SUMMARYWe consider a problem stemming from recent models of phase transitions in stimulus-responsive hydrogels, wherein a sharp interface separates swelled and collapsed phases. Extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly 'fit' the interface geometry are emphasized. Attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface. We show that as the reciprocal interfacial mobility vanishes, it plays the role of a penalty parameter enforcing a pure Dirichlet constraint, eventually triggering oscillations in the interfacial velocity. We also examine alternative formulations employing a Lagrange multiplier to enforce this constraint. It is shown that the most convenient choice of basis for the Lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms, suggesting a lack of stability in the discrete formulation. Under such conditions, neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the Lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux. Fortunately, the adaptation and use of local, domain-integral methodologies considerably improves the flux evaluations. Several example problems are presented to compare and contrast the various techniques and methods.
SUMMARYA hybrid numerical method for modelling the evolution of sharp phase interfaces on ÿxed grids is presented. We focus attention on two-dimensional solidiÿcation problems, where the temperature ÿeld evolves according to classical heat conduction in two subdomains separated by a moving freezing front. The enrichment strategies of the eXtended Finite Element Method (X-FEM) are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary. A new approach with the X-FEM is suggested for this class of problems whereby the partition of unity is constructed with C 1 ( ) polynomials and enriched with a C 0 ( ) function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to signiÿcantly improve estimates for the front velocity. Temporal derivatives of the temperature ÿeld in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a ÿner ÿnite di erence grid, the Level Set method is used to represent the evolution of the phase interface. An iterative procedure is adopted to satisfy the constraints on the temperature ÿeld on the phase boundary. The robustness and utility of the method is demonstrated with several benchmark problems of phase transformation.
We present a theory for the chemically-induced volume transitions of hydrogels. Consistent with experimental observations, we account for a sharp interface separating swelled and collapsed phases of the underlying polymer network. The polymer chains are treated as a solute with an associated diffusion potential and their concentration is assumed to be discontinuous across the interface. In addition to the standard bulk and interfacial equations imposing force balance and solute balance, the theory involves an ancillary interfacial equation imposing configurational force balance. Motivated by experimental observations, we specialize the theory to the situation where the time scale associated with the interface motion is slow compared to those associated with diffusion in the bulk phases. We present a hybrid eXtended-Finite-Element/Level-Set Method (XFE/LSM) for obtaining approximate solutions to the equations arising under this specialization. As an application, we consider the swelling of a spherical specimen whose boundary is traction-free and is in contact with a reservoir of uniform chemical potential. Our numerical results exhibit good qualitative comparison with experimental observations and predict characteristic swelling times that are proportional to the square of the specimen radius. Our results also suggest several possible synthetic pathways that might be pursued as a means to engineer hydrogels with optimal response times.
We present a continuum model for thermally induced volume transitions in stimulus-responsive hydrogels (SRHs). The framework views the transition as proceeding via the motion of a sharp interface separating swollen and collapsed phases of the underlying polymer network. In addition to bulk and interfacial force and energy balances, our model imposes an interfacial normal configurational force balance. To account for the large volume changes exhibited by SRHs during actuation, the governing equations are developed in the setting of finite-strain kinematics. The numerical approximations to the coupled thermomechanical equations are obtained with an extended finite element/level-set method. The solution strategy involves a non-standard operator split and a simplified version of the level-set update. A number of representative problems are considered to investigate the model and compare its predictions to experimental observations. In particular, we consider the thermally induced swelling of spherical and cylindrical specimens. The stability of the interface evolution is also examined.
We present a strategy for obtaining numerical solutions to a system of nonlinear, coupled evolution equations describing volume transitions in stimulus-responsive hydrogels (SRHs). The theory underlying our sharp-interface model of phase transitions in SRHs is provided along with the assumptions leading to the specialized formulation that is the starting point for the numerical method. The discrete formulation is then developed with a hybrid eXtended Finite-Element/Level-Set Method (XFE/LSM). Domain-integral methodologies are used consistently to extract interfacial quantities such as the mechanical driving traction, the jump in the normal component of the solute flux, and requisite geometric information. Several benchmark studies are provided to demonstrate the accuracy and robustness of the numerical strategy. We then investigate various features of SRH kinetics including the regimes of unstable and stable phase transitions, surface pattern formation, and bulk phase separation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.