weakly dissipative source. Using a mass-Lagrangian formulation, the free-boundary is transfor ..."
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Abstract. In this paper we derived a model which describes the swelling dynamics of a gel and study the system in one-dimensional geometry with a free boundary. The governing equations are hyperbolic with a weakly dissipative source. Using a mass-Lagrangian formulation, the free-boundary is transformed into a fixed-boundary. We prove the existence of long time C1-solutions to the transformed fixed boundary problem.
