Analytical and Bioanalytical Chemistry

, Volume 389, Issue 3, pp 863–873

Modeling and simulation of chemo-electro-mechanical behavior of pH-electric-sensitive hydrogel

Authors

  • Rongmo Luo
    • Department of Mechanical EngineeringNational University of Singapore
    • School of Mechanical and Aerospace EngineeringNanyang Technological University
  • Khin Yong Lam
    • School of Mechanical and Aerospace EngineeringNanyang Technological University
Original Paper

DOI: 10.1007/s00216-007-1483-9

Cite this article as:
Luo, R., Li, H. & Lam, K.Y. Anal Bioanal Chem (2007) 389: 863. doi:10.1007/s00216-007-1483-9

Abstract

A chemo-electro-mechanical multi-field model, termed the multi-effect-coupling pH-electric-stimuli (MECpHe) model, has been developed to simulate the response behavior of smart hydrogels subject to pH and electric voltage coupled stimuli when the hydrogels are immersed in a pH buffer solution subject to an externally applied electric field. The MECpHe model developed considers multiphysics effects and formulates the fixed charge density with the coupled buffer solution pH and electric voltage effects, expressed by a set of nonlinear partial differential governing equations. The model can be used to predict the hydrogel displacement and the distributive profiles of the concentrations of diffusive ionic species and the electric potential and the fixed charge density in both the hydrogels and surrounding solution. After validation of the model by comparison of current numerical results with experiment data extracted from the literature, one-dimensional steady-state simulations were carried out for equilibrium of the smart hydrogels subject to pH and electric coupled stimuli. The effects of several important physical conditions, including the externally applied electric voltage, on the distributions of the concentrations of diffusive ionic species, the electric potential, the fixed charge density, and the displacement of the hydrogel strip were studied in detail. The effects of the ionic strength on the bending deformation of the hydrogels under the solution pH and electric voltage coupled stimuli are also discussed.

Keywords

pH-electric-sensitive hydrogelChemo-electro-mechanical modelSmart materialsSolution pH and electric voltage coupled stimuliBending deformationMeshless method

Introduction

As is well known, stimuli-responsive hydrogels have increasingly received considerable attention. They are a class of unique macromolecular crosslinked networks that contain three phases, namely the solid matrix network, interstitial fluid, and ionic species. One of the most notable properties of these materials is that they undergo substantial and abrupt changes in volume in response to external stimuli such as pH, electric field, temperature, etc. [110]. The volume transitions of hydrogels have prompted researchers to investigate the smart materials for application of sensors, actuators, artificial muscles, drug-delivery systems, and separation processes in biomedical engineering [11, 12].

To robustly make the above applications, the stimuli-responsive hydrogels are required with the capability of fast environment-sensitive response and high mechanical strength. However, most of studies have focused more on smart hydrogels responding to a single environmental stimulus, in which their response rate is relatively slow and they have limited axial displacement. Usually two techniques are proposed to overcome the drawbacks. One is geometric reduction of the hydrogels [11]. However, this makes the hydrogel systems fragile and reduces their mechanical strength, which limits their application [13]. The other is the stimulation of the hydrogels by coupled environmental stimuli. Several stimuli-responsive hydrogels were studied to combine two or more stimuli [14, 15]. By selecting the functional groups along the macromolecular crosslinked networks, hydrogels can be designed and synthesized for sensitive response to a change of the surrounding environment [16]. Hydrogels which are prepared from polyelectrolyte gels with relatively high concentrations of ionizable groups fixed on the polymeric crosslinked chains can exhibit pH-responsive characteristics and also respond to electric stimulus simultaneously [12, 1421]. Electric field is an attractive example of the environmental stimulus, because of easy control and manipulation, which induces a volumetric change in a polyelectrolyte gel. Therefore, many experiments were carried out to study the deformation of the smart hydrogels responding to the solution pH and electric field coupled stimuli. For example, using poly(methyl methacrylate)–poly(acrylic acid) (PMMA–PAA) hydrogels, an electrically controlled modulator of solution pH was developed, which may be the primary application of pH-electric-sensitive hydrogels [22]. Shiga investigated the deformation of PAA hydrogels near the phase transition point in the basic solution under the action of DC electric fields, and observed that the type of deformation depends on the concentration of ionic species and the conformation of polymeric chains [23]. At the phase-transition point, the volume of swollen hydrogel could be several times of that at the collapsed state, and the transition can exert a significant force. Because of the good mechanical property, the pH-electro-responsive hydrogel was designed as an actuator in drug-delivery systems [17, 18]. Recently, much attention has been devoted to investigation of the equilibrium swelling and bending behavior of interpenetrating polymer networks (INPs) responsive to pH-electric coupled stimuli, for example hyaluronic acid–poly(vinyl alcohol) (HA–PVA) hydrogels, polymethacrylic acid–poly(vinyl alcohol) (PMAA–PVA) hydrogel, poly(2-acrylamido-2-methylpropane sulfonic acid)–hyaluronic acid hydrogels, poly(vinyl alcohol)–poly(acrylic acid) (PVA–PAA) hydrogel, and poly(acrylic acid)–poly(vinyl sulfonic acid) (AAc–VS) copolymer [21, 2427]. The chemo-electro-mechanical properties of these pH-electric-sensitive hydrogels make them be promising candidates in wide-range biomedical applications [25].

Even though the mechanical behavior of the pH-electric-sensitive hydrogels have been widely studied experimentally, the stimuli-responsive mechanism of the hydrogels remains unclear. Several mathematical models have been developed for simulation of the responsive hydrogels to the single stimulus of externally applied electric field only [2831], but they suffer from several drawbacks. For instance, the model by Gu et al. is able to simulate the distribution of electric potential in the tissue only, instead of the whole system including both the tissue and surrounding solution [28]. The mechanical governing equation in Wallmersperger’s model is expressed by Newton’s second law [29], resulting in inaccurate description of the nonlinear deformation of the hydrogels under higher electric potentials. Crucially, it is difficult for these models to simulate accurately the response of the hydrogels to the pH-electric coupled stimuli. Tamagawa and Taya theoretically investigated the distribution of mobile ions in an amphoteric hydrogel and obtained symmetrical ionic distribution profiles in the pH-electric-sensitive hydrogel [32]. Under the applied electric field, the distributive profiles of the mobile ionic concentrations in the cationic or anionic hydrogel should be heterogeneous with varied gradients [3335]. Because of this controversy and lack of clarity, a more robust model is required to understand further the responsive behavior of stimuli-sensitive hydrogel subject to the pH-electric coupled stimuli.

In this paper, a multiphysics model, termed the “multi-effect-coupling pH-electric-stimuli” (MECpHe) model, has been developed for simulation of the bending and swelling deformation of hydrogels subject to the coupled stimuli of the buffer solution pH and the externally applied electric field. For the present smart multiphase hydrogels with polymeric solid-phase matrix, interstitial fluid, and ionic species, the MECpHe model developed consists of the Nernst–Planck equations for mobile ion concentrations, the Poisson equation for electric potential, and a nonlinear mechanical equation for finite deformation of the hydrogel. The formulation of the fixed charge groups bonded to the crosslinked polymeric network is associated with ambient solution pH and externally applied electric field. The model is solved numerically by the meshless Hermite-cloud method for one-dimensional steady-state simulation of the swelling equilibrium of smart hydrogels responding to the pH-electric coupled stimuli when immersed in a pH buffer solution subject to applied electric voltage. By comparison with experimental data from published literature the model was well validated for the response of the pH-electric-sensitive hydrogels. Simulations were then conducted to predict the effects of buffer solution pH and electric potential coupled stimuli on the distributions of the concentrations of diffusive ionic species and on the displacement of the hydrogel. The effect of ionic strength on the bending deformation of hydrogels subject to solution pH and electric voltage coupled stimuli was also investigated.

Development of MECpHe model

Swelling and bending mechanism

In this study, the pH-electric-sensitive hydrogel is considered as a triphasic hydrophilic mixture including the solid matrix network, interstitial fluid, and ionic species. When the hydrogel is immersed in a pH buffer solution the ion species diffuse between the porous hydrogel and surrounding solution. Because of the presence of the fixed-charge groups, differences are created between ionic concentrations in the interior hydrogels and the exterior solution, resulting in an osmotic pressure. The pressure is the driving force for the swelling mechanism of the hydrogel. The swelling or shrinking of the hydrogel subsequently redistributes the ionic concentrations within the hydrogel. The process repeats until the hydrogel reaches the equilibrium state. When an external electric field is applied, mobile ions will move toward their counter electrode. As a result, the ionic concentration gradient is developed. Osmotic pressure arising as a result of the ion concentration difference is generated at the interfaces between the hydrogel and the surrounding solution, because of the fixed negatively charged groups within the hydrogel. The cations diffuse into the hydrogel more than the anions, and move toward the cathode. Because the increase of osmotic pressure at the interface near the anode is larger than that at the interface near the cathode, the hydrogel near the anode swells more than that near the cathode, which results in bending toward the cathode. The above mechanism of hydrogel deformation can be explained well by Flory’s osmotic pressure theory [36, 37].

Theoretical formulations of MECpHe model

By the law of mass conservation, the change in the number of moles of a diffusive species k in a considered volume with respect to time t can be characterized by the difference between the fluxes entering and leaving the reference volumes. The Nernst–Planck type of continuity equation is thus obtained as [38, 39]:
$$ \begin{array}{*{20}c} {{\frac{{\partial c_{k} }} {{\partial t}} + div{\left( {{\mathbf{J}}_{k} } \right)} + v_{k} r = 0}}{{{\left( {k = 1,2, \ldots ,N} \right)}}} \\ \end{array} $$
(1)
$$ {\mathbf{J}}_{k} = - {\left[ {D_{k} } \right]}{\left( {{\text{grad}}{\left( {c_{k} } \right)} + \frac{{z_{k} F}} {{RT}}c_{k} {\text{grad}}{\left( \psi \right)} + c_{k} {\text{grad}}{\left( {\ln \gamma _{k} } \right)}} \right)} + c_{k} V $$
(2)
in which Jk, [Dk], ck, zk, γk and vk, are the flux (mmol L−1 s−1), diffusivity tensor (m2 s−1), concentration (mmol L−1), valence number, chemical activity coefficient and stoichiometric coefficient in the chemical reaction for the kth diffusive ionic species (k = 1, 2, ..., N). N is the total number of diffusive ionic species, V is the velocity of the solvent flow, r is the rate of chemical reaction which represents a source term, and ψ is the electrostatic potential. F, R, and T are the Faraday constant (9.6487 × 104 C mol−1), the universal gas constant (8.314 J mol−1 K−1), and absolute temperature (K), respectively.

On the right hand side of Eq. (2), the first term represents the diffusive flux because of the concentration gradient, the second the migration flux because of the gradient of the electric potential, the third the chemical flux related to the activity coefficient, and the fourth the convective flux originating from applied convection of the solvent.

It is assumed in this paper that the pore of the hydrogels is very small and that the smart hydrogels are placed in an unstirred solution in a vibration-free experimental device, i.e. diffusion dominates the flux transmitter. As such, the bulk flow of fluid or hydrodynamic velocity can be eliminated and subsequently the convective flux is neglected [4042]. In addition, we also make the assumptions:
  1. 1.

    the effects of water electrolysis and electro-osmosis are neglected;

     
  2. 2.

    all three phases are incompressible, including the polymeric solid matrix, interstitial water, and mobile ions;

     
  3. 3.

    the hydrogel is isotropic and macroscopically homogeneous; and

     
  4. 4.

    the bath solution is ideal so that the variation of the activity coefficients with ionic strength is negligible, i.e. its effect on the concentration profiles is negligible.

     
Based on these assumptions, the governing equation (Eq. 1) can be simplified to:
$$ \begin{array}{*{20}c} {{\frac{{\partial c_{k} }} {{\partial t}} + div{\left( {{\mathbf{J}}_{k} } \right)} = \frac{{\partial c_{k} }} {{\partial t}} + div{\left\{ { - {\left[ {D_{k} } \right]}{\left( {{\text{grad}}{\left( {c_{k} } \right)} + \frac{{z_{k} F}} {{RT}}c_{k} {\text{grad}}{\left( \psi \right)}} \right)}} \right\}} = 0}}{{{\left( {k = 1,2, \ldots ,N} \right)}}} \\ \end{array} $$
(3)
The governing equations (Eq. 3) are coupled with the following Poisson equation for spatial distribution of the electric potential in the domain:
$$ \nabla ^{2} \psi = - \frac{F} {{\varepsilon \varepsilon _{0} }}{\left( {{\sum\limits_k {z_{k} c_{k} + z_{f} c_{f} } }} \right)} $$
(4)
where ɛ is the relative dielectric constant of the surrounding medium and ɛ0 is the vacuum permittivity or dielectric constant (8.85418 × 1012 C2 N−1 m−2). zf and cf are the valence and density of the fixed charge groups in the hydrogel, respectively.
Based on the Langmuir monolayer absorption theory, the fixed charge density can be derived and written as [38]:
$$ c_{f} = \frac{1} {{H + 1}} \cdot \frac{{c^{s}_{f} \cdot K}} {{K + c_{H} }} $$
(5)
where H is defined as the local hydration of the hydrogel, \( H = {V^{w} } \mathord{\left/ {\vphantom {{V^{w} } {V^{s} }}} \right. \kern-\nulldelimiterspace} {V^{s} } \) (Vw and Vs are the volumes of the interstitial fluid and dry hydrogel, respectively), K is the dissociation constant of the fixed charge groups, and \( c^{s}_{f} \) is the total concentration of the fixed charge groups in the hydrogel in the dry state.
From the relationship between the hydration and the phase volume fractions, we have:
$$ 1 + H = 1 + \frac{{V^{w} }} {{V^{s} }} = \frac{{V^{s} + V^{w} }} {{V^{s} }} = \frac{V} {{V^{s} }} = \frac{1} {{\phi ^{s} }} = \frac{1} {{1 - \phi ^{w} }} $$
(6)
the volume fractions of water and solid phases, ϕw and ϕs, are thus obtained as:
$$ \phi ^{w} = \frac{H} {{1 + H}} $$
(7)
$$ \phi ^{s} = \frac{1} {{1 + H}} $$
(8)
It can reasonably be assumed that the volume fraction of ions, ϕi, is vanishingly small compared with ϕw and ϕs, i.e.:
$$ \phi ^{w} + \phi ^{s} \approx 1 $$
(9)
The relationship between the volume fractions of water and solid phases is thus simplified further as:
$$ \phi ^{w} \approx 1 - \phi ^{s} = 1 - \frac{{V^{s} }} {V} = 1 - \frac{{V^{s} }} {{V_{0} }}\frac{{V_{0} }} {V} = 1 - \phi ^{s}_{0} \cdot J $$
(10)
where \( \phi ^{s}_{0} \) is the volume fraction of solid matrix at initial configuration and \( J{\left( { = {{\text{d}}V_{0} } \mathord{\left/ {\vphantom {{{\text{d}}V_{0} } {{\text{d}}V}}} \right. \kern-\nulldelimiterspace} {{\text{d}}V}} \right)} \) is the volume ratio of the apparent solid phase which can be expressed by the Green strain tensor E of the apparent solid phase as follows [43]:
$$ J^{{ - 1}} = {\sqrt {1 + 2F_{1} {\left( {\mathbf{E}} \right)} + 4F_{2} {\left( {\mathbf{E}} \right)} + 8F_{3} {\left( {\mathbf{E}} \right)}} } $$
(11)
where F1(E)=tr(E), F2(E) and F3(E) are the first, second and third invariants, respectively, of Green strain tensor E.
From Eqs. (7) and (10), one obtains:
$$ \phi ^{w} = \frac{H} {{1 + H}} = 1 - \phi ^{s}_{0} J $$
(12)
Then the local hydration of the hydrogel is written as:
$$ H = \frac{{1 - \phi ^{S}_{0} J}} {{\phi ^{S}_{0} J}} $$
(13)
Substituting Eqs. (12) and (13) into Eq. (5), the density of fixed charge groups is rewritten as:
$$ c_{f} = \frac{{c^{s}_{f} \cdot K \cdot \phi ^{s}_{0} }} {{{\left( {K + c_{H} } \right)}{\sqrt {1 + 2F_{1} {\left( {\mathbf{E}} \right)} + 4F_{2} {\left( {\mathbf{E}} \right)} + 8F_{3} {\left( {\mathbf{E}} \right)}} }}} $$
(14)
The present MECpHe model uses the finite elastic deformation theory, instead of the small deformation theory, to construct the mechanical governing equation. Considering the effect of chemo-electro-mechanical coupled domains, the pH-electric-sensitive hydrogels undergo larger deformation when the applied electric voltage is relatively high, for which the linear elastic theory cannot provide sufficiently accurate computation. This is because the difference between the initial (reference) and current (deformed) configurations cannot be neglected as is done for linear elasticity. For geometrically nonlinear analysis, therefore, the mechanical governing equations for large deformation, based on a total Lagrangian description, are required as:
$$ \begin{array}{*{20}c} {{\nabla \cdot {\mathbf{P}} = \nabla \cdot {\left( {{\mathbf{SF}}^{T} } \right)} = \nabla \cdot {\left[ {{\left( {{\mathbf{CE}} - p_{{{\text{osmotic}}}} {\mathbf{I}}} \right)}{\mathbf{F}}^{T} } \right]} = 0}}{{{\text{in}}\,\Omega }} \\ \end{array} $$
(15)
$$ \begin{array}{*{20}c} {{{\mathbf{u}} = {\mathbf{G}}}}{{{\text{in}}}} \\ \end{array} \,\Gamma _{g} $$
(16)
$$ \begin{array}{*{20}c} {{{\mathbf{P}} \cdot {\mathbf{n}} = {\mathbf{W}}}}{{{\text{in}}\,\Gamma _{w} }} \\ \end{array} $$
(17)
where F is the deformation gradient tensor, G is the specified displacement vector on the boundary portion Γg, W is the surface traction vector on the boundary Γw, n is the unit outward normal vector, u is the displacement vector from the initial configuration X to the deformed configuration \( {\mathbf{x}}{\text{ }}{\left( {{\mathbf{x}} = {\mathbf{X}} + {\mathbf{u}}} \right)} \), P is the first Piola–Kirchhoff stress tensor, that is a kind of expatriate and lives partially in the reference configuration X and partially in the deformed configuration x, C is the material tensor, posmotic is the osmotic pressure, I is identity tensor, and E is the Green–Lagrangian strain tensor used as strain measurement.
It should also be noted that deformation of the hydrogels will be small when the applied electric voltage is low. Then the linear theory enables sufficiently accurate computation and Eq. (15) is thus reduced to:
$$ \nabla \cdot {\mathbf{P}} = \nabla \cdot {\left( {\lambda _{s} {\text{tr}}{\left( {\mathbf{E}} \right)}{\mathbf{I}} + 2\mu _{s} {\mathbf{E}} - p_{{{\text{osmotic}}}} {\mathbf{I}}} \right)} = 0 $$
(18)
where λs and μs are the Lamé coefficients of the solid phase.
In terms of the current numerical simulations by the MECpHe model, one-dimensional steady-state computations in this paper are conducted only for the responsive behavior of the pH-electric-sensitive hydrogels with the computational domain covering both the hydrogels and the surrounding solution. Figure 1 illustrates the problem set-up where a hydrogel strip is immersed in a pH buffer solution between positive and negative electrodes. Therefore, we have two types of one-dimensional boundary condition imposed at the electrodes and the hydrogel-solution interfaces:
$$ \left. c \right|_{{{\text{Anode}}}} = \left. c \right|_{{{\text{Cathode}}}} = c^{*} $$
(19)
$$ \left. \psi \right|{}_{{{\text{Anode}}}} = 0.5V_{e} \,{\text{and}}\,\left. \psi \right|_{{{\text{Cathode}}}} = - 0.5V_{e} $$
(20)
$$ p_{{{\text{osmotic}}}} = RT{\sum\limits_k {{\left( {c^{{{\text{in}} - {\text{interface}}}}_{k} - c^{{{\text{out}} - {\text{interface}}}}_{k} } \right)} - p_{0} } } $$
(21)
where c* is the initial ionic concentration of the bath solution and Ve the externally applied electric voltage. \( c^{{{\text{in}} - {\text{interface}}}}_{k} \) is the ionic concentration within the hydrogels near the interfaces between the hydrogels and the surrounding solution, \( c^{{{\text{in}} - {\text{interface}}}}_{k} \) is the ionic concentration within the exterior solution near the interfaces, and p0 denotes the fluid pressure in the reference configuration.
https://static-content.springer.com/image/art%3A10.1007%2Fs00216-007-1483-9/MediaObjects/216_2007_1483_Fig1_HTML.gif
Fig. 1

Schematic diagram of a hydrogel strip immersed in pH buffer solution subject to an externally applied electric field

At the solvent/hydrogel interfaces \( {\left( {x = {{\left( {L \pm h} \right)}} \mathord{\left/ {\vphantom {{{\left( {L \pm h} \right)}} 2}} \right. \kern-\nulldelimiterspace} 2} \right)} \):
$$ \frac{{{\text{d}}u}} {{{\text{d}}x}} = \frac{{p_{{{\text{osmotic}}}} }} {E} $$
(22)
In addition, to prevent the hydrogel from undergoing translational motion in the applied electric field, a point constraint is introduced in the middle of the hydrogel \( {\left( {x = L \mathord{\left/ {\vphantom {L 2}} \right. \kern-\nulldelimiterspace} 2} \right)} \):
$$ u = 0 $$
(23)
Formulation of the MECpHe model is now complete. It is composed of the Nernst–Planck equations (Eq. 3) for the diffusive ionic concentrations ck, The Poisson equation (Eq. 4) for the electric potential ψ, and the nonlinear mechanical governing equation (Eq. 15) for the large deformation u of the hydrogels. For numerical solution of the MECpHe model, several mathematical challenges are encountered and they at least include the multi-energy domains associated with the coupled nonlinear partial differential equations, the computational domain remeshing because of moving boundaries, and the localized high gradient near the hydrogel–solution interfaces. A strong-form meshless technique termed the Hermite-cloud method is employed for simulation of the performance of the hydrogels responding to the pH-electric coupled stimuli [44]. The Hermite-cloud method directly calculates approximate solutions for both the functions and their first-order derivatives simultaneously. Therefore, it may achieve more accurate results, especially for the first-order derivatives of the distributive functions of diffusive ionic concentrations and electrical potential with the localized high gradients over the hydrogel–solution interfaces. In addition, the method is a strong-form numerical technique so that it may efficiently remesh computational domains in each of iterations to solve the present nonlinear coupled partial differential governing equations [45, 46]. By the Hermite-cloud method, the partial differential equations arising from the MECpHe model are first discretized into a set of nonlinear algebraic equations. The computational flowchart is depicted in Fig. 2. Using the fixed charge density cf expressed by Eq. (14), the Nernst–Planck equations (Eq. 3) and Poisson equation (Eq. 4) are solved numerically by the Newton interative technique to obtain the converged ionic concentrations ck and electric potential ψ. By substituting the converged ionic concentrations into the mechanical equilibrium equation (Eq. 15), the hydrogel displacement u is computed accordingly. This newly computed displacement u makes the fixed charge density cf redistributed within the hydrogel, which is used as input for next iterative step for solution of the Poisson–Nernst–Planck system. In this way the iteration is conducted continuously until the ionic concentrations ck, the electric potential ψ, and the hydrogel displacement u are converged simultaneously.
https://static-content.springer.com/image/art%3A10.1007%2Fs00216-007-1483-9/MediaObjects/216_2007_1483_Fig2_HTML.gif
Fig. 2

Computational flowchart of the developed MECpHe model

Results and discussion

Validation of the MECpHe model

To validate the MECpHe model, numerically computed results were compared with experimental bending data obtained from the literature [25] for a PMAA/PVA IPN hydrogel strip immersed in pH buffer solution and measurement of the corresponding deformation, as shown in Fig. 1. When the pH is low, the carboxylic acid groups in PMAA are in the form R–COOH. As the pH increases, R–COOH dissociate to R–COO, and the hydrogel strip swells uniformly. If the electric field is applied, the electric potential causes non-uniform distribution of ionic concentrations, leading to unequal concentration differences at the two interfaces between the hydrogel and the surrounding solution near the anode and cathode, respectively. The unequal concentration differences result in unequal osmotic pressure at the two interfaces near the anode and cathode, which makes the hydrogel bend [47, 48].

The values used for validation of the MECpHe model include R = 8.314 J mol−1 K−1, F = 9.6487 × 104 C mol−1, T = 298 K, ɛ0 = 8.854 × 10−12 C2 N−1 m−2, ɛ = 80, c* = 137.1 mmol L−1, \( c^{s}_{f} = 200\,{\text{mmol L}}^{{ - {\text{1}}}} \), zf = −1, the initial water volume fraction \( \phi ^{w}_{0} = 0.8 \), and K = 10−5.5 mmol L−1. In general, the elastic modulus of the polymer PMAA varies with the pH of the buffer solution [4951]. The elastic modulus is taken here as 3 MPa in accordance with experimental results [49]. The distance L = 30 mm between the two carbon electrodes and the hydrogel strip is tailed in 20 × 5 × 0.2 mm3. The simulated results are shown in Fig. 3 for the bending behavior of the hydrogel subject to the electric potential (Ve = 15 V) and solution pH coupled stimuli. To measure efficiently the bending deformation of the hydrogel, an equilibrium bending angle (EBA), α, is defined as \( \alpha = {45L_{0} {\left( {e_{1} - e_{2} } \right)}} \mathord{\left/ {\vphantom {{45L_{0} {\left( {e_{1} - e_{2} } \right)}} {\pi h}}} \right. \kern-\nulldelimiterspace} {\pi h} \) in degree units [25], where e1 and e2 are the strains of the hydrogel strip at the two ends in the thickness direction, and L0 and h are the length and thickness, respectively, of the hydrogel strip. Under the higher electric voltage Ve = 15 V the distributive profile of α increasing gradually with buffer solution pH value may be divided into three stages. When the pH is lower than pH 4.0 or higher than pH 6.0, α increases gradually. However, it will increase rapidly in the range from pH 4.0 to 6.0. In fact, the Flory’s osmotic pressure theory may be used to explain the mechanism of hydrogel deformation [36, 37]. Mobile ions move toward their counter electrode under the externally applied electric field. The ion concentration gradient over both the interfaces between the hydrogel and the surrounding solution is then developed because of the fixed negative-charged groups within the hydrogel. The osmotic pressure is generated because of the ion concentration difference. The cations Na+ diffuse into the hydrogel more than the anions Cl, and move toward the cathode. Because the increase of osmotic pressure at the interface near the anode is larger than that at the interface near the cathode, the hydrogel near the anode swells more than that near the cathode, which results in bending toward the cathode. Figure 3 shows that the simulated results agree well with the experimental data [25]. This validates the MECpHe model as capable of efficiently simulating hydrogels responsive to the pH-electric coupled stimuli.
https://static-content.springer.com/image/art%3A10.1007%2Fs00216-007-1483-9/MediaObjects/216_2007_1483_Fig3_HTML.gif
Fig. 3

Comparison of numerically computed results with experimental data

Electric voltage effect on response of pH-electric-sensitive hydrogel

For further understanding of the effects of various physical conditions on the responsive behavior of hydrogels subject to solution pH and electric field coupled stimuli, several simulations are carried out numerically with the input values required by the MECpHe model, R = 8.314 J mol−1 K−1, F = 9.6487 × 104 C mol−1, T = 298 K, ɛ0 = 8.854 × 10−12 C2 N−1 m−2, ɛ = 80, c* = 4.0 mmol L−1, \( c^{s}_{f} = 10.0\,{\text{mmol}}\,{\text{L}}^{{ - 1}} \), zf = −1, K = 10−2.1 mmol L−1, \( \phi ^{w}_{0} = 0.8 \), L = 2400 μm, h = 800 μm, and Young’s modulus 3.0 MPa. The simulation results are illustrated in Figs. 4, 5, 6, 7, 8 and 9. The effect of the externally applied electric voltage Ve is discussed in detail in terms of the responsive distribution of the diffusive ionic concentrations ck, the electric potential ψ, the fixed charge density cf and the hydrogel strip displacement u.
https://static-content.springer.com/image/art%3A10.1007%2Fs00216-007-1483-9/MediaObjects/216_2007_1483_Fig4_HTML.gif
Fig. 4

Distribution of diffusive Na+ concentration responding to pH varied with the externally applied electric voltage (Ve = 0.08 V)

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Fig. 5

Distribution of diffusive Cl concentration responding to pH varied with the externally applied electric voltage (Ve = 0.08 V)

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Fig. 6

Distribution of electric potential ψ responding to pH varied with the external electric voltage (Ve = 0.08 V)

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Fig. 7

Distribution of fixed charge density cf responding to pH varied with the external electric voltage (Ve = 0.08 V)

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Fig. 8

Distribution of displacement u responding to pH varied without externally electric voltage

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Fig. 9

Distribution of displacement u responding to pH varied with the external electric voltage (Ve = 0.08 V)

Figures 4 and 5 show the effect of buffer solution pH on the distribution of the concentrations of diffusive ionic species of the system in response to the electric voltage. It is seen that, if an electric field is applied, such as Ve = 0.08 V, the distribution of the concentrations of the diffusive ionic species Na+ and Cl are not uniform in the hydrogels and bath solution, and also not symmetric in the whole domain. Under the applied electric field, the mobile cations Na+ move from the anode region to the cathode region until the equilibrium state is achieved. Because of the negatively charged hydrogel, the diffusive Na+ concentration increases at the hydrogel edge near the cathode and decreases near the anode. To maintain the state of electroneutrality at every local point in both the solution and the hydrogel, the Cl concentration also increases at the hydrogel edge near the cathode and decreases near the anode. It is also observed from the figures that, along the pH-axis direction, the profile of concentration distributions over the hydrogel–solution interface for the cation Na+ forms a slant profile near the anode and a trench profile near the cathode, while the distributive profile of the concentration of the anion Cl forms a trench profile near the anode and a slant profile near the cathode. Moreover, the profiles of the distributions of the ionic concentrations change significantly from pH 1.0 to 6.0 but only gradually from pH 6.0 to 10.0. The simulations shown in Figs. 4 and 5 are consistent qualitatively with the experimental phenomena [52].

Figures 6 and 7 show the effect of buffer solution pH on the distributions of the electric potential ψ and the fixed charge density cf within the hydrogel in response to an electric voltage. As the electric voltage such as Ve = 0.08 V is applied to the system, as shown in Fig. 6, the distribution of the electric potential ψ is non-uniform and gradients of ψ are observed over the hydrogel–solution interfaces. It is found that the distribution of fixed charge density cf within the hydrogel decreases when the voltage Ve = 0.08 V is applied to the system. The increase of the electric voltage Ve will make the hydrogel strip swell, which results in the redistribution of the fixed charge groups within the hydrogel. Along the pH-axis direction, the distributive profiles of fixed charge group increase rapidly from pH 1.0 to 6.0 and then gradually from pH 6.0 to 10.0, similarly to those for the concentrations of the diffusive ionic species Na+ and Cl and the electric potential ψ.

Figures 8 and 9 show the effect of buffer solution pH on the displacement of the hydrogel under different electric potentials. It is seen that the displacement of the hydrogel strip changes dramatically with the electric-pH coupled stimuli. In the range of from pH 1 to 6, if the applied electric voltage Ve = 0.08 V, the hydrogel strip swells noticeably and the displacement is approximately four times larger than that of the hydrogel strip subject to the single stimulus of solution pH without electric stimulus. In the range of from pH 6 to 10, however, the effect of solution pH on the displacement of the hydrogel is insignificant and the displacement is distributed almost uniformly. It can be expected that the smart hydrogels responsive to solution pH and electric voltage coupled stimuli can deform more have better mechanical strength. Thus the hydrogels can be potential candidates for linear actuators in BioMEMS [19].

Equilibrium bending deformation of pH-electric-sensitive hydrogel

As promising materials for sensors, actuators and artificial muscles for biomedical engineering applications, the mechanical properties of pH-electric-sensitive hydrogels are of particularly interest. It is known that the mechanical force is generated by osmotic pressure, which is governed by ionic transport through the system and is affected by hydrogel architecture [53, 54]. To investigate the effect of the ionic strength \( I{\left( {I = 0.5 * {\sum\limits_k {c_{k} z^{2}_{k} } }} \right)} \) on the bending deformation of the hydrogels, several numerical simulations were conducted with the input values for the MECpHe model, R = 8.314 J mol−1 K−1, F = 9.6487 × 104 C mol−1, T = 298 K, ɛ0 = 8.854 × 10−12 C2 N−1 m−2, ɛ = 80, zf = −1, K = 10−2.1 mmol L−1, \( \phi ^{w}_{0} = 0.8 \), L = 2400 μm, h = 800 μm, and Young’s modulus 3.0 MPa. In these simulations, an average curvature, Ka, is defined as \( K_{a} = {2{\left( {e_{1} - e_{2} } \right)}} \mathord{\left/ {\vphantom {{2{\left( {e_{1} - e_{2} } \right)}} {{\left[ {h{\left( {2 + e_{1} + e_{2} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {h{\left( {2 + e_{1} + e_{2} } \right)}} \right]}} \) (where e1 and e2 are the strains of the hydrogel strip at the two ends) at the fixed point of the hydrogel thickness axis for measurement of the bending deformation of the hydrogel [55].

Figures 10a–d show the coupling influences of the solution pH, the electric voltage Ve and the solution ionic strength \( I{\left( {I = 0.5 * {\sum\limits_k {c_{k} z^{2}_{k} } }} \right)} \) on the hydrogel curvature Ka, in which \( c^{s}_{f} = 10.0\,{\text{mmol L}}^{{ - {\text{1}}}} \), the range of solution pH is from 1.0 to 9.0, the range of electric voltages Ve is from 0 to 0.4 V, and the different solution ionic strengths I = 2.0, 4.0, 8.0 and 16.0 mmol L−1. Figure 11 illustrates the effect of the ionic strength I on the average curvature Ka at pH 4.0 and with Ve = 0.4 V. It is observed from Fig. 10a–d that the average curvature Ka of the hydrogel increases with increasing externally applied voltage Ve, i.e. the hydrogel equilibrium bending deformation is larger under higher electric voltage Ve. But the characteristics of the hydrogels in response to the solution pH are different. The average curvature Ka under the same electric voltage increases dramatically from pH 1.0 to 4.0 but only slightly in the pH range 5.0–9.0. The deformation of the hydrogel may be mainly driven by the changes in the osmotic pressure [23], which is generated by the different ion concentrations between the interior hydrogels and exterior solution. The higher voltage results in larger ionic concentration differences over the interfaces between the hydrogel and the solution. The concentration differences cause the difference of the osmotic pressures at the two sides of the hydrogel, which makes the hydrogel bend. The higher voltage Ve thus results in a larger average curvature Ka, as shown in Fig. 10a–d. A numerical observation from the figures reveals that a significant increase in the equilibrium swelling and bending of the hydrogel occurs in the solution pH range 2.0–4.0, which is associated with the dissociation constant pKa = 2.1 for the hydrogels. If the solution pH is larger than 4.0, the carboxyl groups may be totally ionized, which could be the reason for the tiny increase of Ka in the pH range 5.0–9.0. The simulation results are consistent with the experimental phenomena [56]. It is found from the four figures that under the solution pH and externally applied electric voltage Ve coupled stimuli, the average curvature Ka of the hydrogel increases with ionic strength I, in which the Ka increment is not linear and will slow down with increasing ionic strength. It is known that, with increasing ionic strength of the system medium, the osmotic pressure decreases, and ultimately reduces the equilibrium swelling capacity of the hydrogels [57, 58]. However, the increase of the ionic strength (electrolyte concentration) induces an increase in mobile counterions diffusing into the hydrogel. Because of the fixed charge groups existing in the hydrogel under the action of the electric field, redistribution of the counterions results in a larger difference between the ionic concentrations at the ends of the hydrogel, which increases the difference between the osmotic pressures over the two interfaces between the hydrogel and the surrounding solution. The bending deformation of the hydrogel therefore increases further and the average curvature Ka also increases. The effect of ionic strength I may be weakened by the shielding effect of the fixed charge groups [25], and the average curvature Ka decreases gradually with the increasing ionic strength, as shown in Fig. 11. The simulations agree well qualitatively with the published experimental studies [25, 59].
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Fig. 10

Coupling effect of ionic strength I and electric voltage Ve and solution pH on variation of the average curvature Ka (a) I = 2.0 mmol L−1, (b) I = 4.0 mmol L−1, (c) I = 8.0 mmol L−1, (d) I = 16.0 mmol L−1

https://static-content.springer.com/image/art%3A10.1007%2Fs00216-007-1483-9/MediaObjects/216_2007_1483_Fig11_HTML.gif
Fig. 11

Coupling effect of ionic strength I and electric voltage Ve and solution pH on variation of average curvature Ka (pH 4.0, Ve = 0.4 V)

Conclusion

The multiphysics model termed the multi-effect-coupling pH-electric-stimuli (MECpHe) model has been developed for simulation of the performance behavior of smart hydrogels responding to pH and electric voltage coupled stimuli when the hydrogels are immersed in pH buffer solution subject to externally applied electric field. Considering the chemo-electro-mechanical coupled effects and incorporating the fixed charge density with the buffer solution pH and externally applied electric voltage coupled effect, the developed MECpHe model is composed of nonlinear partial differential governing equations with the capability of predicting the hydrogel responsive deformation and the distributions of the concentrations of diffusive ionic species, the electric potential, and the fixed charge density in the whole computational domain, including the hydrogel and surrounding buffer solution. To solve the model consisting of multiple coupled fields, the meshless Hermite-cloud numerical method and a hierarchical iteration technique are employed. The model is examined for accuracy and efficiency by comparison of numerically simulated data with experimental data extracted from the literature. The one-dimensional steady-state simulation is numerically conducted for a smart hydrogel responsive to the pH-electric coupled stimuli. The effect of the externally applied electric voltage on the distributions of the concentrations and electric potential of diffusive ionic species, on the fixed charge density, and on the displacement of the hydrogel strip is discussed in detail. The effect of ionic strength on the bending deformation of the hydrogels in response to solution pH and electric voltage coupled stimuli was also investigated.

Acknowledgement

The authors gratefully acknowledge financial support from the Agency for Science, Technology, and Research (A*STAR) of Singapore through A*STAR SERC Grant – SRP on MEMS Phase II under the project number: 022 107 0009.

Copyright information

© Springer-Verlag 2007