Introduction

Polymer gels are three-dimensional polymer networks crosslinked by physical or chemical interactions and swollen in solvent. One popular method for the formation of polymer gels is the radical polymerization of monomers and crosslinkers.1, 2 Because monomers and crosslinkers react at random, the resultant network has a substantial number of heterogeneities,3, 4 including spatial (heterogeneous distribution of crosslinks), connectivity (dangling chains and elastically ineffective loops) and topological (trapped entanglements) heterogeneities.5, 6 Such heterogeneities are inevitable in conventional polymer gels and prohibit the precise control and prediction of the physical properties of the gels; this lack of control and predictability severely limits the scope of the applications of hydrogels. Thus, the establishment of principles for the construction of ideally homogeneous polymer networks is a significant challenge not only for understanding molecular models for the physical properties of the networks, but also for achieving flexibility in the design of polymer gels.

A popular approach toward the achievement of the ideal network is the ‘end-crosslinking’ method, which forms a polymer network from AB-type polycondensation of telechelic prepolymers and multifunctional crosslinkers (Figure 1a).7 However, the resultant network contains a substantial amount of heterogeneity.8, 9 Thus, ‘end-crosslinking’ is not an appropriate method for the construction of ideal polymer networks.

Figure 1
figure 1

Polymer network structure formed from AB-type polycondensation of prepolymers with different architectures: (a) end-linking reaction, (b) size-mismatched crosslink-coupling and (c) conventional crosslink-coupling. A full color version of this figure is available at Polymer Journal online.

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Recently, we developed Tetra-PEG gels through our new network-formation method, ‘crosslink-coupling;’ the network is formed by the combination of two mutually reactive tetra-arm prepolymers with the same shapes (Figure 1c).10, 11, 12 Our previous study revealed that Tetra-PEG gels form an extremely homogeneous polymer network with a small number of structural defects.13, 14, 15 Although the connectivity and spatial heterogeneities were observed, the degree of heterogeneity is much smaller than that for conventional gels.16, 17, 18, 19

What is the key to suppressing the heterogeneity? As shown in Figure 1a and c, both the end-crosslinking and crosslink-coupling methods, respectively, can lead to an ideal network structure if the reaction proceeds ideally. However, end-crosslinking yields a heterogeneous network, whereas crosslink-coupling yields a homogeneous network. These experimental findings indicate that the prepolymer architecture is vital in suppressing the heterogeneity. The differences in the prepolymer architecture for the end-linking and crosslink-coupling are the ‘size’ and ‘functionality’ of the prepolymers; end-linking uses prepolymers with ‘different sizes’ and ‘different functionalities,’ whereas the crosslink-coupling uses prepolymers with the ‘same sizes’ and ‘same functionalities’.

In this study, we try to investigate the effect of ‘size.’ We form polymer networks from the AB-type crosslink-coupling of prepolymers with ‘different sizes’ but the ‘same functionalities;’ we call these networks size-mismatched Tetra-PEG gels (Figure 1b). We investigate the reaction efficiency (p) of these networks using infrared measurements, the elastic modulus (G) and the ultimate elongation ratio (λmax) using stretching measurements and the fracture energy (T0) using tearing measurements. By comparing these mechanical properties with those of conventional Tetra-PEG gels and with the predictions from models, we investigate the effect of the prepolymer architecture on the network structure formed by AB-type crosslink-coupling.

Experimental Procedure

Characterization of Tetra-PEG modules

Tetra-amine-terminated PEG (Tetra-PEG-NH2) and tetra-NHS-terminated PEG (Tetra-PEG-OSu) were purchased from NICHIYU (Tokyo, Japan). Here, NHS is N-hydroxysuccinimide. The details of the preparation of Tetra-PEG-NH2 and Tetra-PEG-OSu were reported previously.10 The molecular weight, the polydispersity, the end-group functionality (X0), and the overlapping polymer volume fraction (φ*) of the Tetra-PEG gels are shown in Supplementray Table 1 in the Supplementray Information.

Fabrication of Tetra-PEG gels

We combined three different molecular weights (Mw) of Tetra-PEG-NH2 (Mw=5 k, 10 k and 20 k) and Tetra-PEG-OSu (Mw=5 k, 10 k and 20 k) to fabricate Tetra-PEG gels composed of size-mismatched Tetra-PEG-NH2 and Tetra-PEG-OSu (Table 1). Equimolar amounts of Tetra-PEG-NH2 and Tetra-PEG-OSu were dissolved in phosphate buffer (pH 7.4) and phosphate-citric acid buffer (pH 5.8), respectively. The corresponding initial polymer volume fractions, φ0, were between 0.050 and 0.096 (mass density=1.129 g cm−3). The ionic strengths of the buffers were chosen to control the reaction rate (Table 1).20, 21 The two solutions were mixed, and the resulting solution was poured into the mold. We waited for at least 12 h for the completion of the reaction before the subsequent experiment was performed.

Table 1 The combinations of prepolymers, the overlapping polymer volume fraction ( φ *) and the ionic strength of the buffers for each initial volume fraction ( φ0) in the Tetra-PEG gels
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Stretching measurements

The stretching measurements were performed on a dumbbell shape (2 mm thick) at room temperature with a mechanical testing apparatus (Autograph AG-X plus; SHIMADZU, Kyoto, Japan) at a crosshead speed of 60 mm min−1, which corresponds to a strain rate at the gauge section of ∼2 per min. During the stretching, the distance between the gauge points was measured with a CCD camera (XCL-5005CR; SONY, Tokyo, Japan). Because the gel samples were used in the as-prepared state, the stretching was performed in air. The stress–strain relations obtained here corresponded to the equilibrium relations without time effects because no appreciable relaxation was observed for the stress after the imposition of a constant large strain at this crosshead speed. We stretched a sample repeatedly and confirmed that the evaporation of water did not affect the experimental results. At least four samples were tested for each network concentration, and the observed moduli were arithmetically averaged.

Infrared measurements

After the stretching measurements, the gel samples were soaked in H2O for 2 days at room temperature and then were dried. The dry samples were cut into thin films (thickness: 40 μm) with a Microtome (SM2000R, Leica, Wetzlar, Germany). These samples were swollen with D2O until equilibrium was reached. The infrared spectra of these samples were obtained at 25 °C with a JASCO FT-IR-6300 equipped with a deuterated triglycine sulfate (DTGS) detector, in which 128 scans were summed at a resolution of 4 cm−1 for each sample. At least two samples were tested for each network concentration.

Tearing measurements

The tearing measurements were performed at room temperature with a mechanical testing apparatus (Autograph AG-X plus; SHIMADZU, Kyoto, Japan). Because the gel samples were used in the as-prepared state, the tearing was performed in air. The gels were cut into the shape specified by JIS K 6252 at 1/2 size (50 mm × 7.5 mm × 1 mm, with an initial notch of 20 mm). The two arms of the test samples were clamped, and one arm was pulled downward at a constant velocity of 40 mm min−1, whereas the other arm remained stationary. The tearing force F was recorded.

Results and Discussion

Reaction efficiency

Conventional Tetra-PEG gels are fabricated from Tetra-PEG-NH2 and Tetra-PEG-OSu with equal molecular weights.10 To investigate the effect of the size mismatch of the prepolymers, we fabricated 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels, where 5 –10 k indicates that equimolar Tetra-PEG-NH2 (5 kg mol−1) and Tetra-PEG-OSu (10 kg mol−1) were mixed (Table 1).

First, we measured the p of the 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels using infrared measurements (Figure 2). p was almost constant as a function of φ0 and was 0.88–0.93, which is as high as that of conventional Tetra-PEG gels (0.82–0.95). These results suggest that the p only depends on the reaction group and not on the polymer size.

Figure 2
figure 2

The reaction efficiency (p) as a function of the initial polymer fraction (φ0) in the Tetra-PEG gels (5 k–10 k, circles; 5 k–20 k, squares; 10 k–20 k, rhombuses). A full color version of this figure is available at Polymer Journal online.

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Elastic modulus

We performed stretching tests for the 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels. According to the linear elasticity theory,22 we estimated the elastic moduli (G) from the initial slopes of the stress (σ)−elongation (λ) curves. The elastic moduli are predicted from the affine network model (Gaf) and from the phantom network model (Gph) to be as follows:

where kB is Boltzmann’s constant and T is the absolute temperature. The number densities of the active crosslinks (μ) and the elastically effective chains (ν) for the tetra-functional network are predicted based on a tree-like approximation.

Although the Tetra-PEG gel is formed by the AB-type coupling of tetra-arm polymers, we can treat this coupling as AA-type coupling of tetra-arm polymers when we consider the stoichiometric conditions.14 As for AA-type coupling of tetra-arm polymers, the probability that an arm does not lead to an infinite network (P∞) is correlated with p by23

In terms of P∞, the probability that a given tetra-arm polymer has a degree of connection of n (P(Xn)) is given by

where xCy is the usual notation for the number of combinations of x items taken y at a time: x!/ y! (x−y)!. Here, it should be noted that only if n⩾3 the connections do become cross-links. From Equation 4, the probabilities of a tri-functional crosslink (P(X3)) and a tetra-functional crosslink (P(X4)) are calculated as follows:

With Equations 5 and 6, μ and ν are predicted as follows:

where ρ is the number density of the Tetra-PEG prepolymers.

The variations of G, Gaf and Gph as a function of φ0 in the 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels are shown in Figure 3. Gaf and Gph increased linearly as φ0 increased; this increase reflects the constant p. In the 5 –20 k Tetra-PEG gels (Figure 3b), G and Gph agreed with each other throughout the range; this agreement suggests that the elasticities are roughly predicted by the phantom network model. In the 5 –10 k Tetra-PEG gels (Figure 3a), the downward deviation of G from Gph became increasingly pronounced as φ0 decresed; this result suggests that elastically ineffective loops were formed in the low φ0 range.24 It seems, however, that G approaches Gph asymptotically in the high φ0 region, similar to the result for 5 k Tetra-PEG gels. In contrast, in the 10 –20 k Tetra-PEG gels (Figure 3c), G was slightly larger than Gph. In our previous paper, we observed the master relation between G/Gaf and φ0/φ*, where φ* is the overlapping concentration of prepolymers. The shift from the phantom to affine network models was clearly observed from the plot of this relation.25 Thus, we performed the same plot for the 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels (Figure 4). Here, we calculated φ* for the mixed solutions of the 5 k, 10 k and 20 k Tetra-PEG prepolymers from the formula:

Figure 3
figure 3

The value of G (squares) estimated from the stretching measurements, Gaf (circles) and Gph (triangles), estimated from the reaction efficiency (p) as a function of φ0 in the (a) 5 k–10 k, (b) 5 k–20 k and (c) 10 k–20 k Tetra-PEG gels. The dashed line is the guide showing the relation G∼φ0.

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Figure 4
figure 4

The value of G/Gaf as a function of φ0/φ* for the Tetra-PEG gels (5 k–5 k, triangles; 10 k–10 k, left triangles; 20 k–20 k, right triangles; 5 k–10 k, circles; 5 k–20 k, squares; 10 k–20 k, rhombuses). The data for the 5 k–5 k, 10 k–10 k and 20 k–20 k Tetra-PEG gels were taken from the literature.25 The dashed and dotted lines are the guides showing G/ Gaf=0.5 and 1.0, respectively. The experimental errors were smaller than the sizes of the symbols. A full color version of this figure is available at Polymer Journal online.

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where φ*A and φ*B are the overlapping concentrations of the prepolymers A and B, respectively (Table 1). In this plot, the phantom and affine network model predictions are the horizontal lines at G/Gaf=0.5 (dashed line) and 1 (dotted line), respectively. The results for the 5 k, 10 k and 20 k Tetra-PEG gels are reproduced from our previous paper.25 The results of the size-mismatched Tetra-PEG gels, especially the 5 –10 k and 5 –20 k Tetra-PEG gels, fall onto a master relation; this result suggests that the elasticities of these gels can be predicted using the phantom network model. This result corresponds well to Flory’s prediction that phantom-like behavior appears in the fairy dilute regime.26 The deviation below φ* is responsible for the formation of elastically ineffective loops, as discussed in our previous paper. However, in the 10 –20 k Tetra-PEG gels, G/Gaf is slightly higher than the master relation. Although the discrepancy is not large, there is a possibility that the mixing of size-mismatched prepolymers affects the elasticity. To investigate this point in detail, we investigated the fracture energy and the ultimate elongation.

Fracture energy

We performed the tearing measurements for the 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels to investigate the intrinsic fracture energy (T0). The fracture energy is defined as the energy required for the development of a crack with a unit length.27, 28 The Lake–Thomas model is a popular model for the prediction of the fracture energy of elastomers and well predicted the fracture energy of conventional Tetra-PEG gels.25 In the Lake–Thomas model,29 T0 is estimated to be the energy needed to break the chemical bonds on the fracture surface and is given by

where N is the degree of polymerization between neighboring crosslinks, U is the energy required to rupture a monomer unit and d is the displacement length. In our previous treatment, we assumed the absence of trapped entanglement: ν∼φ0/N and d∼N1/2. Hence, T0 scales with the φ0 and N by

The values of T0 for the 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels are shown as a function of φ0N1/2 in Figure 5. The data for the 5 –5 k, 10 –10 k and 20 –20 k Tetra-PEG gels are taken from our previous report.25 The data for size-mismatched Tetra-PEG gels fell onto the master curve showing T0∼φ0N1/2; this result suggests the absence of trapped entanglements and the applicability of the Lake–Thomas model, similar to conventional Tetra-PEG gels. Although there is an ambiguity because of experimental error, these data indicate that all the samples, including the 10 –20 k Tetra-PEG gel, have almost no trapped entanglement.

Figure 5
figure 5

Fracture energy as a function of φ0N1/2 for the Tetra-PEG gels (5 k–5 k, triangles; 10 k–10 k, left triangles; 20 k–20 k, right triangles; 5 k–10 k, circles; 5 k–20 k, squares; 10 k–20 k, rhombuses). The data for the 5 k–5 k, 10 k–10 k and 20 k–20 k Tetra-PEG gels were taken from the literature.25 The dashed lines are the guides showing T0∼φ0N1/2. A full color version of this figure is available at Polymer Journal online.

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Ultimate elongation ratio

Finally, we investigated the ultimate elongation ratio. The ultimate elongation ratio is estimated from the stress–elongation relation with the extended Gent model, whose applicability was confirmed in our previous studies.30 The extended Gent model predicts uniaxial stretching behaviors as follows:31

where λmax is the ultimate elongation, and the parameter α was estimated from the fit to the stress–elongation curves for the Tetra-PEG gels (20 k) under biaxial stretching. Because G is fixed to be the value obtained from the linear fit, the only fitting parameter is λmax.

The fit results for the 5 –10 k, 5 –20 k and 10 –20 k Tetra-PEG gels are shown in Figure 6. Regardless of φ0 and prepolymer combinations, the stress–elongation relations are well predicted by the extended Gent model; this result suggests that the uniaxial stretching behaviors of the size-mismatched Tetra-PEG gels are similar to those of the conventional Tetra-PEG gels.

Figure 6
figure 6

Stress–elongation curves in the (a) 5 k–10 k, (b) 5 k–20 k and (c) 10 k–20 k Tetra-PEG gels (φ0=0.096).

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The values of λmax for the 5 –10 k and 10 –20 k Tetra-PEG gels are shown as a function of φ0 in Figures 7a and b, respectively. λmax increased as φ0 increased (also see Supplementray Figure 1 in the Supplementray Information). These results differ from the prediction of the Kuhn model, which predicts λmax∼N1/2 and ‘no’ dependence on φ0,32 and they correspond to our results for conventional Tetra-PEG gels.30

Figure 7
figure 7

Ultimate elongation ratio as a function of φ0 in the (a) 5 k–10 k and (b) 10 k–20 k Tetra-PEG gels.

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In our previous study, we proposed a novel semi-empirical model that predicts the λmax of conventional Tetra-PEG gels:30

where L is the contour length, lg is the geometrical distance between neighboring crosslinks and a is the monomer length. The only difference between the Kuhn model and our model is the denominator, lg, which is calculated as lg∼ν−1/3.

The prediction from Equation 14 is shown as a guide in Figures 7a and b and corresponds well to the experimental results; the φ0-dependence of λmax is well predicted using our model. Figure 8 shows another prediction, that is, the N-dependence of λmax (also see Supplementray Figure 2 in the Supplementray Information). Data points with the same φ0 but different N values are shown in the same panel. It is clear that the N-dependence of λmax is well predicted using Equation 14. Finally, we replotted all the data for λmax as a function of N2/3φ1/3 in Figure 9. As expected, all of the data fall onto a master curve. These results suggest that the size mismatch of the prepolymers does not affect the ultimate elongation of the Tetra-PEG gels. These results also support the validity of our prediction that the ultimate elongation depends on φ0 and N, and this dependence is λmax ∼φ01/3N2/3. It should be noted that the 10 –20 k samples also obeyed the prediction of Equation 14. Considering all the data, we conclude that the size mismatch of the prepolymers does not affect their structure and mechanical properties.

Figure 8
figure 8

Ultimate elongation ratio as a function of N in the Tetra-PEG gels (a, φ0=0.050; and b, φ0=0.096). The data for the 5 k–5 k, 10 k–10 k and 20 k–20 k Tetra-PEG gels were taken from the literature.30

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Figure 9
figure 9

Ultimate elongation ratio as a function of φ01/3N2/3 for the Tetra-PEG gels (5 k–5 k, triangles; 10 k–10 k, left triangles; 20 k–20 k, right triangles; 5 k–10 k, circles; 5 k–20 k, squares; 10 k–20 k, rhombuses). The data for the 5 k–5 k, 10 k–10 k and 20 k–20 k Tetra-PEG gels were taken from the literature.30 The dashed lines are the guides showing T0∼φ01/3N2/3. A full color version of this figure is available at Polymer Journal online.

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Conclusion

In conclusion, the elastic modulus, the fracture energy and the ultimate elongation ratio of the size-mismatched Tetra-PEG gels have tendencies similar to those of conventional Tetra-PEG gels. From these results, we conclude that the size-mismatched Tetra-PEG gels have the same degree of homogeneity as the conventional Tetra-PEG gels, and the difference in prepolymer size does not affect the homogeneity. It should be noted that the small prepolymer itself induces the formation of elastically ineffective loops, as shown with the 5 k Tetra-PEG gels. The formation of elastically ineffective loops is strongly related to the normalized polymer volume fraction (φ0/φ*). In the range investigated in this study, the effect of elastically ineffective loops is negligible above the overlapping volume fraction (φ0/φ*>1). Thus, we expected to be able to form homogeneous polymer networks from the prepolymers with the ‘same functionalities’ and with molecular weights from 5 k to 40 k above the overlapping polymer volume fraction. The effects of the prepolymer functionality will be discussed in our forthcoming paper.