As described above, the classical triphasic and multi-phasic mixture models are unable to directly calculate the distribution of electrical potentials due to the use of electro-neutrality condition. To overcome the drawbacks mentioned, a novel mathematical model based on modified multi-phasic mixture theories is developed, and the governing equations are derived as follows. By summarizing Equations 6 to 7 for all N ionic species, and neglecting and , when compared with [ 19 - 22 ], we have. Substituting Equation 13 into Equation 14 , the continuity equation of the mixture phase is obtained,.
Where u s is the displacement of the solid phase and B w is set to zero, since its value is very small [ 23 , 24 ]. In order to incorporate the effects of migration and diffusion, the Nernet-Planck type of convection-diffusion equations for each diffusive ionic flux, replacing the continuity equation 2 of the ionic phase, is employed,. Moreover, as the velocity of propagation in the electric field is much higher than the one occurring in the convection-diffusion equation, the Poisson equation is given by,.
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The boundary conditions consist of Dirichlet boundary conditions at the ends of bathing solution,. The interfacial conditions above are based on the assumptions that, for each time step on the hydrogel-solution interface, the chemical potentials of hydrogel membrane are equal to those of bathing solution [ 24 ], and the total mixture stress is equal to zero. A two-level hierarchical iteration technique is carried out for each time step, to solve the coupled nonlinear partial differential governing equations.
The inner iterations are used for the computation of the diffusive ionic concentrations and electric potential simultaneously. Substitute the implemented numerical results into the subsequent outer iteration loop for calculation of the displacements and pressure. The fixed-charge concentration and water volume fraction can also be computed iteratively.
An implicit numerical scheme is conducted to solve the transient nonlinear convection-diffusion equations 15 and A recently developed meshless approach-Hermite-Cloud Method HCM [ 23 ], is employed for the spatial discretization of the governing equations. In this section, meshless Hermite-Cloud Method HCM [ 23 ], is introduced for the spatial discretization of the above nonlinear partial differential initial value problems.
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As an extension of the classical Reproducing Kernel Particle Method RKPM [ 25 - 27 ], this meshless approach employs Hermite theorem for the construction of the interpolation functions, and uses the point collocation technique for discretization of Partial Differential equations with Boundary Value PDBV problems. Where N T and are total numbers of discrete points covering the computational domain, and are the shape functions defined as,.
In which is the kernel function defined as,. For example, in one-dimensional or two-dimensional quadratic PDBV problem, it is given respectively. And is a symmetric matrix associated with the fixed kernel centered point , and is expressed as. In the equation 25 , is called the window function, and defined here as a cubic spline function,. Assuming , the discrete approximations of may be given by,. The auxiliary conditions are required for the additional unknown functions.
By imposing the first-order partial differentials with respect to the variables x and y on the approximation expressed by the equation 22 , and using the equation 29 , we have the following auxiliary conditions. The meshless method employs the Hermite interpolation theorem to construct the approximate unknown function , in form of the equation 22 , and couples the first-order differential functions , by the equation 29 and the auxiliary conditions equations 30 and An implicit numerical scheme is developed to solve the transient nonlinear convection-diffusion equations 15 and 16 , and a Newton iteration process is also implemented for each time step.
In consideration of the general case, the convection-diffusion equation is given by. In which k is the diffusion coefficient, v is the convection coefficient,c 1 , c 2 are known constants, and f x,t and are specified known functions. If we define , respectively, equation 35 can be given by,.
It should be noted if a 2-dimensional matrix A is specified,. Where is the strain in thickness direction. An approximately average curvature Ka, resulting from the pressure difference between the two ends of the thickness h, is defined geometrically at the middle point of the thickness [ 18 ],. Where e 1 and e 2 indicate the swelling strains at the two ends of hydrogels thickness. To investigate the transient nonlinear behaviors of electric-stimulus responsive hydrogels, one-dimensional numerical simulations are carried out for a hydrogel strip immersed into NaCl solution applied by an externally applied electric field.
The steady-state solutions without an externally applied electric field, as shown in figures 3 and 4 are taken as initial condition for numerical simulations. The transient variations of diffusive ionic concentrations and electrical potential are numerically simulated and shown in figures 5 and 6.
Additionally, it is observed from figure 7 that the electric potential is also increasing on the cathode side and decreasing on the anode side of the gel. The constant distribution of concentration of fixed charges is provided in figure 8. The change of electric potential within the hydrogel is smaller than that in the exterior solution due to the higher conductivity of the mobile ions in the hydrogel strip.
The interface concentration difference between the interior hydrogel membrane and exterior bathing solution on the anode side is larger than that on the cathode side. These simulated phenomena were in good agreement with the published experimental results [ 29 ].
The larger interface differences of concentrations and electric potential on the anode side result in the larger pressure, compared with those on the cathode side. This pressure difference makes the hydrogel strip have a bending deformation. You may send this item to up to five recipients. The name field is required. Please enter your name. The E-mail message field is required. Please enter the message. Please verify that you are not a robot.
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You already recently rated this item. Your rating has been recorded. Write a review Rate this item: 1 2 3 4 5. Preview this item Preview this item. This is especially so in advanced biomaterials such as polymeric hydrogels responsive to biostimuli for a wide range of potential BioMEMS applications, where multiphysics and mul- phase are common requirements. These environmental stimuli-responsive hydrogels are often known as smart hydrogels.
In the published studies on the smart or stimu- responsive hydrogels, the literature search clearly indicates that the vast majority are experimental based. In particular, although there are a few published books on the smart hydrogels, none is involved in the modelling of smart hydrogels. For the few published journal papers that conducted mathematical modelling and numerical simulation, results were far from satisfactory, and showed signi?
This has resulted in ad hoc studies of these hydrogel materials mainly conducted by trial and error. This is a very time-consuming and inef? Read more Show all links. Allow this favorite library to be seen by others Keep this favorite library private. How to cite this article If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Download Citation If you have the appropriate software installed, you can download article citation data to the citation manager of your choice.
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Andreas Voigt Andreas Voigt. Uwe Marschner. Andreas Richter. Article information.subccanlazi.tk
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Article Information Volume: 30 issue: 9, page s : Article first published online: December 4, ; Issue published: May 1, Email: andreas. Abstract Full Text Abstract. Keywords active polymers , modeling polymers , hydrogel-based micro-valve , ANSYS , multi-field simulation , concentration-triggered hydrogel , circuit descriptions , lab-on-a-chip. Sign Out. Email required Password required Remember me Forgotten your password? Need to activate? Institutional Access does not have access to this content.
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