Constitutive model of salt concentration-sensitive hydrogel

Mechanics of Materials 136 (2019) 103092

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Constitutive model of salt concentration-sensitive hydrogel Shoujing Zheng, Zishun Liu

⁎

T

International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China

ARTICLE INFO

ABSTRACT

Keywords: Salt concentration sensitivity Hydrogel Donnan equilibrium FEM

Salt concentration-sensitive hydrogel is a material that not only has potential in novel applications of mechanical engineering, but also is very common in nature. Several theoretical studies of this type of hydrogel have been conducted in the past several years. However, existing research on the theory of salt concentration-sensitive hydrogel usually assumes the concentration of the solution to be independent of the deformation, ignoring the fact that correlations exist between them. In this study, we present a theory of salt concentration-sensitive hydrogel aiming at predicting its deformation behavior by linking the concentration of the solution with the deformation based on Donnan equilibrium and large-deformation theory. To address the difficulties raised by the process of free-energy differentiation, a mathematical method to obtain stress without the aforementioned assumption is developed. Then, two typical cases, namely a free-swelling case and a constrained-swelling case, are studied to demonstrate the usefulness of this theory. We also compare the present theory with existing ones to emphasize the suitability of the present theory for the salt concentration-sensitive hydrogel. Furthermore, because of the novel mathematical process, the present theory can be realized with the finite-element method (FEM), whose results are then compared with the analytical results. Finally, to verify the robustness of the FEM formulation, we conduct an interesting bilayer swelling experiment, which shows a reasonable agreement with the FEM result. We also compare the theoretical prediction with the experimental data that are available in the literature.

1. Introduction When exposed to a solvent, a hydrogel can swell to many times its initial size (Zheng et al., 2018). Through the use of appropriate monomers, this swelling mechanism has been shown to be controllable via external stimuli, such as temperature (Zheng and Liu, 2018), salt concentration (Hong et al., 2010), pH (Marcombe et al., 2010), and light (Toh et al., 2014). A detailed description can be found in a recent review (Liu et al., 2015). Among all those types of hydrogels, salt concentration-sensitive hydrogel demonstrates a wide range of applications, such as sensors (Shin et al., 2010) and actuators (Harmon et al., 2003), artificial muscles (Liu and Calvert, 2010), and drug delivery (Hamidi et al., 2008). Recent developments in novel applications such as ionic cables (Yang et al., 2015) and ionic skin (Sun et al., 2015) have utilized salt concentration-sensitive hydrogels as effective ionic conductors, paving ways to the potential development of wearable electronics and soft robotics. Regarding modeling of mechanical and chemical properties, many theories have been developed to understand the deformation behavior of salt concentration sensitive hydrogel. Katchalsky et al. (1952)

⁎

extended the classic Flory–Rehner model built for neutral gels to describe the salt concentration-sensitive hydrogel. Similarly, Dušek and Patterson (2010) used an analogy of the coil-globule transition to study the swelling of gels. Later, Hong et al. (2010) developed a field theory of polyelectrolyte gels by coupling large deformation and electrochemistry, which can be extended to explain the phenomenon of salt concentration-sensitive hydrogel. Similarly, Yu et al. (2017) used the theory to study the phase transition of the gel. Marcombe et al. (2010) also considered the stimuli of salt concentration in their theory. However, the studies mentioned are not direct studies of salt concentrationsensitive hydrogel. For example, Hong et al. (2010) also considered electric displacement in their study, which is unnecessary for salt concentration-sensitive hydrogel. There is also an inappropriate assumption that the ion concentration is independent of the deformation, which cannot be programed in the UHYPER subroutine and thus makes the FEM formulation problematic. To tackle those problems, this paper describes the study of a direct theory of the gel. We first develop the theory of the gel based on Donnan equilibrium and large-deformation theory, which aims at predicting the deformation behavior. To deal with the difficulties that arise

Corresponding author. E-mail address: [email protected] (Z. Liu).

https://doi.org/10.1016/j.mechmat.2019.103092 Received 11 April 2019; Received in revised form 14 June 2019; Accepted 14 June 2019 Available online 18 June 2019 0167-6636/ © 2019 Elsevier Ltd. All rights reserved.

Mechanics of Materials 136 (2019) 103092

S. Zheng and Z. Liu

from the process of free-energy differentiation, which is due to the aforementioned inappropriate assumption, we developed a new mathematical method of differentiation that assumes the ion concentration is dependent of the deformation. Then, we study two typical cases, a freeswelling case and a constrained-swelling case, to demonstrate the usefulness of the present theory. Furthermore, this theory is compared with published theories to highlight the suitability of the present theory for this gel. In addition, we implement the present theory in the FEM code, whose results are then compared with the analytical results. Finally, the theoretical prediction is compared with the experimental data available in the literature. We also conduct an interesting bilayer swelling experiment, which shows reasonable agreement with the FEM result. The article is organized as follows. In Section 2, we develop the theory of salt concentration-sensitive hydrogel based on Donnan equilibrium and in a large-deformation framework. In Section 3, we apply the theory to solve two boundary value problems analytically. In Section 4, we implement the theory into an FEM code. In Section 5, we compare the numerical results with the experimental data. Concluding remarks are provided in Section 6.

provides thermodynamic equilibrium. We also define the nominal concentration Ca as a

It should be noted that the nominal concentration C has been assumed as independent of J in many studies (Hong et al., 2009, 2010, 2008; Marcombe et al., 2010). For neutral gels (Hong et al., 2009, 2008), the assumption is correct because the only concentration in the theory is that of the solvent. However, for salt concentration-sensitive hydrogel (Hong et al., 2010; Marcombe et al., 2010), owing to the restriction of the Donnan equilibrium, Ca is related to J. This relation is described in Section 2.3. Therefore, the original assumption is overthrown in our study. 2.2. Free-energy density function Among the many models describing the hydrogel, the best-known model is that of Hong et al. (2008) for neutral gels, in which the freeenergy density function of the gel is a combination of stretching the network of the polymers and mixing the polymers and solvent molecules. Following Hong et al. (2010), we further extend this approach by adding a term due to the ions:

2. Theory of salt concentration-sensitive hydrogel In this section, we formulate the theory of salt concentration-sensitive hydrogel. We start with the widely accepted continuum theory frequently used in the past decade, which considers the large deformation and mixing of polymers and solvent (Hong et al., 2009). The present theory distinguishes itself from other studies in two main aspects. The first is the differentiation of a part of the free-energy density. The second is its consideration of Donnan equilibrium. The former enables a mathematically strict and stable implementation process of the theory into the FEM, whereas the latter shows a better prediction in more concentrated solutions.

(3)

W = Wnet + Wsol + Wion,

where Wnet, Wsol, and Wion are the free-energy density of stretching the network, mixing the polymers and solvent, and mixing the solvent and ions, respectively. The free-energy density function of stretching the network is specified as (Hong et al., 2008)

Wnet =

1 NkT (I 2

3

2 ln J ),

(4)

where invariant I = tr (F TF) ; N is the density of the number of the polymer chains in the referential state in m − 3, rendering Nv a dimensionless parameter; k is the Boltzmann constant; and T is the absolute temperature. The free-energy density function of mixing the polymers and solvent is (Flory, 1941)

2.1. Thermodynamic equilibrium We take the undeformed dry network of a gel as the reference state, which is described by coordinate X. The current state can be described by x. The deformation gradient of the network is

x F= , X

(2)

C a = c aJ.

Wsol =

(1)

kT vC s vC s ln v 1 + vC s

1 + Cs

,

(5)

s

where C is the nominal concentration of the solvent; v is the volume of the solvent molecule; for simplicity, we assume the volume of mobile ions is also v; and χ is the dimensionless parameter describing the enthalpy of mixing. The free-energy density function of mixing the solvent and ions, which differs from that of the neutral gel, is assumed to be (Marcombe et al., 2010)

where det(F) = J . According to Fig. 1, in the current state, a gel is subjected to geometric constraints and mechanical loads, as well as in equilibrium with the salt solution, whose chemical potential is affected by the salt concentration. Inside the gel, on the other hand, the chemical potential is affected by J and the true concentration ca, the number of mobile ions of species a per unit volume in the gel. The equilibrium of the chemical potential inside and outside of the gel

Wion = kT C+ ln

C+ + cref J

1 +C

ln

C cref J

1

,

(6)

a where cref is a reference value of the a take cref to be c0, the concentration a

concentration of species a; here, we in the salt solution. Ca is in m − 3, rendering vC a dimensionless parameter. The free-energy density function is the cornerstone of this theory. In Section 2.3, we combine it with Donnan equilibrium to analyze the equilibrium of salt-concentration-sensitive hydrogel, as shown in Fig. 1. 2.3. Donnan equilibrium To specify the relationship between the true concentration ca and J, we introduce the well-known Donnan equations (Marcombe et al., 2010):

Fig. 1. A gel is in contact with a solvent with a fixed salt concentration and is subject to a mechanical load and geometric constraint.

c+ c = 0, c0 c 2

(7)

Mechanics of Materials 136 (2019) 103092

S. Zheng and Z. Liu

where c0 is the true concentration of the salt solution. Furthermore, electroneutrality requires that (Hong et al., 2010) (8)

C0 + C = C +, where C0 is the fixed charges in the gel. A combination of Eqs. (2) and (8) yields

C0 + c = c+. J

(9)

With Eqs. (7) and (9), we obtain

c+ =

C02 + 4c02 , J2

C0 1 + 2J 2

C02 + 4c02 . J2

C0 1 + 2J 2

c =

(10) (11)

With the help of the Donnan equation, the function W has been reduced to only two independent variables, W(J, c0). Furthermore, with the condition of molecular incompressibility enforced as a constraint, the theory specifies the true stress as ij

=

FjK W J FiK

µ ij, v

(12)

whereμis the chemical potential in the salt solution and can be specified as

µ=

Fig. 2. (a) Schematic of the free swelling from the dry state; (b) an initially isotropic gel cube subjected to constraints on one pair of parallel sides and swelling in the lateral direction.

(13)

2kTvc0.

From Eqs. (10) and (11), we obtain

c+ = J

C0 2J 2

1 C02 2 J3

1 C02 J2

c C = 02 J 2J

1 C02 2 J3

+ 4c02

1 C02 J2

results of those different considerations are compared in detail in Section 3.3. Herein, we formulated the theory of salt concentration-sensitive hydrogel. In Section 3, the theory is used to produce theoretical prediction, and in Section 4, FEM results can be produced based on this theory.

, (14)

.

+ 4c02

(15)

3. Free- and constrained-swelling cases

Using the function W(J, c0) specified in Section 2.2 and Eqs. (10)–(15), we obtain ij

kT / v

=

(

Nv (FiK FjK J

c+ ln

c+ c0

)

1 +c

ij )

+ ln

(ln

c c0

J

1 J

+

1 J

+

) (

1 +J

J2 c+ J

We now apply the present theory to two typical cases, a free-swelling case (Fig. 2a) and a constrained-swelling case (Fig. 2b). The theory enables us to study the relationships among the stretch, concentration of ions, and stress.

+ 2 c0+ ln

c+ c0

+

c J

ln

c c0

)

. (16)

3.1. Free swelling

It should be noted that the process from the free-energy function to Eq. (16) is a strict mathematical differentiation. The process is fundamentally different from those of Hong et al (2010) and Marcombe et al. (2010) and their following studies (Yu et al., 2017), which assumed that Eq. (2) allows Ca to be independent of J and thus obtained different stress equations. Because of the strict mathematical process, the present theory enables the FEM realization in Section 3.3. The theory by Hong et al. (2010), however, cannot be properly implemented in FEM because of its assumption that Ca is independent of J. Although Marcombe et al. (2010) has FEM realization, we notice in its supporting information that the process in the UHYPER from U(1) to UI (3) allows Ca to be independent of J. Because of this inappropriate assumption, the FEM may work in some cases in Marcombe et al. (2010), but others, like those presented in Section 4, cannot be calculated by their UHYPER code. It should be noted that here, during the past decade, many relevant studies accepted the assumption that Ca is independent of J. Here, we use Donnan equilibrium and large-deformation theory to overthrow this assumption so a proper FEM realization can be achieved. Furthermore, the present theory considered three types of free-energy function, which is suitable for our concerns for salt concentrationsensitive hydrogel. The above studies (Hong et al., 2010; Marcombe et al., 2010), due to their different research concerns, introduce free energy of polarization and free energy of dissociation, respectively. The

Consider the case of the free swelling case and defineλas the linear stretch, so (17)

3

J=

The static equilibrium means that the stress vanishes, so Eq. (16) becomes

N 2

(

1

)+

(

c+ ln

c+ c0

2

ln

3

1 3

)

+

1

(ln

1 +c

+

4+

)

c c0

1 +

3

(

c+ J

ln

c+ c0

+

c J

ln

c c0

)

+2

2

,

c0 = 0 (18) and Eqs. (14) and (15) become

c+ = J

C0 2 6

1 C02 2 9

1 C02 6

c C = 06 J 2

1 C02 2 9

1 C02 6

,

+ 4c02

+ 4c02

(19)

. (20)

By solving Eqs. (18)–(20), we obtain the salt concentration-sensitive 3

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S. Zheng and Z. Liu

Fig. 3. During the free-swelling process, the (a) stretch, (b) volume ratio, (c) true concentration of counter ion, and (d) true concentration of co-ion as a function of salt concentration in the solution with different nominal concentrations of fixed charges.

gel in equilibrium with the salt solution. The results depend on the following dimensionless parameters: the crosslinking density Nv, mixing enthalpy parameterχ, and nominal concentration of the fixed charges vC0. Fig. 3 plots the theoretical results for free swelling with Nv = 10−3 and = 0.1. It can be seen from Fig. 3a and b that increasing the salt concentration in the salt solution results in a continuous decrease in the stretch and thus a continuous shrinking of volume. The ion concentrations in the gel also vary with the salt concentration in the external solution (Fig. 3c and d). Please note that we only consider the concentration of ions from 10−6–10−1, and the range that is too large or too small is out of our consideration.

c+ = J

1)

0

(

2

c+ ln

+ ln c+ c0

0

2 0

)

1 2

1 +c

+

1 0

(ln

2

c c0

+

2 ( 0 2)

)

1 +

kT / v

2

(

c+ J

ln

c+ c0

+

c J

ln

c c0

)

1 2(

C02 2 3 0 )

1 C02

2 ( 0 2)

, + 4c02

1 C02

2 ( 0 2)

(22)

. + 4c02

(23)

=

Nv 0

2

(

2 0

2 ).

(24)

Solving Eqs. (21)–(24), we obtain the salt concentration sensitive gel in equilibrium with the salt solution under constraint. The results are shown in Fig. 4 for 0 = 1.5. Fig. 4 shows similar trends to those in the free-swelling case. The variation in stress with salt concentration can also be observed in Fig. 4d. 3.3. Comparison with existing works Several theories have been developed to describe the salt concentration-sensitive hydrogel. The best-known ones are that of Hong et al. (2010), a theory considering the large deformation and electrochemistry of polyelectrolyte gels; and that of Marcombe et al. (2010), a theory of pH and salt-concentration-sensitive hydrogel. After careful review of those two existing theories and subsequent work (Yu et al., 2017), two major issues have been found. The first is the assumption that Ca is independent of J. For example, a part of

+ 0

0

1 C02 2 ( 0 2)3

The stress caused by the constraint in the gel along the thickness direction can be given by

In this section, we demonstrate the gel's swelling under constraint, as shown in Fig. 2b. When constrained on opposing faces, the gel is free to swell in the lateral directions. Laterally, it is stress-free, whereas there are stresses induced in the longitudinal direction. The swelling stretch is determined by ( 2

2(

2 )2

c C0 = J 2( 0 2)2

3.2. Constrained swelling

N

C0

+2 .

c0 = 0 (21) In this constrained case, Eqs. (14) and (15) become 4

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S. Zheng and Z. Liu

Fig. 4. During constrained swelling, the (a) stretch, (b) true concentration of the counter ion, (c) true concentration of the co-ion, and (d) the nominal stress as a function of salt concentration in the solution with different nominal concentrations of fixed charges.

Fig. 5. Comparison of the theoretical results of the present work and Hong et al. (2010). (a) and (b) have different ranges of salt concentration.

the free energy of ions mixing with the solvent in (Marcombe et al., C 1) , so the result of the stress due to this part of 2010) is kTC+ (log ref+

To solve those two issues, we develop our theory in Section 2. The present work uses Donnan equilibrium to determine the relationship between the ion concentration in the gel, whereas Hong et al. (2010) introduces the free energy of polarization. Eqs. (7.1) and Eq. (7.2) in the paper determine the ion concentrations in the gel, which does not satisfy the Donnan equilibrium. The effect of the difference is shown in Fig. 5. Although Hong et al. (2010) did not provide an explicit equilibrium equation, its following work (Yu et al., 2017), however, provided that (Eq. (3.9)) in its paper, which is used to draw the black line in

c+

energy, with the help of the assumption, is kTc+. However, strict deviation should result in a part in Eq. (16). The assumption of Ca being independent of J makes FEM realization problematic. The second issue is the introduction of free energy of polarization in Hong et al. (2010); this part aims to tackle the electrochemistry of polyelectrolyte gels, so it is unnecessary for salt concentration-sensitive gels, which makes the prediction widely inaccurate in concentrated solution (Fig. 5b). 5

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S. Zheng and Z. Liu

Fig. 6. Comparison of the theoretical results between the present work and Marcombe et al. (2010). When pKa gradually changes from 4.3 to 2 in Marcombe et al. (2010), the results approach the result of the present work.

Fig. 7. Stretch of a free swelling gel and a constrained swelling gel as a function of the salt concentration of the external solution.

used approach in thermodynamics (Hong et al., 2009), we introduce ^ via Legendre transformation: another free-energy function W

Fig. 5. In Fig. 5a, when the salt concentration is relative dilute (vc0= 10−7–10−2), the prediction from both works agrees pretty well. However, in Fig. 5b, when the salt concentration is relative concentrated (vc0= 0.01–1), the effect of electrochemical potential of the ions increases the stretch of gels so much that the result is not accurate in the physical world. For example, when the salt concentration is 1 mol/l (vc0= 0.0602), the stretch of the equilibrium state should stay relatively small, as the present work predicted, instead of increasing substantially as shown in the black line. The ion concentration in the gel in Marcombe et al. (2010) is specified with Donnan equilibrium and the introduction of dissociation due to the consideration of pH values in the solution. Although the present theory does not consider dissociation owing to the nature of salt-concentration-sensitive hydrogel, there is a connection between those two theories. When pKa = 4.3, the value set in Marcombe et al. (2010) representing the dissociation of weak acids, the results are shown in Fig. 6. When pKa decreases gradually to 2, a value of full dissociation with which the effect of dissociation can be neglected, and the results agree very well with those of the present work. However, a difference between the present work and Marcombe et al., 2010) exists as well, which is also the difference between the present work and Hong et al., 2010): the present work takes a strict differentiation from free energy to stress, whereas the existing theories and subsequent works take the assumption of Ca being independent of J in the differentiation process. The main influence of the different approaches is reflected in the FEM realization. The present theory can be implemented into the subroutine code smoothly owing to its strict differentiation from free energy to stress. It is problematic, however, for the existing theories to achieve that. There is no FEM realization by Hong et al. (2010). Although there is FEM realization by Marcombe et al. (2010), the code is not mathematically strict. For example, we notice in its supporting information that the process in UHYPER from U(1) to UI(3) allows Ca to be independent of J. Because of this inappropriate assumption, we are unable to use the subroutine code provided by Marcombe et al. (2010) to obtain the results shown in Fig. 6. However, with rigorous theory, we can deduce a mathematically strict FEM subroutine. The results are detailed in Section 4.

^ =W W

(25)

µCs.

It should be noted that we write J = 03 J in all subsequent implementations in ABAQUS, whereJ denotes the actual swelling ratio and J′ denotes the swelling ratio used in ABAQUS. The non-dimensionalized free energy density in Eqs. (4)–(6) is therefore rewritten as ^ W kT

1

= 2N ln

+ C+ ln c

(

2 2 ¯ 0 J 3 I1

3 1 0J 3 0J

C+ 3 0 0J

+

3 3 0J

2 ln(

3 0J

)

) +(

µ¯ s

1 + C ln c

3 0J

1) .

C 3 0 0J

1

(26)

We have implemented the above theory in ABAQUS by coding the ^ into a user-defined subroutine for a hyperelastic material. function W Details about the FEM implementation can be found in the appendix. We first test our finite-element program using the cases described in Section 3. In Fig. 7, the cases of free and constrained swelling are plotted. In both cases, the results obtained by FEM match well with those of the analytical solutions. To demonstrate the robustness of the developed subroutine (UHYPER), in Section 5, we show a few illustrative numerical

4. FEM implementation In this section, we implement the present theory into subroutine code UHYPER of the commercial finite-element software ABAQUS. The equilibrium condition of a salt concentration-sensitive hydrogel is expressed in Eq. (16), which governs the following independent inhomogeneous fields: xi(X), c+ (X) , and c (X) . Following a commonly

Fig. 8. Relative volume ratio ( = 03/ 3 ) versus NaCl concentration for mixed solvents with various volume fractions of acetone, comparing theoretical predictions (lines) with the data (symbols) from the experiments by Ohmine and Tanaka (1982). 6

Mechanics of Materials 136 (2019) 103092

S. Zheng and Z. Liu

Fig. 9. Comparison of the experiment and the simulation. (a) 1 mol/L NaCl solution, reference state; (b) 0.1 mol/L NaCl solution; (c) 0.01 mol/L NaCl solution. Scale bars are 2.5 mm for all images.

simulation examples compared with the relevant experimental results.

According to Ohmine and Tanaka's study (Ohmine and Tanaka, 1982), the volume fraction of polymer was 0.05, which made the initial state of the gel with 0 = 201/3 = 2.7 and the measured volume ratio = 03/ 3. Following Yu et al.’s approach (Yu et al., 2017), we assume the change in the acetone concentration (φ) is represented by the change in the dimensionless interaction parameterχ, namely,

5. Experimental verification Ohmine and Tanaka (1982) performed a series of experiments to study salt's effects on the swelling behavior of salt concentration-sensitive hydrogels. They (Ohmine and Tanaka, 1982) provided all material parameters except C0 in our model. The comparison is plotted in Fig. 8. The theoretical predictions match well with Ohmine's experimental data for ionized acrylamide gel with four different acetone concentrations by using one fitting parameter, f = 0.09–0.03φ, where φis the acetone concentration. Both the theoretical predictions and the experimental results show that the swelling ratio induced by the change in the salt concentration in the outer solution increases as the acetone concentration increases.

= ¯0 ( ) +

¯1 ( ) , J

(27)

where

¯0 ( ) = 1.6 + 0.01(1 ¯1 ( ) =

2 + 0.6(1

), ).

(28) (29)

We next consider a thin layer of a salt concentration-sensitive 7

Mechanics of Materials 136 (2019) 103092

S. Zheng and Z. Liu

hydrogel bonded to a neutral gel that will not swell under the stimulus of salt concentration. The salt concentration-sensitive hydrogel is allowed to swell, and this causes large bending of the bilayer. The initial dry geometry is taken to be 25 mm in length and 5 mm in width, with 2.5 mm for the swellable gel and 2.5 mm for the non-swellable elastomeric substrate. To verify our proposed model, we compared the corresponding experimental results with the theoretical predictions. First, we conducted the experiment of the swelling process of the bilayer, the swellable part of which imbibes water molecules due to the difference of salt concentration between inside and outside of the hydrogel. We purchased the following substances from Sigma Aldrich: acrylamide (AAm, monomer), N, N’-methylenebis(acrylamide) (MBAA, crosslinker), N, N, N, N’-tetramethylethylenediamine (TEMED, accelerator), and ammonium persulfate (APS, initiator). We prepared the samples by the standard method (Ohmine and Tanaka, 1982). We first prepared an aqueous solution of 2.2 M AAm, then added MBAA, TEMED, and APS in quantities of 0.00156, 0.0046, and 0.0058 times the weight of AAm. This aqueous solution was injected into a 20 × 20 × 1 mm3 acrylic mold, sandwiched between two glass plates, and stored at room temperature for 24 h. The aqueous solution turned into a hydrogel. The synthesized sample was used in the experiment as the neutral gel. Half of the synthesized hydrogels were then placed in a basic solution (pH 12) of 0.01 mol/L NaOH in water for 20 days to hydrolyze a portion of the acrylamide groups into acrylic acid groups. The synthesized sample was used in the experiment as the salt concentrationsensitive hydrogel. The gels were then washed in water and placed in 1 mol/L NaCl solution as the referential state. Then, the neutral gel was dyed black to differentiate it from the salt concentration-sensitive hydrogel, as can be seen in Fig. 9. We then cut both types of hydrogel into rectangles using a laser cutting machine. The rectangle samples with a length of 25 mm and width of 2.5 mm were then bonded together with glue, as shown in Fig. 9. We then put the bilayer into the NaCl solutions with concentrations of 0.1 mol/L and 0.01 mol/L. The bilayer that achieved equilibrium can be seen in Fig. 9a. The case was then modeled for comparison with the experimental results. The non-swellable hydrogel was modeled as an incompressible hydrogel material with a constant predefined field. The interface between the gel and non-swellable gel was taken to be perfectly bonded. As in the previous example, the initial condition for the salt concentration of the swellable gel was taken to be the same as the experiment. We set a thin film outside the gel to constrain the deformation. Fig. 9 shows snapshots of the deformed bilayer immersed in the 0.1 mol/L and 0.01 mol/L NaCl solution. The example further demonstrates the great potential of the proposed FEM method to analyze the deformation of salt concentration-sensitive hydrogel in nature and engineering applications and help us understand their deformation behaviors.

FEM result, an interesting bilayer swelling experiment has been conducted, which shows good agreement with the FEM result. The theoretical prediction has also been compared with the experimental data available in the literature. The salt concentration-sensitive hydrogel is a very common material with great application potential, so the theory and FEM realization developed in this paper can help the gel community understand and predict the deformation behavior of salt concentration-sensitive hydrogels. Acknowledgment The authors are grateful for the support from the National Natural Science Foundation of China through grant numbers 11820101001, 11811530287 and 11572236. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmat.2019.103092. Appendix. UHYPER implementation details SUBROUTINE UHYPER(BI1,BI2,AJ,U,UI1,UI2,UI3,TEMP,NOEL, 1 CMNAME,INCMPFLAG,NUMSTATEV,STATEV,NUMFIELDV,FIELDV, 2 FIELDVINC,NUMPROPS,PROPS) C============================= =============================== === C User defined hyperelastic material subroutine C for gel with Flory-Rehner free-energy function C to be used in Abaqus Standard C Fomulated and written by ShoujingZheng, Nov 25, 2018 C————————————————————— C Material properties to be passed to the subroutine: C PROPS(1) - Nv C PROPS(2) - chi C PROPS(3) - lambda_0 initial swelling C PROPS(4) - cf fixed charges C C State variable: C TEMP - co true concentration in the solvent C C The initial value of PH0 and lambda_0 should match each other C============================ ============================== ===== C INCLUDE 'ABA_PARAM.INC' C CHARACTER*80 CMNAME DIMENSION U(2),UI1(3),UI2(6),UI3(6),STATEV(*),FIELDV(*), 1 FIELDVINC(*),PROPS(*) REAL(8) Nv, chi, lambda0, detF0, cf, c0, mu_kT REAL(8) U UI1 UI2 UI3 c1 c2 c3 c3 cp cn dcp dcn c4 c5 REAL(8) d2cp d2cn c6 c7 d3cp d3cn Nv = PROPS(1) chi = PROPS(2) lambda0 = PROPS(3) cf=PROPS (4) detF0 = lambda0**3 c0 = TEMP ! TEMP is used to store chemical potential here mu_kT=−2*c0 U(1) =Nv/2 * ( lambda0**2*BI1*AJ**(2.0/3.0) - 3 & - 2*(3*LOG(lambda0) + LOG(AJ)) ) - chi/detF0/AJ & - (detF0*AJ-1)*LOG(AJ/(AJ-1/detF0)) - mu_kT*(detF0*AJ-1) U(2) = 0 UI1(1) =Nv/2 * lambda0**2*AJ**(2.0/3.0) UI1(2) = 0

6. Concluding remarks We have developed a theory of salt concentration-sensitive hydrogel based on Donnan equilibrium and large-deformation theory, which aims to predict the hydrogel's deformation behavior. In the theory, to solve the incorrectness arising from the process of free-energy differentiation, which is due to an inappropriate assumption, a strict mathematical differentiation method has been developed by adding the Donnan equilibrium constraint. Then, two typical cases, a free-swelling case and a constrained-swelling case, have been studied to demonstrate the usefulness of the present theory. We have also compared this theory with existing ones to emphasize the suitability of the present theory for the salt concentration-sensitive hydrogel. Furthermore, we have implemented the present theory to the FEM code, whose results are the same as the theoretical results. Finally, to certify the stability of the 8

Mechanics of Materials 136 (2019) 103092

S. Zheng and Z. Liu

UI1(3) = Nv/3*lambda0**2*BI1*AJ**(−1.0/3.0) & + (1-Nv)/AJ - ( LOG(AJ/(AJ-1/detF0)) + mu_kT )*detF0 & + chi/detF0/AJ**2 UI2 = 0 UI2(3) = -Nv/9*lambda0**2*BI1*AJ**(−4.0/3.0) & - (1-Nv)/AJ**2 + 1/AJ/(AJ-1/detF0) - 2*chi/detF0/AJ**3 UI2(5) = Nv/3*lambda0**2 * AJ**(−1.0/3.0) UI3 = 0 UI3(4) = -Nv/9*lambda0**2 * AJ**(−4.0/3.0) UI3(6) = 4*Nv/27*lambda0**2*BI1*AJ**(−7.0/3.0) & + 2*(1-Nv)/AJ**3 - (2*AJ-1/detF0)/(AJ*(AJ-1/detF0))**2 & + 6*chi/detF0/AJ**4 c1=cf/(detF0*AJ) c2=(cf**2/ (detF0*AJ)**2 + 4*c0**2)**(0.5) c3=cf**2/(detF0*AJ)**2 + 4*c0**2 cp=(c1+c2)/2 cn=(-c1+c2)/2 dcp=−0.5*cf/detF0/AJ**2–0.5* cf**2/detF0**2/AJ**3/c2 dcn=0.5*cf/detF0/AJ**2–0.5*cf**2/detF0 **2/AJ**3/c2 c4=cf/detF0/AJ**3 c5=3/2*cf**2/detF0**2/AJ**4/c20.5*cf**4 & / detF0**4/AJ**6*(c3)**(−3.0/2.0) d2cp=c4+c5 d2cn=c4+c5 c6=−3*cf/detF0/AJ**4 c7=−6*cf**2/detF0**2/AJ**5/ c2+9/2*cf**4/detF0**4 & / AJ**7*(c3)**(−3.0/2.0)- 3/2*cf**6/detF0**6 & / AJ**9*(c3)**(−5.0/2.0) d3cp=c6+c7 d3cn=-c6+c7 U(1)=U(1) & + detF0*AJ*(cp*(LOG(cp/c0)−1)+cn*(LOG(cn/c0)−1)) UI1(3)=UI1(3)+detF0*(cp*(LOG(cp/c0)−1) & + cn*(LOG(cn/c0)−1))+ detF0*AJ*(dcp*LOG(cp/c0) & + dcn*LOG(cn/c0)) UI2(3)=UI2(3)+2*detF0*(dcp*LOG(cp/c0)+dcn*LOG(cn/c0)) & + detF0*AJ*(d2cp*LOG(cp/c0)+d2cn*LOG(cn/c0) & + dcp**2/cp+dcn**2/cn) UI3(6)=UI3(6)++3*detF0*(d2cp*LOG(cp/c0) & + d2cn*LOG(cn/c0)+ dcp**2/cp+dcn**2/cn) & + detF0*AJ*(d3cp*LOG(cp/c0)+d3cn*LOG(cn/c0) & + 3*d2cp*dcp/cp+3*d2cn*dcn/cn & - (dcp**3/cp**2+dcn**3/cn**2)) STATEV(1)=c0

STATEV(2)=-TEMP RETURN END. References Dušek, K., Patterson, D., 2010. Transition in swollen polymer networks induced by intramolecular condensation. J. Poly. Sci. Part A 6. Flory, P.J., 1941. Thermodynamics of high polymer solutions. J. Chem. Phys. 9, 660 -660. Hamidi, M., Azadi, A., Rafiei, P., 2008. Hydrogel nanoparticles in drug delivery. Adv. Drug Deliv. Rev. 60, 1638–1649. Harmon, M.E., Tang, M., Frank, C.W., 2003. A microfluidic actuator based on thermoresponsive hydrogels. Polymer (Guildf) 44, 4547–4556. Hong, W., Liu, Z., Suo, Z., 2009. Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. Int J. Solids Struct. 46, 3282–3289. Hong, W., Zhao, X., Suo, Z., 2010. Large deformation and electrochemistry of polyelectrolyte gels. J. Mech. Phys. Solids 58, 558–577. Hong, W., Zhao, X., Zhou, J., Suo, Z., 2008. A theory of coupled diffusion and large deformation in polymeric gels. J. Mech. Phys. Solids 56, 1779–1793. Katchalsky, Lifson, S., Eisenberg, H., 1952. Equation of swelling for polyelectrolyte gels, (J. polymer sci., 7, 571‐574, 1951). J. Poly. Scie. 8, 476 -476. Liu, Z., Toh, W., Ng, T.Y., 2015. Advances in mechanics of soft materials: a review of large deformation behavior of hydrogels. Int. J. Appl. Mech. 07, 1530001. Liu, Z.S., Calvert, P., 2010. Multilayer hydrogels as muscle-like actuators. Adv. Mater. 12, 288–291. Marcombe, R., Cai, S., Hong, W., Zhao, X., Lapusta, Y., Suo, Z., 2010. A theory of constrained swelling of a pH-sensitive hydrogel. Soft Matter 6, 784–793. Ohmine, I., Tanaka, T., 1982. Salt effects on the phase transition of ionic gels. J. Chem. Phys. 77, 5725–5729. Shin, J., Braun, P.V., Lee, W., 2010. Fast response photonic crystal pH sensor based on templated photo-polymerized hydrogel inverse opal. Sensors Actuators B Chem. 150, 183–190. Sun, J.Y., Keplinger, C., Whitesides, G.M., Suo, Z., 2015. Ionic skin. Adv. Mater. 26, 7608–7614. Toh, W., Ng, T.Y., Hu, J., Liu, Z., 2014. Mechanics of inhomogeneous large deformation of photo-thermal sensitive hydrogels. Int. J. Solids Struct. 51, 4440–4451. Yang, C.H., Chen, B., Jing, J.L., Jian, H.Y., Zhou, J., Yong, M.C., Suo, Z., 2015. Ionic cable. Extreme Mech. Lett. 3, 59–65. Yu, Y., Landis, C.M., Huang, R., 2017. Salt-Induced swelling and volume phase transition of polyelectrolyte gels. J. Appl. Mech. 84 051005-051005-051012. Zheng, S., Li, Z., Liu, Z., 2018. The fast homogeneous diffusion of hydrogel under different stimuli. Int. J. Mech. Sci. 137, 263–270. Zheng, S., Liu, Z., 2018. Phase transition of temperature-sensitive hydrogel under mechanical constraint. J. Appl. Mech. Trans. Asme 85.

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