# Decoupling between Translational Diffusion and Viscoelasticity in Transient Networks with Controlled Network Connectivity

^{1}

^{2}

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## Abstract

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## 1. Introduction

_{B}, and T are the number density of the polymer chains composed of N submolecules, the Boltzmann constant, and the absolute temperature, respectively. S(n,t) is called the shear orientational function, defined as:

_{i}(n, t) is the end-to-end vector of the n-th submolecules in the i-th chain at a particular time, t. <...> represents the ensemble average for all the chains. Equation (1) means that entropic tension along each chain always occurs. Still, if the spatial distribution of the chains is isotropic, the overall tension is balanced, and no macroscopic stresses can be observed. When the strain is applied, and the chain distribution becomes anisotropic, the balance is broken, resulting in macroscopic stress. On the other hand, the anisotropic distribution disappears through each chain diffusion, which is the origin of the viscoelastic relaxation in polymeric liquid. In general cases, including the unentangled and entangled polymers, the time required for the clearance of the anisotropic distribution (τ) is described as:

_{rotaional}means the rotational diffusion coefficient. Flexible polymers have random coil conformation in a good solvent and are approximated to be a sphere. Under sphere approximation, the “translational” diffusion coefficient, D

_{translational}, is proportional to D

_{rotational}, which indicates that the time that the polymers translationally diffuse about “self-size” agrees with the viscoelastic relaxation time. Many experimental studies have verified this concept in polymer melts and solutions [22,23,24,25,26].

## 2. Results and Discussion

#### 2.1. Effects of Network Connectivity on Viscoelasticity

^{2}, whereas G″ demonstrated the symmetric power law behaviors G″~ω and G″~ω

^{−1}at high and low frequencies, respectively. These characteristics agree well with the prediction of the Maxwellian model, which suggests that viscoelastic relaxation occurs via a unique process. It should be noted that the upturn at high frequencies was observed in s = 0.20, which is attributed to the Rouse mode of the dangling chains. As the values of s decreased from 0.50, the plateau level of G′ decreased, and the peak of G″ shifted to high frequencies. In Figure S1 in the Supplementary Materials, the viscoelasticity of the Alexa-labeled and neat samples overlapped each other, indicating that the effect of the fluorescence modification on the rheological property is negligible.

_{end}and K are the total end group concentration and the equilibrium constant, respectively. K was previously estimated to be 6.4 × 10

^{2}M

^{−1}by the surface plasmon resonance for the mono-functional PEG-FPBA and PEG-GDL [29,30]. It should be noted that p estimated by Equation (4) is just the reaction efficiency of the end groups, which includes the bridging chains and closed loop structures, especially low-connectivity regions.

_{visco}monotonically decreased with decreasing p and became shorter than the dissociation time of the dynamic covalent bond (dashed line), estimated by the surface plasmon resonance [29,30]. The p-dependence of the viscoelastic relaxation time is a surprising result contrary to the theoretical predictions by Green, Tobolsky [31], and Yamamoto [32], where the viscoelastic relaxation agrees with the dissociation of the bonds. This is because they assumed that the Rouse dynamics of the network strand are much faster than the bond lifetime, and the orientational anisotropy in the whole system disappears immediately after the dissociation of bonds. This deviation from the theoretical prediction suggests that the viscoelastic relaxation is determined not by the single chain dynamics but by the collective “network” dynamics.

#### 2.2. Effects of Network Connectivity on Diffusivity

_{KWW}is the characteristic recovery time, β is the polydispersity index for the recovery time, and γ is the modification factor (=1.5). Generally, the fluorescence photobleaching recovery process for single-component diffusing species is described by a single exponential function for a two-dimensional diffusion. However, the bleaching laser intensity profile is not homogeneous, characterized by a Gaussian function. The exponent of β includes this experimental non-ideality of laser intensity (≈0.6). As a result, the apparent relaxation time in the KWW function is accelerated from the accurate diffusion time, which is modified by the constant γ. The details are previously described in Appendix [3]. Notably, fitting by Equation (5) the immobilized fraction (A in Equation (5)) is estimated to be approx. 0.25–0.3, which roughly agrees with the percolation threshold of the diamond lattice in three-dimensional space. This agreement suggests that the contribution of the percolation networks on diffusion is negligible.

_{KWW}against d

^{2}. The linear relationship between τ

_{KWW}and d

^{2}was observed, indicating that the fluorescence recovery is primarily attributed to the Fickian diffusion of molecules, and the chemical interconversion is negligible. It should be noted that the data in the smallest area in s = 0.5 deviated from the linear relationship due to the non-Fickian (anomalous) diffusion in transient networks [5,33,34]. In the discussion below, we utilized the data showing the linear relationship to estimate the translational diffusion coefficient.

^{2}, as:

^{2}region, D is almost independent of d

^{2}and constant values, reflecting the translational diffusion in two-dimensional space. In the region where <τ> was independent of d

^{2}, the translational diffusion coefficient (D

_{translational}) is defined (represented by dashed lines). The estimated D

_{translational}is plotted against p in Figure 5b. D

_{translational}decreased with increasing p in the semi-logarithmic plot, suggesting the collision between stickers restricts the polymer dynamics. This tendency was observed in the previous reports [3,5,33,34], and is quantitatively consistent with the Sticky Rouse model [36].

#### 2.3. Comparison between Viscoelasticity and Diffusivity

^{ν}; a is the Kuhn length of polyethylene glycol (≈0.65 nm [37,38,39,40]), N is the Kuhn segment number, ν is the excluded volume exponent in good solvent (≈0.57)). RMSD is an order of 10

^{−6}m and approx. 100 times larger than the gyration radius even in the Tetra-PEG slime with the controlled network structures. This significant difference indicates that the viscoelastic relaxation does not proceed through the diffusive motion of each prepolymer. In this experimental system, the equilibrium connectivity is up to 0.6. Therefore, a mobile sol component in which not all arms are connected to the percolated network exists. It is suggested that the FRAP measurement preferentially detects this mobile component.

## 3. Conclusions

## 4. Materials and Methods

#### 4.1. Synthesis of Fluorescence-Labeled Tetra-Armed Prepolymers

_{2}) (M

_{w}= 2.0 × 10

^{4}g mol

^{−1}) was purchased from SINOPEG BIOTECH Co., Ltd. (Xiamen, China), and its end groups were modified with of 4-carboxy-3-fluorophenylboronic acid (FPBA). The details were reported previously [29]. On the other hand, we synthesized the Tetra-PEG-D-(+)-glucono-1,5-lactone (GDL) partially modified with Alexa Fluor 594 (Alexa). The Tetra-PEG-NH

_{2}was dissolved in methanol at a concentration of 50 g L

^{−1}. Separately, Alexa Fluor™ 594 NHS ester (succinimidyl ester) (Alexa-NHS), purchased from Thermo Fisher SCIENTIFIC (Tokyo, Japan), was dissolved in super-dehydrated dimethyl sulfoxide at the concentration of 1 mg/mL. Compared to the molar amount of the amine end group, we added 0.001 times the amount of Alexa-NHS and stirred the mixture overnight at room temperature. Compared to the molar amount of the amine end group, we added 10 times the amount of D-(+)-glucono-1,5-lactone (GDL) and 20 times the amount of triethylamine and stirred the mixture for three days at 35 °C. The resultant solution was poured into a dialysis membrane (molecular weight cut-off: 3500 for 10,000 g mol

^{−1}) and dialyzed for two days with methanol and water. After passing through a syringe filter (0.45 μm), the solutions were collected and freeze-dried. The synthesis scheme is shown in Figure 7.

^{1}H-NMR was used to confirm that the synthesis was complete. The fluorometer confirmed that the modification degree of the Alexa was 0.0444% (shown in Figure S2).

#### 4.2. Sample Preparation

^{−1}. The two polymer solutions were mixed at various stoichiometrically imbalanced fractions of s = 0.20, 0.30, 0.40, 0.50, where s = [Tetra-PEG-GDL]/([Tetra-PEG-GDL] + [Tetra-PEG-FPBA). Each reaction was allowed to proceed for 12 h at 25 °C.

#### 4.3. Dynamic Viscoelastic Measurements

^{−1}) of the storage (G′) and loss (G″) moduli was measured at 30 °C. Before the measurements, the oscillatory shear strain amplitudes were confirmed to be within the linear viscoelasticity range.

#### 4.4. Fluorescence Recovery after Photobleaching (FRAP) Measurements

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Angular frequency dependence of G′ (

**top**panel) and G″ (

**bottom**panel) for the Tetra-PEG slimes with various s at 30 °C.

**Figure 3.**Representative results of fluorescence recovery after photobleaching for Tetra-PEG slime (s = 0.5, d = 2.5 µm) at 30 °C. The dashed line represents the fitting results using Equation (5).

**Figure 4.**Estimated τ

_{KWW}as a function of d

^{2}for the Tetra-PEG slimes with various s (red: s = 0.5, yellow: s = 0.4, green: s = 0.3, blue: s = 0.2).

**Figure 5.**(

**a**) d

^{2}-dependence of diffusion coefficient (D) for the Tetra-PEG slimes with various s (red: s = 0.5, yellow: s = 0.4, green: s = 0.3, blue: s = 0.2). (

**b**) Estimated translational diffusion coefficient (D

_{translational}) as a function of p.

**Figure 6.**Root-mean-square distance the prepolymers diffuse during the viscoelastic relaxation time (RMSD) as a function of p. The dashed line represents the gyration radius of a four-armed precursor chain.

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**MDPI and ACS Style**

Katashima, T.; Kobayashi, R.; Ishikawa, S.; Naito, M.; Miyata, K.; Chung, U.-i.; Sakai, T.
Decoupling between Translational Diffusion and Viscoelasticity in Transient Networks with Controlled Network Connectivity. *Gels* **2022**, *8*, 830.
https://doi.org/10.3390/gels8120830

**AMA Style**

Katashima T, Kobayashi R, Ishikawa S, Naito M, Miyata K, Chung U-i, Sakai T.
Decoupling between Translational Diffusion and Viscoelasticity in Transient Networks with Controlled Network Connectivity. *Gels*. 2022; 8(12):830.
https://doi.org/10.3390/gels8120830

**Chicago/Turabian Style**

Katashima, Takuya, Ryunosuke Kobayashi, Shohei Ishikawa, Mitsuru Naito, Kanjiro Miyata, Ung-il Chung, and Takamasa Sakai.
2022. "Decoupling between Translational Diffusion and Viscoelasticity in Transient Networks with Controlled Network Connectivity" *Gels* 8, no. 12: 830.
https://doi.org/10.3390/gels8120830