# Evaluation of Mesh Size in Model Polymer Networks Consisting of Tetra-Arm and Linear Poly(ethylene glycol)s

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical

#### 2.1. Geometric Blob

#### 2.2. Elastic Blob

#### 2.3. Correlation Blob

## 3. Results and Discussion

_{w}= 20 k) and linear-PEGs (M

_{w}= 0.2–20 k) at the stoichiometric ratio. By changing the molecular weight of linear-PEGs while fixing the molar concentration of tetra-PEGs (${U}_{4}$), we successfully formed a series of polymer gels with the same crosslinker density but different molecular weights between crosslinkers. The sol samples were prepared as controls by using the same tetra-PEGs and linear-PEGs without mutual reactive end-groups. The 4 × 4 gels were also prepared as controls by using mutual reactive tetra-PEGs with M

_{w}20 k. The relaxation time (${\tau}^{*}$) of correlation blobs was measured by DLS. ${\tau}^{*}$ was obtained by fitting the first relaxation mode. Partial heterodyne correction was performed for the relaxation time of polymer gels to obtain the true relaxation time in non-ergodic system [13,32,33]. The size of correlation blob, ${\xi}_{c}$, was calculated with Equation (8).

_{4}= 3.0 mM; Case B, U

_{4}= 1.5 mM) and the same ${M}_{c}$ are shown as controls; the values are cited from the study of Akagi et al. [17]. The values of ${G}^{\prime}$ were almost the same for 2 × 4 gels and 4 × 4 gels. A simulation and NMR study in 4 × 4 gels has revealed that when the tetra-PEG concentration is above its overlapping concentration (Case A), the unfavorable bonds, such as double-link and higher-order defects, are negligible [35]. Therefore, the comparable values of ${G}^{\prime}$ in Case A suggest that 2 × 4 gels are free of defects just as the 4 × 4 gels are [15,18,25]. ${G}^{\prime}$ in Case B increased as the molecular weight of linear-PEGs increased (${G}^{\prime}~{M}_{c}{}^{1.2}$), suggesting that more ideal networks are formed when the linear-PEGs are long enough to connect the nearby tetra-PEGs. A precise measurement for the reaction conversion may give us more information to discuss the scaling law in Case B. But it was difficult to measure the reaction conversion in our system at this stage. We could not measure the reaction conversion in Case A as well because the values of ${G}^{\prime}$ in 2 × 4 gels are very close to those in 4 × 4 gels. Hence, we assume the reaction conversion of 2 × 4 gels in Case A is as high as 4 × 4 gels (reaction conversion ~85% from previous study [17]).

## 4. Experimental

#### 4.1. Sample Preparation

_{w}= 20 k) with amine-terminated linear PEGs (linear-PEGs) having various M

_{w}(=0.2 k to 20 k) in acetonitrile. The chain-overlap polymer volume fraction (${\varphi}_{4}^{*}$) of tetra-PEG 20 k is around 0.035 (=2.1 mM) [17]. The subscript 4 denotes tetra-PEG and 2 denotes linear-PEGs hereafter. We started the gel preparation from two cases: Case A, well-packed system (${\varphi}_{4}=0.050\left(=3.0\mathrm{mM}\right){\varphi}_{4}^{*}$) to form complete networks and Case B, non-packed system (${\varphi}_{4}=0.025\left(=1.5\mathrm{mM}\right){\varphi}_{4}^{*}$) to form incomplete networks. The linear-PEGs with different molecular weights were added into the tetra-PEG solutions by the stoichiometric ratio to form polymer gels with the same crosslinker density but different molecular weights between crosslinkers.

_{w}= 20 k) with amine-terminated tetra-PEG (M

_{w}= 20 k) equivalently in acetonitrile. We fabricated two gels. One gel forms a complete network (${\varphi}_{4}=0.083\left(=5.0\mathrm{mM}\right){\varphi}_{4}^{*}$), and the other gel forms an incomplete network (${\varphi}_{4}=0.017\left(=1.0\mathrm{mM}\right){\varphi}_{4}^{*}$). We note that crosslinker density of 4 × 4 gels is different with 2 × 4 gels here.

#### 4.2. Dynamic Light-Scattering Measurements

#### 4.3. Rheological Measurements

## 5. Conclusions

- (1)
- The concentration dependence of the correlation length, ${\xi}_{c}$, is independent of the molecular weight and the completeness of the network structure, and follows the well-known scaling law, ${\xi}_{c}~{\varphi}^{-3/4}$. The gels essentially possess the semidilute correlations, which is irrelevant to the network structure.
- (2)
- In contrast to the correlation length, the mechanical properties, i.e., the elastic modulus, depend strongly on the completeness of the networks, and two different scaling relations were found.
- (3)
- The correlation blob is definitely not the mesh size in polymer gels, although it is often referred to as the mesh size in polymer networks. The elastic blob is, by definition, close to the mesh size. However, it is found that the molecular weight between crosslinkers brings a complicated effect in estimation of the mesh size.
- (4)
- An interesting correlation was found for the first time between ${G}^{\prime}$ and ${\xi}_{c}$, depending on the complete/incompleteness of the networks, ${G}^{\prime}~{\xi}_{c}^{-1}$ and ${G}^{\prime}~{\xi}_{c}^{-2}$, respectively, for the complete networks and incomplete networks. The Pincus blob may be a clue for explanation of these correlations.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Miller, D.R.; Macosko, C.W. A New Derivation of Post Gel Properties of Network Polymers. Macromolecules
**1976**, 9, 206–211. [Google Scholar] [CrossRef] - Nishi, K.; Chijiishi, M.; Katsumoto, Y.; Nakao, T.; Fujii, K.; Chung, U.I.; Noguchi, H.; Sakai, T.; Shibayama, M. Rubber elasticity for incomplete polymer networks. J. Chem. Phys.
**2012**, 137, 224903. [Google Scholar] [CrossRef] [PubMed] - De Gennes, P.G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, USA, 1979. [Google Scholar]
- Oikawa, H.; Murakami, K. Dynamic light scattering of swollen rubber vulcanizates and the swelling mechanism. Macromolecules
**1991**, 24, 1117–1122. [Google Scholar] [CrossRef] - Sakai, T.; Kurakazu, M.; Akagi, Y.; Shibayama, M.; Chung, U.-I. Effect of swelling and deswelling on the elasticity of polymer networks in the dilute to semi-dilute region. Soft Matter
**2012**, 8, 2730–2736. [Google Scholar] [CrossRef] - Rüchel, R.; Brager, M.D. Scanning electron microscopic observations of polyacrylamide gels. Anal. Biochem.
**1975**, 68, 415–428. [Google Scholar] [CrossRef] - Marmorat, C.; Arinstein, A.; Koifman, N.; Talmon, Y.; Zussman, E.; Rafailovich, M. Cryo-Imaging of Hydrogels Supermolecular Structure. Sci. Rep.
**2016**, 6, 25495. [Google Scholar] [CrossRef] [PubMed] - Tanaka, T. Collapse of Gels and the Critical Endpoint. Phys. Rev. Lett.
**1978**, 40, 820–823. [Google Scholar] [CrossRef] - Shibayama, M.; Tanaka, T. Volume Phase-Transition and Related Phenomena of Polymer Gels. Adv. Polym. Sci.
**1993**, 109, 1–62. [Google Scholar] [CrossRef] - Tanaka, T.; Sato, E.; Hirokawa, Y.; Hirotsu, S.; Peetermans, J. Critical kinetics of volume phase transition of gels. Phys. Rev. Lett.
**1985**, 55, 2455–2458. [Google Scholar] [CrossRef] [PubMed] - Shibayama, M.; Tanaka, T.; Han, C.C. Small angle neutron scattering study on poly(N-isopropyl acrylamide) gels near their volume-phase transition temperature. J. Chem. Phys.
**1992**, 97, 6829–6841. [Google Scholar] [CrossRef] - Pincus, P. Excluded Volume Effects and Stretched Polymer Chains. Macromolecules
**1976**, 9, 386–388. [Google Scholar] [CrossRef] - Shibayama, M.; Norisuye, T.; Nomura, S. Cross-link Density Dependence of Spatial Inhomogeneities and Dynamic Fluctuations of Poly(N-isopropylacrylamide) Gels. Macromolecules
**1996**, 29, 8746–8750. [Google Scholar] [CrossRef] - Watanabe, N.; Li, X.; Shibayama, M. Probe Diffusion during Sol–Gel Transition of a Radical Polymerization System Using Isorefractive Dynamic Light Scattering. Macromolecules
**2017**, 50, 9726–9733. [Google Scholar] [CrossRef] - Sakai, T.; Matsunaga, T.; Yamamoto, Y.; Ito, C.; Yoshida, R.; Suzuki, S.; Sasaki, N.; Shibayama, M.; Chung, U.-I. Design and Fabrication of a High-Strength Hydrogel with Ideally Homogeneous Network Structure from Tetrahedron-like Macromonomers. Macromolecules
**2008**, 41, 5379–5384. [Google Scholar] [CrossRef] - Matsunaga, T.; Asai, H.; Akagi, Y.; Sakai, T.; Chung, U.-I.; Shibayama, M. SANS Studies on Tetra-PEG Gel under Uniaxial Deformation. Macromolecules
**2011**, 44, 1203–1210. [Google Scholar] [CrossRef] - Akagi, Y.; Gong, J.P.; Chung, U.-I.; Sakai, T. Transition between Phantom and Affine Network Model Observed in Polymer Gels with Controlled Network Structure. Macromolecules
**2013**, 46, 1035–1040. [Google Scholar] [CrossRef] - Matsunaga, T.; Sakai, T.; Akagi, Y.; Chung, U.-I.; Shibayama, M. SANS and SLS Studies on Tetra-Arm PEG Gels in As-Prepared and Swollen States. Macromolecules
**2009**, 42, 6245–6252. [Google Scholar] [CrossRef] - Kurakazu, M.; Katashima, T.; Chijiishi, M.; Nishi, K.; Akagi, Y.; Matsunaga, T.; Shibayama, M.; Chung, U.-I.; Sakai, T. Evaluation of Gelation Kinetics of Tetra-PEG Gel. Macromolecules
**2010**, 43, 3935–3940. [Google Scholar] [CrossRef] - Li, X.; Tsutsui, Y.; Matsunaga, T.; Shibayama, M.; Chung, U.-I.; Sakai, T. Precise Control and Prediction of Hydrogel Degradation Behavior. Macromolecules
**2011**, 44, 3567–3571. [Google Scholar] [CrossRef] - Katashima, T.; Urayama, K.; Chung, U.-I.; Sakai, T. Strain energy density function of a near-ideal polymer network estimated by biaxial deformation of Tetra-PEG gel. Soft Matter
**2012**, 8, 8217–8222. [Google Scholar] [CrossRef] - Li, X.; Khairulina, K.; Chung, U.-I.; Sakai, T. Electrophoretic Mobility of Double-Stranded DNA in Polymer Solutions and Gels with Tuned Structures. Macromolecules
**2014**, 47, 3582–3586. [Google Scholar] [CrossRef] - Nishi, K.; Fujii, K.; Katsumoto, Y.; Sakai, T.; Shibayama, M. Kinetic Aspect on Gelation Mechanism of Tetra-PEG Hydrogel. Macromolecules
**2014**, 47, 3274–3281. [Google Scholar] [CrossRef] - Li, X.; Kondo, S.; Chung, U.-I.; Sakai, T. Degradation Behavior of Polymer Gels Caused by Nonspecific Cleavages of Network Strands. Chem. Mater.
**2014**, 26, 5352–5357. [Google Scholar] [CrossRef] - Kamata, H.; Akagi, Y.; Kayasuga-Kariya, Y.; Chung, U.I.; Sakai, T. “Nonswellable” Hydrogel Without Mechanical Hysteresis. Science
**2014**, 343, 873–875. [Google Scholar] [CrossRef] [PubMed] - Kondo, S.; Hiroi, T.; Han, Y.S.; Kim, T.H.; Shibayama, M.; Chung, U.I.; Sakai, T. Reliable Hydrogel with Mechanical “Fuse Link” in an Aqueous Environment. Adv. Mater.
**2015**, 27, 7407–7411. [Google Scholar] [CrossRef] [PubMed] - Li, X.; Watanabe, N.; Sakai, T.; Shibayama, M. Probe Diffusion of Sol–Gel Transition in an Isorefractive Polymer Solution. Macromolecules
**2017**, 50, 2916–2922. [Google Scholar] [CrossRef] - Li, X.; Hirosawa, K.; Sakai, T.; Gilbert, E.P.; Shibayama, M. SANS Study on Critical Polymer Clusters of Tetra-Functional Polymers. Macromolecules
**2017**, 50, 3655–3661. [Google Scholar] [CrossRef] - Kondo, S.; Chung, U.-I.; Sakai, T. Effect of prepolymer architecture on the network structure formed by AB-type crosslink-coupling. Polym. J.
**2013**, 46, 14–20. [Google Scholar] [CrossRef] - Kondo, S.; Sakurai, H.; Chung, U.-I.; Sakai, T. Mechanical Properties of Polymer Gels with Bimodal Distribution in Strand Length. Macromolecules
**2013**, 46, 7027–7033. [Google Scholar] [CrossRef] - Nishi, K.; Noguchi, H.; Sakai, T.; Shibayama, M. Rubber elasticity for percolation network consisting of Gaussian chains. J. Chem. Phys.
**2015**, 143, 184905. [Google Scholar] [CrossRef] [PubMed] - Pusey, P.N.; Van Megen, W. Dynamic light scattering by non-ergodic media. Phys. A Stat. Mech. Appl.
**1989**, 157, 705–741. [Google Scholar] [CrossRef] - Joosten, J.G.H.; McCarthy, J.L.; Pusey, P.N. Dynamic and static light scattering by aqueous polyacrylamide gels. Macromolecules
**1991**, 24, 6690–6699. [Google Scholar] [CrossRef] - Rubinstein, M.; Colby, R.H. Polymer Physics; OUP Oxford: Oxford, UK, 2003. [Google Scholar]
- Lange, F.; Schwenke, K.; Kurakazu, M.; Akagi, Y.; Chung, U.-I.; Lang, M.; Sommer, J.-U.; Sakai, T.; Saalwächter, K. Connectivity and Structural Defects in Model Hydrogels: A Combined Proton NMR and Monte Carlo Simulation Study. Macromolecules
**2011**, 44, 9666–9674. [Google Scholar] [CrossRef] - Katashima, T.; Asai, M.; Urayama, K.; Chung, U.-I.; Sakai, T. Mechanical properties of tetra-PEG gels with supercoiled network structure. J. Chem. Phys.
**2014**, 140, 074902–074910. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Schematic illustration of polymer networks prepared by mutual reactive tetra-functional poly(ethylene glycol)s (PEG) macromonomers (4 × 4 gels). Blue and red polymer refer to the tetra-functional PEG macromonomers with different end-groups.

**Figure 2.**Schematic illustration of 2 × 4 gels formed by mixing mutual reactive tetra-functional PEG and linear-PEG polymers. (

**Case A**) Complete network. The molar concentration of tetra-PEG (U

_{4}) in the initial solution is set to be 3.0 mM, higher than its overlapping concentration (U

_{4}

^{*}= 2.1 mM for tetra-PEG with molecular weight 20 k) [17]. Linear-PEG with different molecular weights were used as a spacer chain to connect tetra-PEGs. These 2 × 4 gels have the constant crosslinker density but different molecular weights between the crosslinkers. (

**Case B**) Incomplete network. U

_{4}of tetra-PEG in the initial solution is set to be 1.5 mM, lower than its overlapping concentration (U

_{4}

^{*}= 2.1 mM). Polymer gels with many defects are expected to be formed for short linear-PEGs and complete network to be formed for long linear-PEGs.

**Figure 3.**Size of correlation blob in 2 × 4 gels as a function of the total polymer volume fraction ($\varphi ={\varphi}_{2}+{\varphi}_{4}$). The full symbols represent the gels and empty symbols represent the sols. Data of Case A are shown in red and those of Case B are shown in blue. Sol samples were prepared as controls by using the non-reactive tetra-PEGs and linear-PEGs. 4 × 4 gels are also shown as controls. The solid line illustrates the fitting curve, ${\xi}_{c}~{\varphi}^{-0.56}$.

**Figure 4.**Shear modulus (${G}^{\prime}$) of 2 × 4 gels as a function of molecular weight between crosslinkers (${M}_{c}$ ). Data of Case A are shown in red and those of Case B are shown in blue. 4 × 4 gels with corresponding tetra-PEG concentration are shown as controls. The solid lines denote the fitting curves of ${G}^{\prime}~{M}_{c}{}^{0.69}$ and ${G}^{\prime}~{M}_{c}{}^{1.2}$ for Cases A and B, respectively. The values of ${G}^{\prime}$ of 4 × 4 gels are cited from the study of Akagi et al. (tetra-PEG 20 k (M

_{c}10 k), $\varphi =0.051\left(=3.0\mathrm{mM}\right)$ and tetra-PEG 40 k (M

_{c}= 20 k), $\varphi =0.096\left(=3.0\mathrm{mM}\right)$ as controls for Case A; tetra-PEG 40 k (M

_{c}= 20 k), $\varphi =0.051\left(=1.5\mathrm{mM}\right)$ as a control for Case B) [17].

**Figure 5.**Various blob sizes of 2 × 4 gels as a function of molecular weight between crosslinkers (${M}_{c}$). (

**a**) Case A: complete network; (

**b**) Case B: incomplete network. The values in horizontal axis are the molecular weights of the chain between crosslinkers.

**Figure 6.**Schematic of various blobs in 2 × 4 gels in Case A complete network. (

**a**) Network with short chains between crosslinkers; (

**b**) Network with long chains between crosslinkers.

**Figure 7.**Double logarithmic plot of shear modulus and correlation blob of 2 × 4 gels. The solid lines show the fitting curves of ${G}^{\prime}~{\xi}_{c}^{-1}$ and ${G}^{\prime}~{\xi}_{c}^{-2}$ for Cases A and B, respectively.

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**MDPI and ACS Style**

Tsuji, Y.; Li, X.; Shibayama, M.
Evaluation of Mesh Size in Model Polymer Networks Consisting of Tetra-Arm and Linear Poly(ethylene glycol)s. *Gels* **2018**, *4*, 50.
https://doi.org/10.3390/gels4020050

**AMA Style**

Tsuji Y, Li X, Shibayama M.
Evaluation of Mesh Size in Model Polymer Networks Consisting of Tetra-Arm and Linear Poly(ethylene glycol)s. *Gels*. 2018; 4(2):50.
https://doi.org/10.3390/gels4020050

**Chicago/Turabian Style**

Tsuji, Yui, Xiang Li, and Mitsuhiro Shibayama.
2018. "Evaluation of Mesh Size in Model Polymer Networks Consisting of Tetra-Arm and Linear Poly(ethylene glycol)s" *Gels* 4, no. 2: 50.
https://doi.org/10.3390/gels4020050