1 Introduction

Hydrogel is a kind of soft matter. It can absorb a large number of solvent molecules, which is of both solid-like and liquid-like properties. It is a new kind of polymer materials with various applications.

The industrial and biomedical applications of hydrogels made from either natural or synthetic sources are strongly limited by their poor mechanical properties. A hydrogel with normal structure (NC) breaks under a low stress because there are very few energy dissipation mechanisms to slow crack propagation. Many efforts have been focused on increasing the mechanical strength of hydrogels^[1], but the robustness still remains unsatisfactory.

Currently, there are various research works of chemical experiments^[2-6] and references therein. In chemical experiments, a variety of polymers are used in different physical conditions to design various hydrogels, which are to test the physical properties, e.g., the strainstress relationship^[4]. In recent years, three kinds of novel hydrogels with unique structures and high mechanical strength have been developed^[7]. Topological (TP) gels have figure-of-eightcrosslinkers that can slide along polymer chains^[2]. The gel swells to about 500 times of its original weight and can be stretched to nearly 20 times of its original length. The nanocomposite (NC) hydrogel is made from specific polymers with a water-swellable inorganic clay^[3]. Most of the macromolecules are grafted onto nanoparticles, indicating that the nanoparticle clay acts as a highly multifunctional crosslinking agent.

Macromolecular microsphere composite (MMC) hydrogel is a new way of synthesizing hydrogels with a novel, well-defined network structure and high mechanical strength^[4]. In this method, a peroxidized MMS acts as both an initiator and a crosslinker. The mechanisms for the formation of the peroxide and the initiation of polymerization, as well as for the formation of a hydrogel, are proposed in Fig. 1. The mechanical properties of the NS and MMC hydrogels were measured by compression tests. Not surprisingly, the NS hydrogel (87.5% water content) fractured under low deformation. However, the MMC hydrogel (A6, 89% water content) did not break, even under an extremely high strain (most of the hydrogel was squeezed out of the plates), and it quickly recovered its original shape after the release of its load (see Fig. 2). Figure 3 shows the corresponding stress-strain curves of the NS and the MMC (A6) hydrogels. The NS hydrogel broke at a stress of 0.08 MPa and a strain of 45.5%. However, the MMC (A6) hydrogel did not break even at a stress of 10.2 MPa and a strain of 97.9%. The stress sustained by the MMC hydrogel was 120 times more than the stress sustained by the NS hydrogel. Here, we hope to build a mathematical model to investigate the physical property of the MMC hydrogel. Along this line, there are some related mathematical works to analyze and simulate these new materials, e.g., Refs. [1] and [7]-[14]. For example, the phase transformation and deformation in different external environments were described by the mathematical models from the viewpoint of macroscopic^[15-19], microscopic^[20-21] and mesoscopic^[22-23], respectively.

Some additional effects due to the strain-stress have been reported. Mooney^[24] and Rivlin^[25] built up an energy functional to derive the strain-stress relationship which was widely used for its simplicity and validity. However, they did not consider the specific macrostructure and physical characteristics of the hydrogel enough, and the parameters could be determined only by data fitting. Recently, compression and elongation under large deformation were highly concerned. Miquelard-Garnier et al.^[26] and Webber and Creton^[27] did a series of compression experiments with different hydrogels, and they also investigated the compressive hysteresis experimentally but lacking for theories of mechanism to support. The understanding of their macroscopic mechanical properties remains an important challenge.

In Ref. [23], Zhai and Zhang used the time-dependent Ginzburg-Landau (TDGL) mesoscopic model to simulate the phase transition process of macromolecule MMC hydrogel. They proposed a free energy for such a reticular structure according to the structures of MMC hydrogel and the entropy theory. A spectral method was adopted to numerically solve the MMC-TDGL equation. The numerical results were consistent with the chemical experiments, showing the network structure. Furthermore, it was understood that the system shows intermittent phenomenon of polymerization and dissolution with the increasing reaction temperature, which is a very good explanation of chemical experiments. Following it, they investigated the kinetic model for the large deformation theory of hydrogel under the outside stimulations^[26]. They presented the large deformation dynamical model and the time developing process. Considering the force of the large deformation, they introduced the heat equation to express the transformation of the chemical potential. The extended model can be used to describe the development of the large deformation. They presented numerical examples of the cylindrical hydrogel for oneand two-dimensional cases under pressure and stretch. Here, we will continue the work^{[16, 23]} to investigate the property of the MMC hydrogel, such as the strain-stress for elongation and extension and the impact of the macroscopic property with respect to the microscopic parameters, e.g., the polymerization degree and the size of the macromolecular microspheres (MMs). Numerical computation shows that the MMC hydrogel is increasing hard to be compressed as the polymerization degree is high. The larger the size of the MMs is, the more difficultly the MMC hydrogel is compressed. Thus, the approach is shown that it is a novel way to investigate the relation of the macro-property and micro-parameters of the MMC hydrogel. It can be extended to other soft matters.

The outline of the paper is as follows. In Section 2, we state the free energy functional of hydrogel linked to the macrostructure of the MMC hydrogel. Then, we investigate the deformation of the cylindrical MMC hydrogel in Section 3. Some conclusions are drawn in Section 4.

2 Free energy functional of hydrogel

Here, we introduce the free energy functional for general hydrogels. For the self-consistence, we present the sketch of the related theory of the free energy functional density of hydrogels although a part of the content is from Ref. [16]. With it, we can obtain the free energy of the MMC hydrogel based on the microscopic theory of the MMC hydrogel set up by Ref. [23]. Thus, we can investigate the strain-tress property of the MMC hydrogel.

We consider that the hydrogel at the position X and the time t moves to the new position x(X, t). Then, the deformation gradient F is

(1)

where x_i and X_K denote the components of the vectors x and X, respectively. As we all know, the hydrogel contains three phases: solid phase, liquid phase, and ion phase. The solid phase is mainly formed by the polymer network structure. The liquid phase is mainly solvent, and the ion phase is free electronic in the solvent. We assume that the ion size is negligible. Therefore, the volume of the hydrogel mainly contains the solid phase and the liquid phase. By the definition of F(X, t), we know that the determinant of the deformation tensor det F(X, t) is the ratio of volume in the position X at the time t, i.e.,

where dv(x) is the volume element of the gel at the time t, and dV (X) is the volume initial element (i.e., the volume of the gel at t=0). Now, we take the dry gel (solid phase) as the reference state. The quantity can describe the volume fraction of the solid phase at the position X and the time t.

All of the long polymers and small molecules are assumed to be incompressible, and there is no void space in the gel. Thus, the incompressibility is expressed as

(2)

where n is the number of solvent molecules in the volume element at the time t, ν is the volume of each solvent molecule, and the number density of the solvent molecules is .

Now, the free energy density functional of the hydrogel W is required. The above two external roles F and C make contribution to the free-energy density functional together. Therefore, the free energy is a functional which is with respect to the deformation gradient and the nominal concentration of solvent molecules, namely, W=W(F, C). Thus, the derivative variation is

(3)

Suppose that the local arrangement is instant. Then, the nominal strain s_iK and the chemical potential μ are

(4)

The above model talks about the free-energy functional concerning the deformation gradient F and the nominal concentration C. However, in practice, we always know about some information of the chemical potential μ but the nominal concentration C. We can see that the chemical potential μ and the nominal density C are a pair of functions from (4). By the Legendre transformation, we can transfer the free energy functional into the following functional with respect to the chemical potential μ^[11]:

(5)

Therefore, corresponding to (4), we have

(6)

Now, assume that the dry gel immerses in the solvent with the chemical potential μ₀ firstly and swells freely. Since the gel is soaked in the solvent and there is no any constriction, we consider that it is homogenous. Thus, the swelling ratios λ_i (i=1, 2, 3) of the hydrogel in each direction are the same, i.e., . As the gel swells freely, we can get λ₀ by solving (4), . Thus, the deformation tensor F₀ of the free swelling hydrogel^[11] is F₀=diag(λ₀, λ₀, λ₀).

It is our goal to investigate the property of the hydrogel soaked, that is, the hydrogel is the reference state in the following. However, the above theory is based on the reference state by the dry gel. Therefore, we have to translate some quantities, e.g., the deformation tensor and the free energy functional, into the ones based on the reference state by the dry gel. Now, we denote and to be the deformation tensor and the free energy functional of the freely swelling hydrogel, respectively. Then, the deformation tensor is when it takes the dry gel as the reference state.

(7)

in virtue of , and . Moreover, the nominal strain is . Up to now, we have obtained the free energy and strain when we take the freely swelling hydrogel as the reference state. Therefore, we will choose the free swelling hydrogel as object to research in the following.

3 Deformation of cylindrical MMC-hydrogel

In this section, we mainly discuss the deformation behavior of the cylindrical hydrogel under different external stimuli. Firstly, the dry gel swells freely in the solvent with the chemical potential of μ₀ and forms a hydrogel. We investigate the deformation by the model discussed above and compare it with the result from the chemical experiment.

We set the coordinates of the cylindrical hydrogel to be the height H, the radius R, and the circle direction θ in Ref. [7], and assume that the deformation ratios are the same for different points in one direction. Thus, the deformation tensor is

(8)

where λ_i (i=1, 2, 3) denote the ratios of the three directions^[18], that is,

Besides, det F=λ₁λ₂λ₃ is the ratio of deformation volume^[1]. Firstly, the dry gel swells freely to form hydrogel in the solvent with the chemical potential μ₀, and then we take this hydrogel as the object of study. Then, we will act certain stimulation, e.g., the pressure in some direction. Our aim is to investigate its deformation, as shown in Fig. 4. Such a hydrogel is not homogenous because there is a constriction in some place.

Now, we choose the free energy to be

(9)

where W _s(F) is the elastic free energy of the network, and W_m(C) is the mixed entropy between the solvent molecules and polymeric chains. Here, the Flory-Rehner form is chosen as the elastic free energy^[3],

(10)

where k is the Boltzmann constant, T is the temperature, and N is the number of polymeric chains per reference volume.

Next, we introduce the mixed entropy W_m(C) of the MMC hydrogel in detail. In Ref. [23], based on the Flory-Huggins lattice theory and the Boltzmann entropy theory, Zhai and Zhang obtained the mixed entropy for the MMC hydrogel as

(11)

where ø_s is the volume fraction of the segments in the polymer, and ν is the volume per molecule. α, β, τ, and ρ are the parameters depending on the number M of lattices occupied by the MMs, the polymerization degree N₁, the number R of MMs around an MM, and the number L of chains between MMs. χ is the parameter of the interaction between MMs and the solvent. These parameters are important to the macroscopic property of the MMC hydrogel^[23]. They satisfy the relations as follows:

Moreover, we know that the volume fraction ø_L of MMs satisfies^[23]

In virtue of (2), we have

(12)

Here, what we must mention is that the mixed entropy F(ø) for the MMC hydrogel is not nominal but it is in the Eulerian coordinates. Therefore, we translate it as the nominal mixed entropy through the relation

where dv=(1 + νC)dV. Then, substituting (12) into (11) yields the nominal mixed entropy of the MMC hydrogel,

(13)

With the elastic free energy and the mixed entropy for the MMC hydrogel, we can modify the free energy into the form of the ratios of deformation. Similar to that in Section 2, we also induce the Legendre transformation . Then, we can obtain the free energy with respect to the ratios of deformation by (2), (8), (9), (10), and (13). Together with (7), the total free energy is derived as

(14)

Next, we investigate the strain-stress relation of the MMC hydrogel. Taking the hydrogel with the swelling rate λ₀ as the reference state (later, we will explain such a λ₀), we put the hydrogel on the rigid plate and give a force on the top of the cylindrical MMC hydrogel. The steady state equations are

(15)

Here, S is the nominal stress, and the true stress σ=Sλ₁. By (14), these two equations in (15) are

(16)

(17)

where

(18)

The volume of the MMC hydrogel is conserved. Then,

(19)

Combination of (16) and (17) with (19) yields the relation of the strain-stress for the MMC hydrogel,

(20)

Here, NkT is the parameter group concerned with the shear modulus, which varies from 10⁴ N/m² to 10⁷ N/m² associated with the hydrogel properties. By numerical calculation, we can obtain the relation of the strain and stress, as shown in Fig. 5 with λ₀=2.5 and three different values of NkT. With the low shear modulus NkT, the hydrogel can be easily compressed. Otherwise, the maximal strain of compression is limited.

Comparing the compressible case with the chemical experiment result in Fig. 3(a) especially, we can see that when NkT=5 × 10⁵ N/m², they are completely consistent with each other. Through this method, the ratio of the shear modulus of the MMC hydrogel to the swelling stretch λ₀ can be approximated to be 5 × 10⁵. The new MMC hydrogel is not broken when it undergoes a large pressure. The deformation is shown in Fig. 2.

By the relation (20), the uniaxial elongation under large deformation is also performed. We integrate the elongation and compression into Fig. 6. It is shown that the hydrogel is extended as λ₁ > 1. Here, σ < 0 means that the direction of the force is downward. From the curve for λ₁ > 1 in Fig. 6, we can see that the strain-stress relation is almost linear for the extension.

In the last part, we turn to investigate the solvent. In this work, we consider a simple case and take a parameter μ₀, the chemical potential of the solvent, as the surrounding condition of the MMC hydrogel. Thus, there is no surface energy term in the free energy. In fact, μ₀ depends on the parameter λ₀ through the following relation (21).

As in Section2, if we change the reference state, we put the dry gel into the solvent with the chemical potential μ₀ to freely swell. Then, it satisfies λ₁=λ₂=λ₃=λ₀. Therefore,

Now, we can obtain that λ₀ satisfies the relation

(21)

By the numerical calculation of (21), we can get that λ₀ depends on μ₀, as shown in Fig. 7. It can also be seen that the dry gel swells smaller with the larger chemical potential solvent in Fig. 7. It shows that the larger the chemical potential of the solvent is, the smaller the dry gel swells. The calculation parameters are as follows: Nν=1.25 ×10⁻², χ=0.1, M=0.02, R=6, and N₁=400.

From the relation (20), we can see that the relation of strain-stress depends on the polymerization degree N₁ and the swelling ratio of the reference states λ₀. Furthermore, we know that λ₀ depends on the parameters α, β, τ, and ρ from (21). Here, we focus on the polymerization degree N₁ and the number M of lattices occupied by the MMs, which is the size of the MMs. Take three cases of N₁=5, N₁=50, and N₁=500 to compare. When 1 − λ₁ < 0.7, the curves of the strain-stress are almost coincident for these three parameters. In Fig. 8, we draw the case for 1 − λ₁ > 0.7, which means large deformation. It shows that the MMC hydrogel is increasingly hard to compress when N₁ is high. Therefore, the polymerization degree has direct influence on the property of the strain-stress under large deformation. Similarly, we also investigate the effects of the size M of the MMs in Fig. 9. This presents that the larger M is, the more difficultly the hydrogel is compressed.

4 Conclusions

Here, we set up a large deformation model of the MMC hydrogel. We can obtain the strain-stress relation, which completely agrees with the result from the chemical experiment. The approach is a way to investigate the relation of the macroscopic property and microscopic parameters of the MMC hydrogel, e.g., the number M of lattices occupied by the MMs, the polymerization degree N₁, the number density R of MMs around an MM, the number density L of chain between MMs, and the interaction χ of MMs and the solvent. It is verified that this free energy can describe the MMC hydrogel. Therefore, we will focus on this topic in the following.

Acknowledgements I am grateful to Prof. Huiliang WANG of the College of Chemical Sciences, Beijing Normal University for giving me this problem to study and for helpful discussion. Specially, I thank Prof. Huiling WANG for presenting all experimental figures in this paper, some of which have not been published. I also thank Xiaomei YAO who helps me to carry out some numerical computations and draw figures.