As such, the multi-state characteristics of the smart polymer hydrogels and their wide range of multiphysics applications make the multi-disciplinary and multi-phase the basic requirements for the mathematical models. A complete theoretical platform outlining the fundamental theory for the smart polymer hydrogels is established.
Introduction
Definition and Application of Hydrogel
Examples of the synthetic hydrogels include N-isopropylacrylamide (NIPA), poly(acrylic acid) (PAA), poly(acrylonitrile) (PAN), poly(acrylamide) (PAM), poly(acrylonitrile)/poly(pyrrole) (PAN/ PPY ), poly(vinyl alcohol) poly(acrylic acid) (PVA–PAA), poly(hydroxyethyl methacrylate) (PHEMA). However, the eventual stability of the hydrogel depends on the interaction between the polymer matrix network and the aqueous medium in which the hydrogel is immersed.
Historical Development of Modelling Hydrogel
- Steady-State Modelling for Equilibrium of Smart HydrogelsHydrogels
- Mathematical Models and Simulations
- Key Parameters in Steady-State Modelling for Equilibrium of Hydrogels
- Transient Modelling for Kinetics of Smart Hydrogels
- Mathematical Models and Simulations
- Key Parameters in Transient Modelling for Kinetics of Hydrogels Although there are many parameters having influences on the kinetics of volume
- A Theoretical Formalism for Diffusion Coupled with Large Deformation of Hydrogelwith Large Deformation of Hydrogel
- Remarks
The volume transition mechanism of the poly(AAc) hydrogel is dependent on the preparative concentration of AAc in solution under γ-irradiation in addition to the pH of the medium for the volume transition (Jabbari and Nozari, 2000). The swelling of the poly(NIPAAm) hydrogel in the gelled aqueous corn starch solution is mainly determined by the ratio of hydrated polymer chains instead of the water diffusion rate (Zhang and Zhuo, 2000c).
About This Monograph
They provide a simple and qualitative approach to predict the volume transition equilibria of various hydrogels. Accordingly, parametric studies are conducted to further discuss the electrosensitive hydrogels in equilibrium and kinetics, respectively.
On the water swelling behavior of poly(N-isopropylacrylamide) [P(N-iPAAm)] poly(methacrylic acid) [P(MAA)] their random copolymers and consecutive interpenetrating polymer networks (IPNs). Synthesis and swelling kinetics of poly(dimethylaminoethyl acrylate methyl chloride quaternary-co-itaconic acid) hydrogels.
Multi-Effect-Coupling pH-Stimulus (MECpH) Model for pH-Sensitive Hydrogel
Introduction
Development of the MECpH Model
- Electrochemical Formulation
- Ionic Flux
- Electrical Potential
- Fixed Charge Group
- Mechanical Formulation
As is known, the Nernst-Planck equation can characterize the ionic fluxes in the hydrogel in terms of the gradients of the ionic concentration, electric potential and pressure. Dk(m2/s) is the k-th type diffusivity tensor, i is the direction of the flux current. Therefore, the contribution of the chemical activity coefficient is relatively small and thus negligible.
The Poisson equation is a more rigorous approach to characterize the spatial distribution of the electric potential in the domain. It is observed that the assumptions of constant field and electroneutrality are actually the special cases of the Poisson equation.
Computational Domain, Boundary Condition and Numerical Implementationand Numerical Implementation
It is absolutely impossible for the MECpH model to have any closed-form analytical solutions composed of the PNP system (2.2) and (2.24) coupled with the mechanical equation (2.52) or (2.57). As a result, the non-dimensional form of the MECpH governing equations can be written to simplify numerical simulation. Due to the deformation u, the fixed charge density cfis is redistributed in the hydrogel, and it therefore requires a new calculation again.
The swelling/deswelling of the pH-sensitive hydrogel at equilibrium can be predicted by the steady-state simulation based on the Nernst-Planck equations, the Poisson equation and the mechanical equilibrium equation, collectively known as the MECpH model. Second, the convection term in the Nernst-Planck flux equations (2.2) is negligible, since the fluid pressure across the hydrogel remains constant and the fluid velocity remains unchanged upon swelling of the hydrogel.
Model Validation with Experiment
Finally, it is assumed that the properties of the hydrogel under consideration are homogeneous, so that only isotropic equilibrium swelling occurs. We observed the volume changes of the hydrogel and measured the diameter of the cylindrical hydrogel after reaching equilibrium at different environmental pH values (circular markers in Figure 2.4). When the swelling of the hydrogel becomes larger, for example, when the ionizable fixed charge concentration increases, the linear strain theory gives a slightly larger swelling rate than that of the finite strain theory.
As is known, the change of ambient pH changes the degree of ionization of the fixed charge groups and the state of equilibrium swells simultaneously. As the pH of bath solution increases, the ionization of the pendant poly(HEMA) carboxyl.
Parameter Studies by Steady-State Simulation for Equilibrium of Hydrogelfor Equilibrium of Hydrogel
- Influence of Initially Fixed Charge Density of Hydrogel
- Influence of Young’s Modulus of Hydrogel
- Influence of Initial Geometry of Hydrogel
- Influence of Ionic Strength of Bath Solution
- Influence of Multivalent Ionic Composition of Bath Solutionof Bath Solution
It appears that the degree of swelling of the hydrogel decreases by following the sequence of csm and 1200 mM. The characteristic profiles of the ion concentrations and electric potential are similar to those in Fig. The increase in Young's modulus exponentially reduces the swelling of the hydrogels for the three.
As the ionic strength of the bath solution increases, the ratio of ionic concentrations between the inner hydrogel and the outer bath solution decreases (Siegel, 1990). The responsive properties of the pH-sensitive hydrogel decrease the swelling rate exponentially as the ionic strength of the environment.
Remarks
The MECpH model can easily handle the large deformation of the pH-sensitive hydrogels, based on the geometrically nonlinear finite deformation theory. The pendant charges fixed on the backbone of the polymer network of the hydrogels, e.g. Meanwhile, the elastic retraction force of the polymer network balances with the expanding network.
The Robinson buffer system has a lower ionic strength than that of the phosphate buffer system and the HCl/NaCl solution. The effect of sodium ions on the electrical activity of the giant squid axon.
Multi-Effect-Coupling Electric-Stimulus
MECe) Model for Electric-Sensitive Hydrogel
Introduction
Development of the MECe Model
- Formulation of the MECe Governing Equations
- Boundary and Initial Conditions
1+tr(E)) (3.4) whereφ0 is the volume fraction of the polymeric solid phase at reference configuration. Equation (3.6) is then rewritten as 3.2) and (3.7), the continuity condition of the mixture of the hydrogel is obtained as. If the internal energy of the hydrogel mixture U can be expressed by the Helmholtz energy function F as.
3.28) where γα is the rate of heat generation per unit mass of phase α and qα is the heat flux vector. As is well known, pressure results from the difference in diffusive ion concentrations between the hydrogel and the surrounding solution.
Steady-State Simulation for Equilibrium of Hydrogel
- Numerical Implementation
- Model Validation with Experiment
- Parameter Studies
- Influence of Externally Applied Electric Voltage
- Influence of Initially Fixed Charge Density of Hydrogel
- Influence of Concentration of Bath Solution
- Influence of Ionic Valence of Bath Solution
In this section, the influence of externally applied electric field on the response behaviors of electrosensitive hydrogels is discussed. Furthermore, to investigate the influence of the externally applied electric field Ve on the variation of the mean curvature Ka with the bath solution concentration c∗ , Figs. As the fixed charge density cf0 increases, the more mobile ions diffuse into the hydrogel. , the higher the conductivity the hydrogel reaches.
It is observed that, with the increase of the externally applied electric voltage Ve, the differences of both the diffusive ions. The influence of the concentration of the surrounding bath solution c∗ on the distribution profiles of the diffuse ionic concentrations and the electric potential, as well as the hydrogel displacement, is shown in Fig.
Transient Simulation for Kinetics of Hydrogel
- Numerical Implementation
- Model Validation with Experiment
- Parameter Studies
- Variation of Ionic Concentration Distribution with Time
- Variation of Electric Potential Distribution with Time
- Variation of Hydrogel Displacement Distribution with Time
- Variation of Hydrogel Average Curvature with Time
Briefly, the trends of kinetics of the diffuse ionic concentration distributions shown in Figs. It is suggested that the downward step of the electrical potential propagated within the hydrogels generally becomes larger with increasing time. 3.41, 3.44 and 3.45 that, with the increase of the fixed charge density, the variation of the electric potential distributions becomes small with time.
It turns out that the average curvature at a given instant increases with the increase of the fixed charge density. This is consistent with the previous steady-state studies, where the increase in solid charge density causes greater deformation of the hydrogel.
Remarks
This results in an almost linear distribution of the electric potential and reduces the critical time of the kinetics of the hydrogels. Finally, it should be noted that the response time of electrically sensitive hydrogels to an externally applied electrical trigger is generally always very short, normally shorter than 2 min in the cases of the present simulations. The simulated phenomena agree well with the experimental findings and prove the great promise of electrically sensitive hydrogels in biotechnology and bioengineering applications.
Therefore, for the analysis of the responsive properties of the electrically sensitive hydrogels in more accuracy, it is necessary to make the two-dimensional or three-dimensional analyzes in the future. Numerical simulation of the steady-state deformation of a smart hydrogel under an external electric field.
Introduction
With the emergence of various applications that rely on the swelling behavior of hydrogels, there comes a compelling need for physically accurate theories and numerical simulations capable of capturing, analyzing, and predicting the behavior. Theories and models should provide a sufficient physical understanding of the various processes involved in the swelling of hydrogels (Dolbow et al., 2005). The Multi-Effect Coupling Electrical pH Stimuli (MECpHe) model was first developed for hydrogels that respond to coupled pH-electrical stimuli.
After discretization of the MECpHe determining the equations and boundary conditions, the current MECpHe model is examined by comparison between the simulation results and the published experiments. It is followed by parameter studies on the influences of several key hydrogel material properties and environmental solution conditions on the response characteristics of the smart hydrogels (Fig. 4.1).
Development of the MECpHe Model
If chemical flux is neglected, the Nernst-Planck type of mass conservation is In the current MECpHe model, the total concentration of fixed charge groups within the hydrogel in the relaxed state is defined as . The volume fractions of the interstitial and polymeric water solid phases are thus written as.
The ratio between the volume fractions of interstitial water and solid phases of the polymer network is then given by Due to the absence of symmetry in the first Piola–Kirchhoff stress tensor P, it is rarely used in constitutive equations.
Numerical Implementation
The boundary condition of the mechanical deformation is imposed on the hydrogel-. ck−c0k) at X=(L±h)/2 (4.32) To prevent the hydrogel from undergoing rigid body motion, a point constraint is requested at the center of the hydrogel. The non-dimensional form of the nonlinear coupled partial differential governing equations of the MECpHe model and the auxiliary conditions for one-dimensional steady-state simulation are finally discretized as .
Model Validation with Experiment
In addition, H-bonding interactions occur between the carboxyl groups in the hydrogel, yielding a compact H-bonded structure to the hydrogel. This generates the extensive swelling of the hydrogel, as indicated by the higher swelling ratio of the hydrogel (Bajpai and Dubey, 2005). The distance between the two carbon electrodes L=30 mm, and the hydrogel strip is custom-made in 20×5×0.2 mm3.
Basically, the mechanism of hydrogel deformation can be explained by Flory's osmotic pressure theory (Shiga and Kurauchi, 1990; Yang and Engberts, 2000). As a result, a diffusible ion concentration gradient develops, which generates the osmotic pressure due to the change in diffusible ion concentrations across the interfaces between the hydrogel and the surrounding solution.
Parameter Studies by Steady-State Simulation for Equilibrium of Hydrogelfor Equilibrium of Hydrogel
- Influence of Solution pH Coupled with External Electric VoltageElectric Voltage
Because the increase in osmotic pressure at the interface near the anode is greater than that at the interface near the cathode, the hydrogel near the anode swells more than that near the cathode, resulting in the bending toward the cathode, as shown. in fig. It can be seen from the figures that the phenomenon of electroneutrality exists in the bath solution and the hydrogel strip. The concentrations of the diffuse ionic species Na+ and Cl− are distributed uniformly within the hydrogels and symmetrically over the entire system domain.
The same profiles of the concentration differences are observed across the two hydrogel-solution interfaces near the anode and cathode. However, once the electric field is applied, such as Ve=0.16 V, the distributions of the diffusible ionic species Na+ and Cl− concentrations are no longer uniform in the hydrogel and bath solution, nor are they symmetrical in the entire domain.
