Poromechanical modeling of a stimuli-responsive hydrogel in a chemical sensor with force compensation

Abstract

In the category of smart materials, stimuli-responsive hydrogels are of great interest due to their significant stimuli-induced volume change. This ability inspires their application in sensors, actuators, and other electronic devices. In this work, a chemo-thermo-responsive hydrogel is used in a chemical sensor setup, where the thermal stimulus compensates for the gel deformation caused by the chemical stimulation. By keeping the hydrogel volume constant, creep and relaxation effects can be largely suppressed. Since diffusion processes are slow, the desired volume constancy cannot be achieved immediately at all individual points of the hydrogel. To predict the sensor response, we implement a chemo-thermo-poromechanical model in COMSOL Multiphysics^® to simulate the swelling-deswelling behavior of the hydrogel in the sensor. The numerically computed swelling kinetics benefit the further investigation of the sensor optimization.

Introduction

As defined by Buenger et al. (2012) and Liu et al. (2020), a sensor is a device or a machine that detects and responds to stimuli in the environment. In view of the structure with hydrophilic polymer networks, stimuli-responsive hydrogels can adsorb and release solvent under physical and chemical stimuli. Owing to the ability of active deformation and the fast response to environmental change, stimuli-responsive hydrogels have attracted particular attention in the developing field due to their novel sensor-actuator functions (Buenger et al., 2012; Chen et al., 2025; Guenther et al., 2009; Liu et al., 2022).

The capacity of stimuli-responsive hydrogels to convert chemical energy into mechanical energy significantly impacts the advancement of chemical sensors (Gerlach et al., 2004, 2021; Guenther et al., 2009). The further investigated hydrogel-based sensors can respond not only to pH (Lee and Braun, 2003; Richter et al., 2008) and to ions (Dave et al., 2010; Holtz and Asher, 1997) but also to certain inorganic and organic species, that is, ammonia (Erfkamp et al., 2019), glucose (Zhai et al., 2013) and proteins (Cai et al., 2017). Furthermore, due to the high sensitivity to pH, temperature and moisture, hydrogel-based biosensors exhibit enhanced capabilities for managing wound healing processes, enabling the early identification of wound infection (Jankowska et al., 2017; Meier et al., 2011; Rahimi et al., 2016).

As responsive hydrogels respond to biochemical molecules, this advantage inspires the development of novel biosensing approaches (Tavakoli and Tang, 2017). The technology for detecting cellular metabolites is applied to monitor changes occurring at the cell site (Li et al., 2015; Wightman, 2006). In the early detection of cancer and chronic diseases, the monitoring of small molecules, e.g. the inflammatory factor such as cytokines (Ngoepe et al., 2013), the superoxide anions released by cancer cells (Crulhas et al., 2017) offers important opportunities for both diagnosis and treatment (Tavakoli and Tang, 2017).

In the field of engineering, the research focuses on the swelling behavior of responsive hydrogels induced by the coupled physical effects (stimuli). Over the past few decades, the approach to model hydrogel swelling has evolved from the classical polymer-solution theory (Flory and Rehner, 1943; Huggins, 1941), through the theory of mixture (Bowen, 1980), to the modern poromechanical framework (Gebhart and Wallmersperger, 2019; Hong et al., 2008). Coupled-field models are applied within the framework to depict the behavior of hydrogels with various stimuli (Keller et al., 2011). The free energy density functions describe the individual contributions of physical fields within the system, and these fields are coupled through particular field variables (e.g. deformation, electrochemical potential, temperature). This framework establishes a link between the continuum mechanics and the field theory.

Principally, the compensation method in the sensor technology aims to counterbalance the output signal generated by the input (i.e. measurement). Through the determination of the compensation input, an overall increase in measurement precision is achieved (Dyer, 2004; Schulz et al., 2012). The appropriate compensation method can reduce the sensor response time. For example, the temperature is used to compensate for the chemical concentration in a chemical hydrogel sensor (Schreiber et al., 2023). The response time of the sensor is determined by the response behavior of the hydrogel. The introduced force compensation method generates an external counterforce that counteracts the hydrogel’s deflection. It virtually eliminates swelling and significantly improves the dynamic response. By maintaining the hydrogel’s volume constant, the overall response time of the sensor can be significantly reduced (Binder and Gerlach, 2019; Han et al., 2002; Schulz et al., 2012).

Combining two different stimuli-responsive hydrogels forms a bisensitive copolymer. The two different stimuli acting on both hydrogels can be utilized in the bisensitive copolymer. For example, when one part of the hydrogel swells in response to the analyte, the other part of the hydrogel can counteract the swelling and keep the volume constant, see Figure 1.

Figure 1.

Schematic figure of hydrogel swelling. The hydrogel is initially chemically stimulated. Without compensation (- - -), the hydrogel swells; while the swelling can be counteracted by the applied force compensation (—). Symbol $Q$ indicates the swelling degree.

In our case, the temperature change, which is also the new measurement signal, can compensate for the chemically induced swelling. In this way, the sensing time, that is the response time to equilibrium, can be drastically reduced.

In the present paper, we utilize a thermo-chemo-poromechanical framework to compute the temporal behavior of a hydrogel-based chemical sensor with force compensation. The pertinent issues and configurations of the hydrogel sensor are introduced in Section: Problem Statement. Then, the poromechanical model regarding the chemo-thermally coupled field is presented in Section: Poromechanical modeling. In Section: Numerical simulation, we implement our thermo-chemo-mechanical framework by using the Finite Element Method (FEM) to simulate the hydrogel behavior under chemical and thermal stimuli, where the numerical solutions are compared with the experiment. Finally, the conclusions of this work are presented in Section: Conclusions.

Problem statement

The cross-section of the chemical hydrogel sensor is presented in Figure 2. In the sensor configuration, a thin cylindrical hydrogel is attached to a square silicon wafer, which functions as a bending plate. The hydrogel is synthesized from the monomers 2-acrylamido-2-methylpropane sulfonic acid (AMPS) and N-isopropylacrylamide (NIPAAm), crosslinked with N,N’-methylenebisacrylamide (BIS). According to the work of Schreiber et al. (2023), the hydrogel has a thickness of 20  $\mu$ m and a diameter of 1.65 mm. The bending plate has the same thickness as the hydrogel, with an edge length of 3.3 mm.

Figure 2.

Setup of a chemical sensor with the stimuli-responsive hydrogel. The area encircled by the dashed border is to be modeled. Symbols $d$ , $b$ and $h$ denote the diameter, the edge length and the thickness, respectively.

The AMPS-co-NIPAAm hydrogel combines the characteristics of its constituents, where this copolymer is both salt-responsive and thermo-responsive. The swelling volume of the hydrogel decreases with increasing salt concentration, while the temperature-dependent swelling kinetics is governed by the applied thermal field. According to the temperature range applied in the sensor, the co-hydrogel swells with increasing temperature.

In order to obtain a reliable sensor behavior, the simulation of the hydrogel swelling should consider the realistic material and model parameters. Regarding the constituents of AMPS-co-NIPAAm hydrogel (90.9 mol% NIPAAm + 5.4 mol% AMPS +3.7 mol% BIS) applied in the sensor, the material properties of the hydrogel can be predicted based on the previous investigations (Fei et al., 2013; Saikia et al., 2013a; Shi and Wallmersperger, 2022). The study of Fei et al. (2013) ¹ measured the mechanical properties of AMPS-co-NIPAAm hydrogels with different NIPAAm-AMPS ratios, where one of the measured hydrogels corresponds pretty well to the composition in our case.

Based on the setup of the developed chemical sensor (see Figure 2), the swelling and deswelling behavior of the integrated chemo-thermo-responsive hydrogel is to be modeled. The poromechanical framework must provide quantitative predictions of the hydrogel’s mechanical response under the combined influence of chemical and thermal stimuli. The numerical simulation is expected to reproduce the experimentally determined behavior in dependence of salt concentration and temperature. The force compensation, driven by coupled chemo-thermal stimulation, aims to maintain a constant hydrogel volume during operation.

Poromechanical modeling

As a macroscopic approach, the poromechanical modeling builds upon the principles of continuum mechanics and provides a theoretical framework for describing the behavior of porous media. Within this framework, the governing equations are formulated for both the solid and fluid phases, while the constitutive theory can be formulated separately according to the respective constituents and their associated physical fields. This modeling approach is particularly appropriate for coupling multi-field problems with large deformation in polymer gels (Chester and Anand, 2010, 2011; Hong et al., 2008). The chemo-thermal responses to be captured within the sensor application fall precisely within the scope of this theory.

On the basis of the poromechanical theory, the entire gel body consists of a polymeric matrix structure and a pore-filling solution. Owing to the microscopic polymer chains, ions exist as fixed ion groups within the matrix skeleton. In the solution, dissolved ions are mobile in the carrier fluid. According to this fundamental consideration, the poromechanical modeling as well as the numerical implementation concerning the chemo-thermal stimulation are further introduced.

In this section, we review the fundamental equations for the modeling. In addition, the equations are formulated in the Lagrange geometry setting with respect to the solid skeleton. Consider a closed thermodynamic system for the mixture (skeleton and solution), the solution phase $\beta$ is divided into its constituents: the fluidic phase f and the ionic phase $\gamma$ . The corresponding balance equations are referred by the following formulations:

$$\begin{array}{c}{\stackrel{·}{C}}^{\beta }\hfill \end{array}+{\mathbf{\nabla }}_{\mathit{X}}·{\mathcal{J}}^{\beta }=0\mathrm{Mass}\mathrm{balance}$$ (1)

$$\begin{array}{c}{\mathbf{\nabla }}_{\mathit{X}}·{\mathbf{P}}^{\mathrm{tot}}=0\mathrm{Momentum}\mathrm{balance}\hfill \end{array}$$ (2)

$$\begin{array}{c}{\mathbf{P}}^{\mathrm{tot}}·{\mathbf{F}}^{\mathrm{T}}=\mathbf{F}·{({\mathbf{P}}^{\mathrm{tot}})}^{\mathrm{T}}\mathrm{Angular}\mathrm{momentum}\mathrm{balance}\hfill \end{array}$$ (3)

$${\mathbf{\nabla }}_{\mathbf{X}}·\mathbb{D}=Q_{\mathbb{D}}\mathrm{Gauss}\prime \mathrm{law}$$ (4)

$${\mathbf{\nabla }}_{\mathbf{X}}\times \mathbb{E}=0\mathrm{Faraday}\prime \mathrm{s}\mathrm{law}$$ (5)

In equation (1), $C^{\beta }$ and ${\mathcal{J}}^{\beta }$ denote the material concentration and the mass flux of the corresponding constituent, respectively. The operator ${\mathbf{\nabla }}_{\mathbf{X}}$ indicates the gradient with respect to the material point $\mathit{X}$ . The $\mathrm{operator}(\stackrel{.}{•})$ denotes the material derivative with respect to time $t$ . The total 1. Piola-Kirchhoff stress tensor ${\mathbf{P}}^{\mathrm{tot}}$ is chosen as the stress measure in formulations relative to the reference configuration concerning the finite deformation in equations (2) and (3), where the effects of inertia and body forces are neglected. Herein, $\mathbf{F}$ indicates the deformation gradient tensor. Gauss’ law and Faraday’s law are presented to introduce the electrostatics, where $\mathbb{D}$ , $\mathbb{E}$ and $Q_{\mathbb{D}}$ respectively denote the electric displacement, the electric field and the free charge density in the reference configuration.

As an essential part in the thermo-chemo-mechanical coupling, the heat equation is derived from the energy balance (Chester and Anand, 2011; Shi and Wallmersperger, 2022). According to the thermodynamic definitions of the specific heat $c_p$ and the entropy, we obtain the heat equation according to the energy balance regarding the chemical dissipation so that

$$\begin{array}{c}c\stackrel{·}{\vartheta }+{\mathbf{\nabla }}_{\mathit{X}}·\mathcal{Q}-Q_r+{\mathbf{\nabla }}_{\mathit{X}}{\mu }^{\mathrm{f}}·{\mathcal{J}}^{\mathrm{f}}\hfill \\ \multicolumn{1}{c}{+\sum _{\gamma }{\mathbf{\nabla }}_{\mathit{X}}{\mu }^{\gamma }·{\mathcal{J}}^{\gamma }+H=0.}\end{array}$$ (6)

Herein, $\vartheta$ indicates the temperature; $\mathcal{Q}$ and $Q_r$ denote the heat flux per unit reference area and the external heat supply per unit reference volume, respectively. The electrochemical potentials for the fluid and the ion species are presented by ${\mu }^{\mathrm{f}}$ and ${\mu }^{\gamma }$ . The coupling terms are substituted by the variable $H$ .

The functions of mass flux outline the way of particle transport between the hydrogel and the solution:

$${\mathcal{J}}^{\beta }=-{\hat{m}}^{\beta }{\mathbf{\nabla }}_{\mathit{X}}{\mu }^{\beta }.$$ (7)

The mobility of the corresponding constituent

$${\hat{m}}^{\beta }=D^{\beta }{C}^{\beta }/\left(\mathrm{JR}\vartheta \right)$$ (8)

results from the expression of Fick’s first law in dependence of the chemical potential, where $D^{\beta }$ denotes the diffusion coefficient of the constituent $\beta$ . $J$ is the determinant of the deformation gradient tensor $\mathbf{F}$ and $R$ denotes the ideal gas constant.

As the main mode of the heat transfer in our case, heat conduction is modeled by the Fourier ansatz that the heat flux is given by:

$$\mathcal{Q}=-k{\mathbf{\nabla }}_{\mathit{X}}\vartheta ,$$ (9)

where $k$ denotes the heat conduction coefficient.

Following, the constitutive equation establishes a relationship between two conjugate field variables. In this context, the free energy function, which includes the field variables, serves as the basis for the constitutive formulation. Through the additive decomposition, the total Helmholtz energy of the coupled fields is formed into separate contributions. Hence, as per the Coleman-Noll procedure (Coleman and Noll, 1974) in deriving the stress, the total stress tensor ${\mathbf{P}}^{\mathrm{tot}}$ is given by

$${\mathbf{P}}^{\mathrm{tot}}={\mathbf{P}}_{\mathrm{el}}+{\mathbf{P}}_{\mathrm{mix}}+{\mathbf{P}}_{\mathrm{ion}}+{\mathbf{P}}_{\mathrm{elec}}.$$ (10)

Herein, ${\mathbf{P}}_{\mathrm{el}}$ , ${\mathbf{P}}_{\mathrm{mix}}$ , ${\mathbf{P}}_{\mathrm{ion}}$ , ${\mathbf{P}}_{\mathrm{elec}}$ respectively denote the stress contributions of elasticity, polymer mixing, osmotic pressure and electricity. Similarly, the other field variables are derived from the corresponding free energy functions. The detailed formulations can be referred to the preceding studies (Shi and Wallmersperger, 2022, 2024). It has to be noted that the osmotic contribution ${\mathbf{P}}_{\mathrm{ion}}$ is influenced by the proportion of polyelectrolyte hydrogel (e.g. AMPS used in this work).

Numerical simulation

By applying the presented thermo-chemo-poromechanical framework, we numerically determine the swelling behavior of an AMPS-co-NIPAAm hydrogel. The numerical investigation is performed by using the Finite Element Method (FEM). The key of the numerical simulation lies in the weak formulation of physical balance equations. The balance equations in the weak form are subsequently implemented into the commercial software, COMSOL Multiphysics^®. Taking into account the constituents of the currently used hydrogel, we modify the experimentally determined material parameters obtained by Ehrke et al. (2022), Shi and Wallmersperger (2022).

According to further considerations in the aspects of the application, it is assumed that there is no external applied electric field. Since the accumulation of ion species is weak with neither an external electric field nor electrochemical reactions, the effect of the electric double layer will not be taken into account. For the bending plate, the infinitesimal strain is assumed to compute the deflection.

In the context of modeling the swelling behavior of PNIPAAm hydrogels, the Flory-Huggins interaction parameter $\chi$ is treated as a temperature-dependent variable. The fluidic chemical potential can be interpreted as the Lagrange multiplier. Additionally, it is assumed that the initial value of the chemical potential ${\mu }_0^{\mathrm{f}}$ is adopted from the previous experimental result in Shi and Wallmersperger (2022).

It is worth mentioning that the preswollen state of the hydrogel is treated as the initial state in our poromechanical modeling. The deformation gradient at the preswollen state ${\mathbf{F}}_0$ is assigned by

$${\mathbf{F}}_0=J_0^{1/3}\mathbf{I},$$ (11)

where $J_0$ is the initial value of the determinant $J$ . $\mathbf{I}$ is the unit tensor. The initial stretch can be assumed to be dependent on the constituents of the hydrogel composition.

The swelling degree $Q$ describes the temporal swelling behavior of the hydrogel, which is defined as the volume ratio between the current state and the initial state of the observed body, expressed as

$$Q\left(t\right)={\int }_{\mathcal{B}}\frac{J}{J_0}\mathrm{d}V.$$ (12)

As the test case, we consider the swelling-deswelling of the hydrogel with its supporting plate, where the bimorph effect is taken into account. In the sensor structure, the bending motion of the active layer (stimuli-responsive hydrogel) causes the deformation of the conductive passive layer (silicon plate). ²

Boundary conditions

A quarter model of the complete hydrogel-wafer structure will be utilized in the FEM analysis to reduce the computing efforts (see Figure 3). The associated symmetric surfaces are supported in their normal directions ( $u_{\mathrm{normal}}=0$ ). To completely constrain the rigid motion, the bottom edges of both the cylindrical hydrogel and the silicon wafer are supported in $z$ -direction, ensuring $u_z=0$ on the edges.

Figure 3.

Top: Hydrogel-wafer model with its dimensions. The colored section is chosen as quarter model. Bottom: Quarter model of the sample with symmetric surfaces used for the calculation. Indicators denote the positions of boundary conditions. 1 - front surface (hydrogel and plate); 2 - right surface (hydrogel and plate); 3 - top surface (hydrogel);4 - bottom surface (hydrogel); 5 - lateral surface (hydrogel);6 - bottom edge (hydrogel).

The boundary conditions of the chemical field are derived with respect to the chemically induced hydrogel bending (Shi and Wallmersperger, 2024). The ambient temperature within the sensor is regulated by an integrated Peltier element, resulting in an assumed homogeneous boundary condition for the thermal field. Table 1 presents the boundary conditions for coupled fields on each related boundary.

Table 1.

DIRICHLET and NEUMANN boundary conditions used for the simulation of the swelling behavior. ³

Position	$B{C}_{\mathrm{mech}}$	$B{C}_{\mathrm{fluid}}$	$B{C}_{\mathrm{chem}}$	$B{C}_{\mathrm{therm}}$
1	$u_x=0$	${\hat{\mathcal{J}}}^f=0$	${\hat{\mathcal{J}}}^{\gamma }=0$	$\mathcal{Q}=\mathbf{0}$
2	$u_y=0$	${\hat{\mathcal{J}}}^f=0$	${\hat{\mathcal{J}}}^{\gamma }=0$	$\mathcal{Q}=\mathbf{0}$
3	$\mathit{T}=\mathbf{0}$	${\mu }^f={\mu }_0^f$	${\mu }^{\gamma }={\mu }^{\gamma }\left({\overline{c}}^{\gamma }\right)$
4	$\mathit{T}=\mathbf{0}$	${\hat{\mathcal{J}}}^f=0$	${\hat{\mathcal{J}}}^{\gamma }=0$	$\vartheta ={\vartheta }_{\mathrm{BC}}$
5	$\mathit{T}=\mathbf{0}$	${\mu }^f={\mu }_0^f$	${\mu }^{\gamma }={\mu }^{\gamma }\left({\overline{c}}^{\gamma }\right)$
6	$u_z=0$
7	$u_z=0$

The numbers in the first column indicate the positions of the boundary conditions according to Figure 3.

Parameters

The swelling behavior of a stimuli-responsive hydrogel incorporates finite deformation, mass transport, and heat transfer. The related model parameters are necessary for implementing the field variables and the governing equations.

According to the studies of constituents in AMPS-co-NIPAAm hydrogels (Fei et al., 2013; Saikia et al., 2013a), the shear modulus $G$ is determined for the incompressible hyperelastic solid skeleton (Rivlin, 1948).

As one of the most important characteristics of PNIPAAm hydrogels, the effect of the temperature-dependent volume phase transition (VPT) should be considered as the thermal field is applied. The temperature, at which the VPT occurs, is called volume phase transition temperature (VPTT). VPTT shifts to a higher value with increasing AMPS proportion owing to the shift of hydrophilic/hydrophobic balance toward a more hydrophilic nature (Saikia et al., 2013b; Turan et al., 2008).

Volume phase transition is mathematically related to the temperature-dependent interaction parameter $\chi$ , which influences the Gibbs free energy change for polymer-solution mixing. During the phase transition, the hydrogel shrinks with rising temperature owing to the increase of $\chi$ ; when the temperature decreases, the hydrogel behavior is opposite. Additionally, if the temperature is not in the range of phase transition, $\chi$ is assumed to be constant. In the case that $\chi$ stays constant, the thermally induced hydrogel swelling and deswelling are governed by the temperature: the hydrogel swells with increasing temperature, whereas it shrinks with decreasing temperature.

The determination of $\chi$ involves fitting the numerical results to the experimentally measured swelling curve. In our experiment, the synthesized hydrogel undergoes alternating heating and cooling cycles in pure water to investigate the thermally stimulated swelling and deswelling. We assume that the development of $\chi$ corresponds to a hyperbolic ansatz (Chester and Anand, 2011) so that

$$\chi \left(\vartheta \right)=\frac{1}{2}({\chi }_L+{\chi }_H)-\frac{1}{2}({\chi }_L-{\chi }_H)\tanh \left(\frac{\vartheta -{\vartheta }_t}{\mathrm{\Delta }}\right).$$ (13)

Herein, the symbols ${\chi }_L$ , ${\chi }_H$ and $\mathrm{\Delta }$ are model parameters. By considering the difference of used crosslinker, the VPTT ${\vartheta }_t$ of the hydrogel investigated in this work is estimated to be 43°C (Saikia et al., 2013b; Zhang et al., 2004). Since PNIPAAm hydrogels also have limits of hydrophilicity and hydrophobicity, ${\chi }_L$ and ${\chi }_H$ indicate the minimum and the maximum value of $\chi$ at extremely low and extremely high temperatures, respectively. $\mathrm{\Delta }$ implies a range of temperature around VPTT, the hydrogel undergoes a transition from hydrophilic to hydrophobic within this range, and vice versa.

In Table 2, we list the material parameters used for the numerical simulation. In addition, no external heat supply $Q_r$ and no coupling heat $H$ are assumed. The listed parameters are associated to the poromechanical framework, where the interaction parameter $\chi$ influences the polymer-solution mixing, resulting in the change in stress tensor ${\mathbf{P}}_{\mathrm{mix}}$ . The computation of stress tensors is referred to the constitutive theory applied in Shi and Wallmersperger (2022, 2024), which solves the coupled chemo-thermo-mechanical problem.

Table 2.

Parameters used for the numerical simulation of the swelling behavior.

Parameter	Value
Material concentration(fixed ions) $C^{\mathrm{fix}}$	765.5 mol m−3
Diffusion coefficient (fluid) $D^{\mathrm{f}}$	$8\times {10}^{-8}$  m2 s−1
Diffusion coefficient (ions) $D^{\gamma }$	$5\times {10}^{-10}$  m2 s−1
Faraday’s constant $F$	$96,485.3$  sAmol−1
Shear modulus (hydrogel) $G_h$	70 kPa
Shear modulus (wafer) $G_w$	50 GPa
Heat conduction coefficient $k$	0.35 Wm−1 K−1
Ideal gas constant $R$	$8.314$  Jmol−1 K−1
Charge (anion) $z^{-}$	−1
Charge (fixed ion) $z^{\mathrm{fc}}$	−1
Charge (cation) $z^{+}$	1
Model parameter $\mathrm{\Delta }$	2 K
Total permittivity $ϵ$	$7.083\times {10}^{-10}$  Fm−1
Transition temperature ${\vartheta }_t$	316 K
Initial fluidic chemical potential ${\mu }_0^{\mathrm{f}}$	−2.55 Jmol−1
Model parameter ${\chi }_L$	0.478
Model parameter ${\chi }_H$	0.5
Molar volume (fluid) ${\mathrm{\Omega }}^{\mathrm{f}}$	$1.807\times {10}^{-5}$  m3 mol−1

Electrochemical parameters are referred to Shi and Wallmersperger (2022, 2024). Parameters related to the heat transfer are referred to Chester and Anand (2011).

Results

The chemical sensor based on chemo-thermally responsive hydrogels takes advantage of its short response time, because in this application, the compensation method controls temperature to counteract the deflection induced by chemical stimulation. The boundary condition of temperature ${\vartheta }_{\mathrm{BC}}$ is defined by a ramp edge function. Since the Peltier element can rapidly regulate the temperature on the microscopic dimension, a short ramp time is possible, see Figure 4(a).

Figure 4.

(a) Stimuli-responsive hydrogel is heated from 20°C (293.15 K) to 30°C (303.15 K). (b) Comparison between the numerically simulated hydrogel swelling and the experiment results by Schreiber et al. (2023).

By calibrating the sensor signals from the experiments (Schreiber et al., 2023), we compare the numerical results of the thermally stimulated hydrogel swelling with the experiment, see Figure 4(b). The hydrogel swells with increasing temperature since the volume phase transition does not occur. This comparison benefits the modification of the model parameters in Table 2.

Within the fluid chamber of the sensor, the flow rates of both the NaCl solution and the pure water are regulated in order to achieve the required chemical stimulation. In our simulation, the external solution concentration is specified via a ramp edge function, where the ramp time depends on the flow rate in the chamber.

Figure 5 interprets the simulation of the hydrogel swelling-deswelling induced by the chemical stimulation and the thermal stimulation in sequence. For the chemical simulation, the concentration of NaCl is regulated from $1{\mathrm{molL}}^{-1}$ to $0{\mathrm{molL}}^{-1}$ . Afterward, the temperature is reduced from 40°C to 5°C to compensate for the swelling. The ramp time for both stimulations is set to be 30 s. The temperature compensates for the chemically induced swelling in 35 s. In comparison to the duration of swelling (45 s), the duration of recovery is reduced by 22.2%, which can be interpreted as a remarkable advantage. The absolute difference is more significant in the larger hydrogel since the mass transfer takes a longer time. Due to the chemical stimulation, the hydrogel swelling kinetics highly depend on ion transport since the fluid penetration is influenced by the osmotic pressure. This mechanism results in slower deformations formed by chemical stimulation than by using thermal stimulation.

Figure 5.

(a) Chemical stimulation and (b) compensation by thermal stimulation. The ramp time for both stimulations is 30 s.(c) The hydrogel swells due to the concentration decrease and shrinks due to the temperature decrease.

The sensor developed by Schreiber et al. (2023) is operated in the temperature range between 5°C and 40°C to perform the compensation mode. According to Figure 6, the heating from 20°C to 42°C still causes an observable hydrogel swelling. However, the heat-induced phase transition occurs implicitly: the equilibrium swelling degree at 42°C has a lower value than at 30°C, which is indicated by equation (13) and the chosen parameters in Table 2. When the temperature approaches or exceeds VPTT (43°C), the hydrogel initially undergoes swelling and subsequently shrinks due to the volume phase transition. This effect results from the coupling of the temperature and the temperature-dependent parameter $\chi$ . At the beginning (the initial temperature is 20°C), $\chi$ is low and the rising temperature causes water absorption. Subsequently, $\chi$ increases and dominates the fluid penetration, leading to the deswelling of the hydrogel. It is attributed to the fact that the phase transition governs the fluid penetration at VPTT. This interpretation explains the non-obvious swelling when the cooling begins at $t=60$ s, see Figure 5(c).

Figure 6.

Influence of thermal stimulus on hydrogel response with different range of heating. Ramp time is 1 s.

The compensation method (Schreiber et al., 2023) aims at maintaining a certain hydrogel swelling by setting a target voltage. Hence, the temperature course interprets the compensation of the hydrogel swelling by this mode. In the aspect of the numerical simulation, chemical stimulation and thermal stimulation occur simultaneously to keep the swelling degree constant. Additionally, chemical conditions are unchanged based on the test case interpreted by Figure 5(a). In this case, we discuss the influence of temperature course on the compensation mode. The temperature ramp is modified so that it best compensates for the chemically stimulated swelling as shown in Figure 5(c). The highlight is that the compensation mode reduces the measuring time. In comparison to the test case presented by Figure 5, the sensing duration is reduced from 45 to 30 s, which corresponds to the ramp time. As shown in Figure 7, in comparison to the quasi-linear temperature course, the modified temperature input presents a better result as expected. The quasi-linear temperature ramp can be regarded as a delay of the force compensation, resulting in an extended measurement time. The volume change of hydrogel requires additional time to be balanced and the thermal sensor also needs more time to capture the final signal. In Figure 5(c), the force compensation occurs after the swelling and requires 35 s, exceeding 30 s for an immediate compensation. These results collectively demonstrate that the immediate force compensation significantly enhances the sensing performance of the hydrogel-based sensor.

Figure 7.

(a) Temperature inputs applied to compensate for the chemically induced swelling. Temperature is reduced from 40°C (313.15 K) to 5°C (278.15 K). (b) Hydrogel responses in compensation mode with temperature inputs from (a).

Conclusions

In this work, we have modeled the swelling behavior of an AMPS-co-NIPAAm hydrogel within a chemical sensor by using a poromechanical framework, where a coupled chemo-thermal stimulation was considered. Using weak-form formulations, we implemented this framework in COMSOL Multiphysics^® to simulate the hydrogel responses to each form of stimulation. Then, we compared the numerical results with experimental data.

To ensure the quick response of the sensor, the temperature is controlled by the integrated Peltier element to compensate for the chemically induced hydrogel response. In the application, the hydrogel volume positively correlates to the temperature since the arranged temperature does not provoke the phase transition of the used co-hydrogel, which leads to the opposite phenomenon. However, we also discussed the influence of phase transition by simulations as a numerical test case.

The poromechanical framework serves a dual purpose in this study, presenting both the modeling of responsive hydrogels and the exploration of related material parameters. Specific physical parameters, such as the Flory-Huggins parameter and the diffusion coefficient, can be estimated through numerical methods instead of complex laboratory measurements. This is made possible by applying the swelling degree as a reference for parameter modification. Importantly, this approximation remains physically reliable, given that the poromechanical approach integrates continuum mechanics with constitutive laws. Since no optimization procedure was considered at this stage, further improvement can be expected by advancing sensor setup and control regime.