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Because the swelling of polymer gels is a diffusion process of the polymer network, the swelling rate is determined by the diffusion coefficient of a polymer network. The diffusion coefficient (D) includes the polymer–solvent mixing contribution (Dmix) and the elastic contribution from the polymer network (Del). In this study, we experimentally investigate D, Dmix, and Del, using tetra gels whose network structural parameters are systematically controllable. We measured D and the shear modulus (G) using dynamic light scattering and dynamic viscoelastic measurement, respectively. According to the Tanaka–Fillmore theory, we evaluate Dmix and Del using the linear relationship between D and G in the as-prepared state. Consequently, we demonstrate a relation of D = Dmix + Del in the as-prepared, swelling, and equilibrium swollen states.

Dynamics of a polymer gel network is described by the theory proposed by Tanaka, Hocker, and Benedek (THB) that gives the diffusion coefficient of a polymer network (D=K+43Gf). Here, K is the osmotic bulk modulus, G is the shear modulus, and f is the friction coefficient between the polymer network and the solvent. Although several experimental studies investigated the dynamics of a polymer gel network, the THB theory has yet to be quantitatively validated. In this study, we quantitatively validate the THB theory by comparing D and G of Tetra-poly(ethylene glycol) gels measured by dynamic light scattering and dynamic viscoelastic measurement. The THB theory well described the linear increase in D with increasing G and gave K and f from the slope and the intercept. The gel network structural parameter dependences of the given K and f were well explained by the scaling laws. These results suggest that the THB theory is almost valid.

For the application of polymer gels, it is necessary to control independently and precisely their various physical properties. However, the heterogeneity of polymer gels hinders the precise control over the structure, as well as the verification of theories. To understand the structure-property relationship of polymer gels, many researchers have tried to develop a homogeneous model network. Most of the model networks were made from polymer melts that did not have a solvent and had many entanglements in the structure. Because the contribution of entanglements is much larger than that of chemical crosslinking, it was difficult to focus on the crosslinking structure, which is the structure considered in conventional theories. To overcome such a situation, we have developed a new model network system that contains much solvent. Specifically, we fabricated the polymer gel (Tetra-PEG gel) by mixing two types of solutions of tetra-armed poly(ethylene glycol) (Tetra-PEG) with mutually reactive end groups (amine (-PA) and activated ester (-HS)). Because the existence of a solvent strongly reduces the effect of entanglements, the effect of the crosslinking structure on the physical properties can be extracted. In this review, we present the structure-property relationship of Tetra-PEG gel. First, we show the structural homogeneity of Tetra-PEG gels. Then, we explain gelation reaction, elastic modulus, fracture energy and kinetics of swelling and shrinking of Tetra-PEG gels by comparing the theories and experimental results., Graphical abstract

Unlike hard materials such as metals and ceramics, rubbery materials can endure large deformations due to the large conformational degree of freedom of the crosslinked three-dimensional polymer network. However, the effect of the branching factor of the network on the ultimate mechanical properties of rubbery materials has not yet been clarified. This study shows that tri-branching, which entails the lowest branching factor, results in a large elastic deformation near the theoretical upper bound. This ideal elastic limit is realized by reversible strain-induced crystallization, providing on-demand reinforcement. The findings indicate that the polymer chain is highly orientated along the stretching axis, whereat enhanced reversible strain-induced crystallization is observed in the tri-branched and not in the tetra-branched network. A mathematical theory of structural rigidity is used to explain the difference in the chain orientation. Although tetra-branched polymers have been preferred since the development of vulcanization, these findings highlighting the merits of tri-branching will prompt a paradigm shift in the development of rubbery materials., 20+3 pages, 4+5 figures; published version

Cooperative diffusion coefficient (Dcoop) describes the dynamics of a polymer network in a gel, and is estimated by three independent methods. We measured three Dcoop's of a model polymer network system (Tetra-PEG gels), and obtained the experimental evidence to fundamentally understand the dynamics of polymer gels.

Friction between polymer chains and water is considered to govern the water retention and permeation in hydrogels. This concept has been examined by means of water permeation measurements through hydrogels. Previous studies used hydrogels in the equilibrium swollen state and supported the scaling relationship between the friction coefficient (f) and the blob size (ξ) proposed by Tanaka and de Gennes (f ∼ ξ–2). However, the experiments on equilibrium-swollen hydrogels are not enough, and those on as-prepared hydrogels are needed to fully examine the validity of the scaling relationship. In this study, we established a novel water permeation apparatus, which well prevented the swelling of gels and enabled the water permeation experiments on gels in the as-prepared state. Using this novel apparatus, we measured f of a model polymer network system, i.e., Tetra-PEG gels. By comparing f measured by the water permeation experiment and ξ measured by a small-angle neutron scattering experiment, we confirmed the sc...

Hydrogels have attracted attention as biomaterials because of the similar property to biological soft tissues. One of the applications of hydrogels for biomaterials is drug release carrier. For the application, it is important to understand and to control the diffusion behavior of molecules in hydrogels. There are three major models describing diffusion behavior of small molecules in the polymer solution. First is the Ogston model, in which polymer network is considered to behave as an obstacle preventing the diffusion of particles. Second is the hydrodynamic model, in which the existence of polymer increases the effective viscosity and decreases diffusion speed of particles. Third is the free volume model, in which particles diffuse passing through free volume of solvent and polymer. Although the validity of these models has been investigated by many researchers, there is no clear experimental evidence to support any models. Thus, even now, many models are left without validation. The muddling of models was assisted by the heterogeneity in conventional hydrogels. To understand the diffusion behavior of small molecules in a hydrogel, the validation of these models using a homogeneous hydrogel is necessary. Recently, we have developed a homogeneous hydrogel, Tetra-PEG gel, by combining two kinds of tetra-armed pre-polymers. It has been revealed that Tetra-PEG gel has an extremely homogeneous network structure, and network structure parameters (polymer volume fraction : φ 0, molecular weight between crosslinks : M w, binding ratio : p) of Tetra-PEG gel can be controlled independently. In this study, we investigated the effect of network structures on the self-diffusion coefficient of water molecules (D) in Tetra-PEG gels and in pure water (D 0) by Pulsed field-gradient spin-echo Nuclear Magnetic Resonance (PGSE-NMR). The values of D/D 0 decreased as an exponential function with an increase in φ 0 (0 - 0.15), and was independent of M w (10k, 20k g/mol) (Figure 1). This result well agreed with three models, and the difference among three models was negligible. On the other hand, D/D 0 decreased with a decrease in p (0 - 1) (Figure 1). This result was not reproduced by any three models, suggesting these models were not valid to describe diffusion behavior of small molecules in a hydrogel. Here, we tested another model proposed by M. Tokita et al. In the model, the diffusion of probe molecules is mainly determined by the correlation length of the polymer network (blob size : ξ) and the hydrodynamic radius of the probe molecules (R h) as, D/D 0 = exp(-R h/ξ). In Figure 2, D/D 0 in Tetra-PEG gels were plotted against 1/ξ. The values of xwere estimated by our previous small angle neutron scattering experiment. As shown in the figure, all the data including p-tune results fall onto a guide showing D/D 0 = 0.97exp(-2.8/ξ). The experimental values of the front coefficient (0.97) and R h (2.8Å) were close to the theoretical value (1.0) and the hydrodynamic radius of a water molecule (3.0Å), respectively. These results suggest that Tokita’s model is the correct model to describe the diffusion behavior of small molecules in a hydrogel. Figure 1

The swelling dynamics of polymer gels are characterized by the (collective) diffusion coefficient $D$ of the polymer network. Here, we measure the temperature dependence of $D$ of polymer gels with controlled homogeneous network structures using dynamic light scattering. An evaluation of the diffusion coefficient at the gelation point $D_{\mathrm{gel}}$ and the increase therein as the gelation proceeds $\Delta D\equiv D-D_{\mathrm{gel}}$ indicates that $\Delta D$ is a linear function of the absolute temperature with a significantly large negative constant term. This feature is formally identical to the recently discovered "negative energy elasticity" [Y. Yoshikawa et al., Phys. Rev. X 11, 011045 (2021) (arXiv:1912.13191)], demonstrating a nontrivial similarity between the statics and dynamics of polymer networks., Comment: 6+4 pages, 3+6 figures; published version