Hydrogels consisting of permanent chemical bonds plus transient physical bonds are known to have self-healing properties, high toughness, and to be highly viscoelastic. These mechanical responses derive from the ability of the physical bonds to break and reform during deformation. The rate of bond breaking and reformation is found experimentally to be independent of stress but is expected to depend on temperature. Using a poly(vinyl alcohol) hydrogel with chemical and physical cross links, the temperature dependence of the mechanical response is measured using large strain load/unload tension tests and torsional rheometry for temperatures from 13 to 50 °C. The rheometry data show that time-temperature superposition can be used to condense the data to a master curve. It is found that allowing the model parameters to be temperature dependent, a previously developed constitutive model fits the tension and rheometry data well. Horizontal (time) and vertical (amplitude) shift factors calculated directly from the rheometer test data and from tension test data agree well with each other showing that the constitutive model describes the mechanical behavior well over the temperature range explored and that the material closely follows simple time-temperature superposition.

A hydrogel is essentially a network of crosslinked polymer chains swollen in water. An issue with conventional covalently crosslinked hydrogels is that they are too brittle to use in load bearing structures. The area of hydrogel mechanics has advanced rapidly since several new synthetic routes of mechanical reinforcement of hydrogels have been developed [1–4]. Gong et al. introduced the concept of sacrificial bonds: They demonstrated that it is possible to make stiff and tough hydrogels with a double network structure by breaking internal sacrificial bonds [1,5]. In these gels having interpenetrated networks, the first network is brittle as it is highly cross-linked and swollen (thus dilute and stretched) and serves as a sacrificial network. The second network is loosely cross-linked and highly extensible. During mechanical loading, the first network bears most of the load and undergoes progressive irreversible damage, while the second network prevents the formation and growth of macroscopic cracks. Because the first network is chemically cross-linked and is irreversibly broken, this gel does not fully recover its original state after damage [5]. Nevertheless, such gels have a good resistance to crack propagation in fatigue, as shown by the recent works of Suo and coworkers [6].

Recently, several research groups have used noncovalent transient interactions as sacrificial bonds to synthesize highly stretchable and tough hydrogels [4]. These physical interactions are reversible and can break and reform. These gels exhibit time dependent viscoelastic behavior and recover to their original state upon load removal. Sun et al. [4] reported tough physically crosslinked polyampholyte hydrogels having ionic bonds. It was claimed that the polymer network is formed by self-healing weak bonds as well as strong bonds which maintain the network integrity over a sufficiently long time scale [7]. For the interested readers, two reviews on the relative merits of these different hydrogels have recently been published [8,9].

Varying the temperature changes the breaking and reattaching rates of the physical bonds. Thus, for hydrogels with transient physical bonds, temperature will have significant effects on the mechanical response. For many polymers, the relation between strain rates and temperature is found to obey the time-temperature superposition principle (TTS). Since time dependent gels have only recently been introduced, there are few studies on the effects of temperature and loading rate on the mechanical behavior of these gels. Recently, Sun et al. [10] showed that it was possible to use vertical and horizontal shift factors to collapse small strain rheology data onto a master curve for polyampholyte hydrogels. Further, they measured the hysteresis from uniaxial cyclic tests carried out at moderately large strains. They found that the hysteresis from tests with different stretch rates and temperatures can be collapsed onto a master curve using the same shift factors obtained from the rheology.

The goal of the current work is to build on prior models to develop a complete, three-dimensional (3D), constitutive model that incorporates the effect of temperature on the rate of breaking/healing of physical bonds as well as on other physical parameters. With this model, one will be able to compute the mechanical response over a range of stretch rates and temperatures and to directly compare data from constant strain rate, large stretch tension tests to cyclic, torsional rheology, and small strain tests.

In this paper, we study the effects of temperature and loading rate on the mechanical response of poly(vinyl alcohol) (PVA) dual cross-link hydrogel with PVA chemically crosslinked by glutaraldehyde and physically crosslinked by borate ions [11]. Although this gel has a lower resistance to fracture in comparison to the polyampholyte gels developed by Gong et al. [1], it has a well-defined simple chemical structure with only one type of physical crosslink between PVA chains, thus providing an ideal platform for understanding the interplay between microscopic bond kinetics and macroscopic mechanics. In addition, we have previously developed a 3D constitutive model which combines the finite strain elasticity of elastomers with the kinetics of bond breaking and reattachment [12–14] and demonstrated that our model can accurately capture the behavior of both uniaxial tension and shear (cyclic, small strain torsion) tests with complex loading histories at room temperature. This constitutive model developed here enables us to compare data obtained from rheometer tests at different temperatures directly with data obtained from uniaxial, constant stretch rate loading/unloading tests performed at different loading rates and temperatures.

The outline of this paper is as follows: In Sec. II, we present the experimental procedure and the test setup for temperature control. The experimental results from uniaxial tension tests at different temperatures are presented in Sec. III along with the comparison to the predictions of our constitutive model. Results of rheology tests at different temperatures are presented in Sec. IV. These results show that the PVA gel closely obeys the TTS with a horizontal (time) and vertical (amplitude) shift factors. This section also gives a detailed discussion on the relation between parameters in our constitutive model and the shift factor. Section V summarizes the findings.

The dual-crosslink PVA hydrogels were prepared by incorporating borate ions in a chemically cross-linked PVA gel. Details of synthesis are provided in previous works [11,15]. Here, we briefly summarize the procedure. We first made the chemically cross-linked gel by mixing a glutaraldehyde solution into a PVA solution at pH = 1.4, adjusted by hydrochloric acid (HCl). The PVA concentration in the solution was 12%, and the molar ratio of chemical cross-linker and PVA monomers was 1:500. The solution was then injected into a mold. After 24 h, the chemically cross-linked gel was removed from the mold and washed with plenty of water to neutralize the pH. Then, the chemical gel was soaked in a NaCl/Borax solution (Borax, 1 mM/l; NaCl 90 mM/l) for 3 days. Infusion of the ionic solution into the gel causes physical bonds to form. Once the NaCl/Borax soaking was completed, the samples were stored in the same solution for up to 2 weeks prior to testing. Two batches of material were used, one for the large strain tension tests and the other for the small strain rheology tests. Despite minor batch to batch variation, model parameters obtained from large strain tests of the first batch result in a model that fits the rheology data from the second batch well indicating that we are synthesizing consistent materials and demonstrating the robustness of the model.

We performed uniaxial tensions tests on the hydrogel specimens at different loading rates and temperatures. The tests were performed using a custom built tensile tester. Translational motion was provided by a Zaber X-LSM200A-E03 translational stage. The load was measured by an Interface SMT1-1.1 load cell (4.9 N capacity) and displacement by an OMEGA LD620 linear variable displacement transducer. The load and displacement signals were recorded using a Keithley Model 2701 multiplexing digital voltmeter at rates of 25 pairs of data per second. During the tests, the hydrogel samples were immersed in mineral oil to prevent drying. The oil exerts a buoyant force on the grips. This force was experimentally calibrated and subtracted from the measured load to avoid systematic error.

The temperature of the test environment was maintained by a custom built system which is schematically shown in Fig. 1. The system is insulated by gluing insulation foam to the outer surfaces of the acrylic oil container. Prior to testing, we preheated the oil and poured the preheated oil into the tank. Two low power immersion heaters (total power of 15 W) were placed inside the tank to maintain the target temperature. The temperature of the oil near the sample was measured using a digital thermometer (Dallas Semiconductor DS18B20) with ±0.5 °C accuracy. Temperature data were sent to an Arduino board which switched the heater on/off using a simple controller. The control scheme is straightforward: For a target temperature T, we programmed the Arduino controller so that if the temperature was below T +0.1 °C, the heater would turn on. If the temperature was above T +0.3 °C, the heater would turn off. Using this method, we were able to hold the temperature within to ±0.5 °C of the target. We also incorporated an oil circulation system which drew the oil from the top of the tank and injected it to the bottom of the tank at a rate of 100 ml/min. This makes the temperature more uniform in the vertical direction. Without the circulation system, we observed up to a 3 °C difference in the temperature between the top and the bottom of the specimen. With oil circulation, we have verified that the temperature difference between the top and the bottom of the sample is less than 1 °C. For tests below room temperature, we first cool the oil to the test temperature in a refrigerator and then fill the tank with the cooled oil. As the test progresses, additional cool oil at the test temperature is added. We have verified that by using this method, the temperature change during a 30-min test does not exceed 2 °C. It should be noted that for most tests, the testing time was much shorter.

FIG. 1.

Schematic view of the experimental setup.

FIG. 1.

Schematic view of the experimental setup.

Close modal

We performed uniaxial tension tests on hydrogel specimens at five different temperatures, 13, 22.5 (defined as room temperature), 30, 40, and 50 °C. The specimens were 12 mm in width, 2 mm in thickness, and the initial distance between the grips was 24 mm. At each temperature, we loaded the specimens to a stretch ratio λ of 1.3 at five different stretch rates: 0.001/s, 0.003/s, 0.01/s, 0.03/s, and 0.1/s (except for 13 °C where we tested at four rates excluding 0.003/s). Note that our all tension tests are conducted at constant stretch rates λ̇ (loading/unloading); thus, since the true strain rate is λ̇/λ, it decreases with the stretch during loading. The unloading rate for all tests was 0.01/s. We also performed stress relaxation tests on these samples at each temperature. In these relaxation tests, we stretched the sample at a fast rate (0.5/s) to a stretch ratio of 1.3 and held the stretch constant for 30 min. We repeated the tests twice at each temperature to check the consistency of the results.

We performed small strain rheology tests using a TA Instruments DHR3 torsional rheometer with parallel plates. Hydrogel sheets with a thickness of 2.1 mm were cut into 20 mm diameter circular specimens using a stainless-steel punch. To eliminate the slip between the specimen and the parallel plates, we applied precompression by setting the gap between the parallel plates to 1.6 mm. We observed that the normal force is constant to within 5% during a cycle and approximately constant over all frequencies (see supplementary material) [16]. No anomalies were seen in the raw or processed data that would suggest slipping. The samples were cyclically loaded to 0.3% strain at frequencies from 0.01 to 10 Hz. To test the limit of linearity, we performed amplitude sweeps and found that the storage and loss modulus data are constant for strains less than 0.3% (see supplementary material). Three data points were taken per decade of frequency. As in the cyclic tension tests, these rheology tests were performed at 13, 22.5, 30, 40, and 50 °C. In all tests, the maximum phase angle was less than 40  (see supplementary material). To reduce drying of the gel at elevated temperatures, a solvent trap was used to achieve a high-humidity environment. The trap holds the sample in a small, closed space at close to 100% relative humidity during the tests. We weighed the sample before and after the tests and found that even for the 50 °C tests, the maximum weight loss is less than 5%. The tests were repeated three times to check consistency.

1. Test results

The nominal stress, σ , versus stretch ratio, λ, curves at different loading rates and temperatures are shown in Figs. 2(a)–2(e). In these figures, the maximum stretch ratio is 1.3 which is considerably higher than typical strains imposed in linear rheology. We have carried test with larger stretch ratios up to 5.5, and these results are reported in earlier works [13]. Even with larger stretch ratios that clearly fall outside the linear regime, our model with strain hardening fits the cyclic test data well with the same set of parameters [13]. During the cycle tests shown here, for all the temperatures and loading rates tested, we found a large hysteresis with a small residual deformation right after unloading the specimen to zero stress. At a fixed temperature, the stiffness of the gel increases with the loading rate. This is because the unbroken physical bonds can accumulate more stretch at faster loading rates (see also Sec. IV). Figure 2(f) shows the load versus stretch response at a stretch rate of 0.01/s at different temperatures. The gel is stiffer at a lower temperature and softer at a higher temperature. This behavior can be explained by the kinetics of the physical bonds and will be discussed in Sec. IV.

FIG. 2.

Uniaxial tension tests data: Nominal stress versus stretch at different stretch rates and temperatures: (a) 13 °C, (b) 22.5 °C, (c) 30 °C, (d) 40 °C, (e) 50 °C, and (f) effects of temperature on the stress-stretch curve at a fixed stretch rate of 0.01/s and varying temperature.

FIG. 2.

Uniaxial tension tests data: Nominal stress versus stretch at different stretch rates and temperatures: (a) 13 °C, (b) 22.5 °C, (c) 30 °C, (d) 40 °C, (e) 50 °C, and (f) effects of temperature on the stress-stretch curve at a fixed stretch rate of 0.01/s and varying temperature.

Close modal

2. Review of the constitutive model

Our test results can be explained quantitatively by a constitutive model developed in our earlier work [12,13]. Here, we briefly review this constitutive model. Note that the nomenclature here differs slightly from [12] and [13]. Macroscopically, the gel is assumed to be isotropic and incompressible. The model assumes two independent types of bonds: The chemical crosslinks form a permanent elastic network. Physical crosslinks form transient bonds within this network that can break and reattach with rates independent of the stress or strain acting on it. We assume that chemical cross-links do not break. When a strand in the transient network breaks, it instantaneously releases the strain energy it carries. When a broken physical strand reattaches at time τ, it has zero strain energy at that time, even though the network is strained. The total strain energy is equal to the sum of the strain energy carried by the strands in both networks. We also assume that the breaking and reattaching of the physical strands reaches a dynamic equilibrium soon after the gel is synthesized. This steady state assumption implies that the breaking and reattachment rates are equal and independent of time. This rate is denoted by χ̇.

As in our previous works, we assume that the strain energy densities of the undamaged networks for both permanent and transient bonds are the same. Since the maximum stretch ratio used in all the experiments reported here is less than or equal to 1.3, strain hardening effects can be neglected, and we use the neo-Hookean strain energy function. With these assumptions, the nominal stress in uniaxial tension is

σ=Gρ[λ(t)1λ2(t)]+Gn(t)[λ(t)1λ2(t)]+Gχ̇0tϕB(tτtB)[λ(t)λ2(τ)λ(τ)λ2(t)]dτ,
(1a)

where G is the small strain shear modulus, ρ is the molar fraction of the chemical crosslinks, and n(t) is the fraction of physical crosslinks that are attached at t=0 and survive until current time t>0 and is given by

n(t)χ̇0ϕB(tτtB)dτ=χ̇tB2αB[1+(αB1)ttB]2αB1αB.
(1b)

The “survivability” function ϕB

ϕB(tτtB)=[1+(αB1)(tτ)/tB]1/(1αB)
(1c)

denotes the fraction of physical crosslinks that are formed at time τ and remain attached at time tτ, where tB is the characteristic time for breaking and 1<αB<2 is a material parameter. The dimensionless material parameter αΒ controls the average survival time of a newly healed transient bond.

We briefly summarize the physical meaning of Eqs. (1a)–(1c). The first term on the right hand side of Eq. (1a) is the stress carried by the chemical network, the second term represents the loss of stress due to the breaking of the initially attached physical strands, while the integral term describes the recovery of stress from the reattachment of physical strands. For example, in a test with continuous loading, the effect of reattachment is to recreate temporary elastic strands attached at both ends but with a different reference configuration than the original chains, which gives rise to the term λ(t)λ2(τ)λ(τ)λ2(t) in Eq. (1a). This term is consistent with our assumption that the energy of the newly reattached strand carries zero strain energy. Finally, in Eq. (1c), we used a power law function with power exponent αB to describe the breaking kinetics. This is a phenomenological model motivated by the expectation that there exists a spectrum of relaxation time scales associated with the breaking of physical strands due to the statistical nature in strand life time or the polymer network structure adjacent to a physical strand; empirically, such processes can be well described by power law functions.

Equations 1(a)–1(c) imply that our constitutive model is completely determined by four material parameters: Gρ, Gχ̇, αB, and tB. These four parameters are determined by a uniaxial relaxation test where we first load a sample at a constant stretch rate to a fixed stretch ratio, then hold it at this stretch until the stress reaches a plateau. The plateau stress, the log –log plot of the stress vs time at large time (use both the slope and the intercept from the linear fit), and the initial slope of the loading part provide us four conditions for the four material parameters, so that the material parameters can be estimated. Then, the parameters can be iterated to give the best fit to loading/unloading tests carried out at constant stretch rates. Further details of the model identification procedure are described in our previous work [13]. In [13], we have shown that χ̇ depends on tB as well as the characteristic healing time of physical bond tH by

χ̇=1ρtH+tB2αB;

hence χ̇ is expected to be temperature dependent.

3. Determination of material parameters and comparison to experiments

Previously, we have shown that our constitutive model is able to accurately predict uniaxial tension test data at room temperature [12,13]. Here, we demonstrate that it can also accurately predict the uniaxial tension tests at different temperatures. For each temperature, we first use the stress relaxation test data to estimate the four material parameters (details of method of fitting were given in our previous paper [13]). These parameters are then fine-tuned to match both the cyclic tension test data and the rheology data (see Sec. III B) and are summarized in Table I. It can be seen that temperature has different effects on these four parameters. The parameters Gρ and αB are insensitive to temperature, while Gχ̇ increases and tB decreases with increasing temperature.

TABLE I.

Model fitting parameters.

Temperature (°C)Gρ (kPa)Gχ̇ (kPa/s)αBtB (s)Gχ̇tB2αB (kPa)
13 3.34 25.21 1.62 0.49 32.62 
22.5 3.14 27.85 1.62 0.42 30.94 
30 3.33 34.30 1.62 0.32 28.85 
40 3.11 46.12 1.61 0.21 25.15 
50 3.16 53.60 1.61 0.16 22.55 
Temperature (°C)Gρ (kPa)Gχ̇ (kPa/s)αBtB (s)Gχ̇tB2αB (kPa)
13 3.34 25.21 1.62 0.49 32.62 
22.5 3.14 27.85 1.62 0.42 30.94 
30 3.33 34.30 1.62 0.32 28.85 
40 3.11 46.12 1.61 0.21 25.15 
50 3.16 53.60 1.61 0.16 22.55 

Next, we use these parameters to predict the stress-stretch curve for the uniaxial constant stretch rate tension tests (see Sec. III A 1) and compare these predictions with test data. Figures 3(a)–3(c) compare model predictions with experiments for three temperatures and three loading rates, and Fig. 3(d) shows the comparison to relaxation test data for the three temperatures. There is excellent agreement between model predictions and experimental data. The results for the other temperatures and stretch rates are just as good and are given in the supplementary material.

FIG. 3.

Comparison of model and experimental tests at three different stretch rates and temperatures of (a) 13 °C, (b) 30 °C, and (c) 50 °C, and (d) is the comparison of model and experimental tests for the above three tests for relaxation test, where the sample is first loaded at a constant stretch rate to a stretch ratio at 1.3 then held at this stretch. “E” stands for experimental data and “M” stands for model prediction using parameters from Table I. In the experiments, samples are unloaded to zero stretch, therefore, some samples go into compression and buckle. The model calculation does not account for buckling.

FIG. 3.

Comparison of model and experimental tests at three different stretch rates and temperatures of (a) 13 °C, (b) 30 °C, and (c) 50 °C, and (d) is the comparison of model and experimental tests for the above three tests for relaxation test, where the sample is first loaded at a constant stretch rate to a stretch ratio at 1.3 then held at this stretch. “E” stands for experimental data and “M” stands for model prediction using parameters from Table I. In the experiments, samples are unloaded to zero stretch, therefore, some samples go into compression and buckle. The model calculation does not account for buckling.

Close modal

1. Experimental data and shifting

The storage modulus G′, loss modulus G″, and tanδ=G/G obtained from rheology tests at linear-viscoelastic region are shown in Figs. 4(a)–4(c). First, we investigate whether these data can be collapsed into a single master curve using TTS [17] defined by bTGT,ωaT=GTRef,ω, where ω is the frequency, T is temperature, and aT and bT are the horizontal and vertical shift factors. The same shifting is applied to G″. We use room temperature TRef = 22.5 °C as the reference temperature. In conventional polymer melts where the dynamics is due to monomer friction, tan δ data at different temperatures depend only the horizontal shift factor aT, because both G′ and G″ are affected by the same vertical shift factor. Thus, we first use the tan δ data to obtain the horizontal shift factors at each temperature. The shifted tan δ data are plotted in Fig. 5(a). We apply these horizontals shifts to G′ followed by a vertical shift to collapse these data onto a master curve. The same vertical shift is then applied to G″. The results are shown in Figs. 5(b) and 5(c). The shift factors are tabulated in Table II. Comparing Figs. 5(a)–5(c) with the preshifted data in Figs. 4(a)–4(c), we conclude that the TTS works well for our gel for all the frequencies in the tests. The horizontal shift factors have a wide range, from 1.20 to 0.41 over the 13–50 °C temperature range, while the vertical shift has a smaller range, 0.96–1.16.

FIG. 4.

Rheology test results: (a) tan δ, (b) G′, and (c) G″.

FIG. 4.

Rheology test results: (a) tan δ, (b) G′, and (c) G″.

Close modal
FIG. 5.

Master curves of: (a) tan δ, (b) G′, and (c) G″.

FIG. 5.

Master curves of: (a) tan δ, (b) G′, and (c) G″.

Close modal
TABLE II.

Shift parameters from model fitting and rheology results.

Temperature (°C)tBaT from tension testsaT from rheologybT from tension testsbT from rheology
13 0.49 1.17 1.20 0.95 0.96 
22.5 (ref.) 0.42 1.00 1.00 1.00 1.00 
30 0.32 0.77 0.79 1.07 1.05 
40 0.22 0.52 0.53 1.23 1.12 
50 0.16 0.39 0.41 1.37 1.16 
Temperature (°C)tBaT from tension testsaT from rheologybT from tension testsbT from rheology
13 0.49 1.17 1.20 0.95 0.96 
22.5 (ref.) 0.42 1.00 1.00 1.00 1.00 
30 0.32 0.77 0.79 1.07 1.05 
40 0.22 0.52 0.53 1.23 1.12 
50 0.16 0.39 0.41 1.37 1.16 

2. G′ and G″ from constitutive model

In our previous work [14], we derived G,G based on our constitutive model

G(ω)=Gρ+GρK(T)I1(ωtB,αB),
(2a)
G(ω)=GρK(T)I2(ωtB,αB),
(2b)

where

I1(ωtB,αB)12αBαB10cos(ωtBαB1x)(1+x)1αB1dx,
(2c)
I2(ωtB,αB)2αBαB10sin(ωtBαB1x)(1+x)1αB1dx,
(2d)
K(T)=χ̇tB(2αB)ρ=(1ρ)/ρ1+(2αB)(tH/tB),
(2e)

where ω is the frequency in rad/s. Using the parameters listed in Table I, we can also calculate the storage modulus G′, loss modulus G″, and tanδ=G/G in the rheology tests. Comparison between the model and the experiment at room temperature is shown in Fig. 6. The model agrees well with the experimental data. Results for other temperatures fit equally well and are shown in the supplementary material.

FIG. 6.

G′, G″, and tan δ at room temperature. (a) G′ and G″ evaluated using Eqs. (2a) and (2b) and parameters from Table I with rheology data (model results are shifted by a vertical constant to account for precompression, this procedure was also used in our previous work). (b) tan δ from model and rheology data.

FIG. 6.

G′, G″, and tan δ at room temperature. (a) G′ and G″ evaluated using Eqs. (2a) and (2b) and parameters from Table I with rheology data (model results are shifted by a vertical constant to account for precompression, this procedure was also used in our previous work). (b) tan δ from model and rheology data.

Close modal

In Sec. III, we showed that the uniaxial cyclic test data at different temperatures and loading rates are captured well by our constitutive model. There are four independent material parameters in our model, and they are given in Table I for different temperatures. In a stress relaxation test, the modulus at long time is Gρ. Here, ρ is the molar fraction of chemical crosslinks, which we do not expect to have a strong temperature dependence. The small strain shear modulus G is, according to rubber elasticity, proportional to the absolute temperature, so the fractional change in Gρ should be very small for the range of temperature used in our experiments. This is consistent with the results of our relaxation test which shows that the long time moduli are quite insensitive to temperature (see supplementary material).

The other two material parameters, Gχ̇ and tB, describe the dynamics of chain breaking and reattaching. As the temperature increases, the dynamics accelerate, resulting in a shorter average chain life; thus, one would expect the characteristic time for breaking, tB, to decrease with temperature. Finally, Table I shows that the exponent αB of the survivability function ϕB is insensitive to temperature.

To make further contact with our model, consider the storage modulus G′ and loss modulus G″ given in Eqs. (2a)–(2d). In order for TTS to work perfectly, G′ and G′ should have the separable form as given below

G=b(T)ϕS(ωa(T)),
(3a)
G=b(T)ϕL(ωa(T)),
(3b)
tanδ=ϕL(ωa(T))/ϕS(ωa(T)),
(3c)

where b(T) and a(T) are functions that depend only on temperature T, ϕS and ϕL are functions that depends only on the combination ωa(T). In Eqs. (3a) and (3b), b(T) and a(T) are responsible for vertical and horizontal shifts, respectively. Since ω appears only in the factor ωtB in I1(ωtB,αB) and I2(ωtB,αB) [see Eqs. (2a)–(2d)], tB can be identified with a(T) and the horizontal shift as long as αB is independent of temperature. Our results in Table I support this feature of our model. Next, consider the tanδ=G/G obtained by dividing Eq. (2b) by Eq. (2a), it is

tanδ=K(T)I2(ωtB)1+K(T)I1(ωtB),
(4)

where we have excluded αB from the argument of I1 and I2 since it is found to be independent of temperature. Equation (4) implies that tanδ is independent of the vertical shift if and only if K(T)=K0 is independent of temperature. Indeed, if this is the case, then the storage and loss moduli have the same form as Eqs. (3a) and (3b), i.e.,

G=GρK0b(T)[1K0+I1(ωtB)]ϕS(ωtB),G=GρK0b(T)[I2(ωtB)]ϕL(ωtB).
(5)

From Eq. (5), we see that GρK0= b(T) and is given by

GρK0=Gχ̇tB2αB=b(T).
(6)

In the following, we identify the vertical and horizontal shift factors as bT(T)b(T)/b(T0) and aT(T)a(T)/a(T0), respectively.

In the constitutive model, tB, the characteristic breaking time in the survivability function decreases with temperature as a result of accelerated breaking kinetics. As Eqs. 2(a) and 2(b) show, the frequency and temperature dependence are given by the product ωtB. Since the frequency is the inverse of time, aT should be given by tB/tBRef, where tBRef is the breaking time at a reference temperature (room temperature for our case). Using the breaking times, we obtained from the tension tests, we can compute the shift factor tB/tBRef and compare these with the shift factors obtained from rheology test. We also compare the vertical shift factor computed using Eq. (6) from rheology and tension data. These comparisons are shown in Table II and Fig. 7. It can be seen that the horizontal factors computed from both sources are highly consistent. The vertical shift factors agree less well at the higher temperatures (40 and 50 °C). This is because K(T) is not a constant independent of temperature. From Eq. (2e), K(T) depends on ρ and on the ratio tH/tB. This ratio need not be independent of temperature since the characteristic time of healing tH depends on the availability of binding sites, which is not the case for breaking. In addition, since the number of “free” physical crosslinkers which are not bound to the network increases with temperature, the molar fraction of the chemical crosslinks ρ could increase with temperature. For this reason, we cannot expect K(T) to be independent of temperature, and therefore, TTS is not completely satisfied by our model. Indeed, the last column of Table I shows that K(T) decreases very slowly for the first three temperatures but decreases somewhat faster at higher temperatures.

FIG. 7.

(a) Horizontal shift factors: log10aT from the rheology tests, calculated from the tension tests and from Arrhenius fitting of the shift factors, using Ea = 2.56 × 104 J/mol. (b) Vertical shift factors: log10bT from the rheology tests, calculated from the tension tests and from Arrhenius fitting of the shift factors, using Eb = 6.63 × 103 J/mol.

FIG. 7.

(a) Horizontal shift factors: log10aT from the rheology tests, calculated from the tension tests and from Arrhenius fitting of the shift factors, using Ea = 2.56 × 104 J/mol. (b) Vertical shift factors: log10bT from the rheology tests, calculated from the tension tests and from Arrhenius fitting of the shift factors, using Eb = 6.63 × 103 J/mol.

Close modal

Finally, our data suggest that log aT and log(1/(bTT)) are approximately inversely linear in temperature, i.e.,

log(aT)=Ea2.303R(1T1T0),
(7a)
log(1TbT)=Eb2.303R[1T1T0],
(7b)

where R =8.314 J/mol K is the gas constant, Ea and Eb are the activation energies associated with bond breaking and the enthalpy of association, respectively. Figures 7(a) and 7(b) compare Eqs. (7a) and (7b) with the shift factors we obtained from rheology test and tension test data, using Ea = 25.6 kJ/mol and Eb = 6.63 kJ/mol. Within the range of temperatures considered here, the shift factors fit the Arrhenius law, Eqs. (7a) and (7b), well with some deviation at low temperature. The value of Ea is reasonable for a PVA-borax system; for solutions (without chemical crosslinks), a value of 42 kJ/mol was previously reported [18,19]. The value of Eb is much smaller than Ea which is to be expected since the vertical shift is much less. This is the reason why TTS works in this frequency range. We expect TTS to break down at very low frequencies when the gel is elastic and dominated by chemical cross-linking.

The constitutive model was developed specifically for PVA gels with a fixed composition of chemical and physical cross-links. Recently, we have varied these compositions and found that our model works just as well. Hence, we expect our model should work well for other gels with chemical and physical cross-links. However, the key assumption that the breaking and healing kinetics of physical bonds are independent of the stress acting on these bonds will limit its applicability. Preliminary work in our group shows that by modifying our model such that the survivability function in Eq. (1c) depends on stress we are able to predict the mechanical behavior of the tough, polyampholyte gels of Gong et al. [1].

The model is not expected to apply to purely physical gels (associative polymer solutions) in a long time range and/or at a high strain, because the model does not take into account of the terminal dynamics of the polymer chains. For purely chemical gels, the model reverts to a simple Neo-Hookean material, which works reasonably well for the corresponding chemical gel.

Cyclic tension and rheology tests are carried out on a PVA dual cross-link hydrogel at different temperatures. The tension tests are performed under moderately large strains, while the rheology tests are performed at small strains. The storage and loss moduli from the rheology tests are found to obey TTS. Model parameters from large strain tests of one batch of material are found to accurately predict the small strain rheology results from a second match of the material.

Of considerable significance is that the tension cyclic test and rheology test data at all test temperatures can be explained by a nonlinear viscoelastic constitutive model with four independent material parameters {Gρ,Gχ̇,αB,tB}. These parameters are determined independently by performing relaxation tests at different temperatures and after a slight amount of fine-tuning are found to match both tension and rheology data at different temperatures. We find that αB and Gρ are approximately independent of temperature, while Gχ̇ and tB are strongly temperature dependent. It should be noted that TTS imposes very stringent requirements on our constitutive model. We find that the constitutive model approximately satisfies the stringent requirements needed for simple TTS to be valid.

In summary, our constitutive model is capable of quantifying complex loading histories in uniaxial tension test and at the same time correctly captures linear rheology of a PVA dual-crosslinked hydrogel at different temperatures. The general framework of the constitutive model is applicable to other gels with chemical and physical cross links. When no physical crosslinks are present the model reverts to a simple Neo-Hookean material, which works reasonably well for the corresponding chemical gel. The model is not expected to capture the behavior of a purely physical gel.

J.G., M.L., C.Y.H., and A.Z. are supported by the National Science Foundation under Grant No. CMMI-1537087. This work made use of the Cornell Center for Materials Research Shared Facilities, which are supported through the NSF MRSEC Program (No. DMR-1719875). The authors thank an anonymous reviewer for asking us to check the limit of linear rheology.

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Supplementary Material