Introduction

Soft materials generally encompass spatial hierarchical structures of different scales, including monomeric structures (approximate scale: 0.1 nm), main-chain conformations (1–10 nm), entanglement or network topologies (10–100 nm), microphase separation structures (100–1000 nm), and crystalline and domain structures (1–10 µm) [1,2,3,4,5,6,7]. However, our understanding of the influence of these structures on the mechanical properties of soft materials remains incomplete for two primary reasons. First, it is difficult to control hierarchical structures independently because of the notable correlations among them. Second, studies on local mechanical information related to each structure are scarce. This is complicated by the fact that conventional rheometers record macroscopic stress/strain responses to applied macroscopic strains/stresses, resulting in average data and lost local information. Thus, further work is needed to elucidate the relationships among these hierarchical structures to fully understand the molecular mechanisms underlying the mechanical properties of soft materials.

To address the first problem, we recently developed transient network models with well-controlled network structures (Tetra-PEG slimes) [8,9,10,11,12]. These Tetra-PEG slimes comprise two types of tetra-armed polyethylene glycol (PEG), whose end groups are modified with phenylboronic acid or diols, which produce dynamic covalent bonds in aqueous solutions. Owing to the design of precursors with narrow dispersity and a definite branching structure, the resulting networks exhibit uniform strand lengths and functionalities. Network formation from mutually associative precursors allows us to precisely control the network connectivity, p, by mixing each precursor with an unbalanced stoichiometry [13]. Here, we define p as the fraction of bonded end groups in all reactive end groups in the equilibrium state, which can be expressed as [13]:

$$p=\left\{1+\frac{1}{{C}_{{{{\rm{end}}}}}K}\right\}+{\left[{\left\{1+\frac{1}{{C}_{{{{\rm{end}}}}}K}\right\}}^{2}-4s\left(1-s\right)\right]}^{\frac{1}{2}}$$
(1)

where Cend is the molar concentration of the reactive end groups, K is the equilibrium constant between the phenylboronic acid and diol end groups, and s is the mixing fraction of the tetra-PEG molecules with phenylboronic acid end groups to the total number of tetra-PEG molecules. The most important advantage of Tetra-PEG slime is that it provides precise control over the network topology while maintaining all other structural features.

In this study, we employed a microrheological approach for Tetra-PEG slimes with precisely controlled network connectivities. Microrheology involves various techniques for obtaining local viscoelastic information. A representative example is the particle tracking method [14,15,16,17,18,19,20,21], which can provide viscoelastic information at the micrometer scale via the assessment of the thermal diffusion of embedded particles used as probes. Notably, analyzing individual particles can provide insight into the heterogeneity in particle dynamics of a given system [22,23,24,25]. The results obtained here provide a definite basis for the relationship between the local viscoelasticity and network topology, which is the most basic hierarchical structure of soft materials.

Materials and methods

Sample preparation

Fluorescent polystyrene particles (Fluoresbrite® YG Microspheres; diameter a: 0.51 µm) were purchased from Polysciences, Inc. (Warrington, USA). The particles were diffused in a phosphate buffer (pH 7.4, 200 mM) at a volume fraction of 1.0 × 10−3 vol%. To remove agglomerated particles after mixing, the buffer was filtered through a syringe filter with a pore size of 5.0 µm.

Two types of four-armed PEG, whose end groups were modified with 4-carboxy-3-fluorophenylboronic acid (FPBA) or d-(+)-glucono-1,5-lactone (GDL), were purchased from Sinopeg Biotech Co., Ltd. (Xiamen, China; tetra-PEG-FPBA and tetra-PEG-GDL, respectively); the mass-average molar mass is 2.0 × 104 g mol−1. Tetra-PEG-FPBA and tetra-PEG-GDL were dissolved separately in the phosphate buffer containing fluorescent particles. The polymer concentration was set to 60 g L−1, which is approximately equal to the overlapping concentration of tetra-PEG. The two polymer solutions were then mixed in different fractions s:

$$s=\frac{{N}_{{{{\rm{FPBA}}}}}}{{N}_{{{{\rm{FPBA}}}}}+{N}_{{{{\rm{GDL}}}}}}$$
(2)

where NFPBA and NGDL are the molar concentrations of tetra-PEG-FPBA and tetra-PEG-GDL, respectively, and the mixing fraction, s, ranges from 0 to 0.5. Following mixing, the samples were maintained at 25 °C overnight to ensure reaction completion.

Dynamic viscoelastic measurements

Dynamic viscoelastic measurements were performed via a stress-controlled rheometer (MCR302; Anton Paar, Graz, Austria) with a cone–plate fixture (diameter: 25 and 50 mm; cone angle: 2°). Storage (G′) and loss (G′′) moduli were measured at 25 °C with angular frequencies of 0.1–100 rad s−1, and the applied strain varied between 0.01 and 1. Prior to the measurements, the oscillatory shear‒strain amplitudes were confirmed to occur within the linear viscoelasticity range.

Multiple particle tracking

Samples containing fluorescent probe particles were placed in a silicone mold (thickness: 5 mm) and sandwiched between glass slides to prevent drying. The trajectories of the probe particles were monitored via an inverted microscope (ECLIPSE Ti-U; Nikon, Tokyo, Japan; objective: 40×). The particles were observed under fluorescent light (wavelength λ: 505 µm), and videos of particle dynamics were captured via a charge-coupled device camera (ORCA-Flash4.0, Hamamatsu Photonics, Shizuoka, Japan). The average stack interval was 0.023 s, and the duration ranged from 100 to 10,000 s. The videos were analyzed to obtain the particle trajectories via Fiji software. For each condition, more than 20 probe particles were tracked over 10 intervals, resulting in data for more than 200 particle observations.

Results and discussion

The storage (G′) and loss (G′′) moduli of Tetra-PEG slimes with different network connectivities (p) are shown in Fig. 1. The plateau modulus decreased with decreasing p, reflecting a decrease in the density of the elastically active strands [26]. However, the terminal relaxation time shifted to higher frequencies with decreasing connectivity. According to our previous studies [8, 11, 13], the connectivity dependence of the terminal relaxation time is determined by the reconstruction of the backbone, which is the primary stress-supporting strand [13].

Fig. 1
figure 1

Angular frequency (ω) dependence of the storage (G′) and loss (G′′) moduli for Tetra-PEG slimes with different network connectivities (p): (black) p = 0; (purple) p = 0.14; (blue) p = 0.25; (green) p = 0.30; (orange) p = 0.35; and (red) p = 0.40. The circles and triangles represent G′ and G′′, respectively

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The two-dimensional trajectories (x and y displacements) of the probe particles (diameter a: 0.51 nm) embedded in Tetra-PEG slimes with various s values are shown in Fig. 2. In all cases, the duration of the trajectories was 100 s. Regardless of the value of s, the particles demonstrated erratic and nondirectional tracks, reflecting their thermal diffusion. In the lower-p regions, the particles diffused over distances of approximately 10 μm. However, particle motion was increasingly restricted with increasing p, as indicated by the increasingly localized trajectories. These results correspond to the macroscopic observation that viscosity increases with increasing p.

Fig. 2
figure 2

Representative two-dimensional trajectories over 100 s of multiple probe particles (diameter a = 0.51 nm) embedded in Tetra-PEG slimes with different network connectivities (p): (a) p = 0; (b) p = 0.14; (c) p = 0.25; (d) p = 0.30; (e) p = 0.35; and (f) p = 0.40

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The relationships between the mean square displacement of the particles (the average change in the square displacement, r2, after lag time τ, given as \(\langle \Delta {r}^{2}\left(\tau \right)\rangle\)) and τ in Tetra-PEG slimes with p = 0–0.42 are shown in Fig. 3. Notably, \(\langle \Delta {r}^{2}\left(\tau \right)\rangle\) was calculated as follows:

$$\left\langle \Delta {r}^{2}\left(\tau \right)\right\rangle =\left\langle {\left[x\left(t+\tau \right)-x\left(t \right)\right]}^{2}+{\left[y\left(t+\tau \right)-y\left(t \right)\right]}^{2}\right\rangle$$
(3)

where x(t) and y(t) are the time-dependent coordinates of the centroids of the tracked particles, τ is the lag time, and the angular brackets indicate an average over many starting times for the ensemble of particles.

Fig. 3
figure 3

Double logarithmic plots of mean square displacement (\(\langle \Delta {r}^{2}\left(\tau \right)\rangle\)) against lag time (τ) for probe particles embedded in Tetra-PEG slimes with different network connectivities (p): (a) p = 0; (b) p = 0.14; (c) p = 0.25; (d) p = 0.30; (e) p = 0.35; and (f) p = 0.40. (n is the total number of particle trajectories observed.)

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In the region of p ≤ 0.14, \(\langle \Delta {r}^{2}\left(\tau \right)\rangle\) increased with increasing τ, indicating that \(\langle \Delta {r}^{2}\left(\tau \right)\rangle\) ≈ τ1 across all time scales monitored. This linearity suggests that the diffusion behavior of the particles is based on a random-walk model, indicating that the microenvironment surrounding the particles is liquid [27]. Moreover, in the region of 0.14 < p ≤ 0.42, \(\langle \Delta {r}^{2}\left(\tau \right)\rangle\) increased more gradually (slope < 1) at short timescales, which is referred to as subdiffusive behavior and can be attributed to the elastic trapping of particles within the network structures of materials. In other words, the microenvironment traversed by the particles was a solid-like environment [27]. The timescale of this subdiffusive behavior increased with increasing p, which is qualitatively consistent with the macroscopic viscoelastic observation that terminal relaxation is increasingly delayed with increasing p. Notably, the particle motions in this study can be approximated as two-dimensional trajectories using particles in water. These findings are detailed in the Supporting Information (Figure S1). The generalized Einstein–Stokes equation [15] indicates that the inflection point of \(\langle \Delta {r}^{2}\left(\tau \right)\rangle\) should correspond to the terminal relaxation time estimated via macrorheology. However, our results do not conform with this determination, which may be attributed to the low spatial resolution of our system. Thus, we avoid a detailed discussion of the subdiffusive mode and focus only on the normal diffusive region at longer timescales, where τ-dependence is negligible.

As shown in Fig. 3, the variation in \(\langle \Delta {r}^{2}\left(\tau \right)\rangle\) increased with increasing p, suggesting heterogeneity in the particle dynamics. To examine this heterogeneity, the probability distribution of the diffusion coefficient D was evaluated (Fig. 4). Here, D was normalized by Dave, which is the average value of D for all the particles. When the distribution follows a Gaussian distribution (Eq. (4)), the shape of the histogram is symmetric with a central value of 1, indicating spatial homogeneity.

$$f\left(x\right)=\frac{1}{\sqrt{2\pi }\sigma }\exp \left\{-\frac{\left(x-1\right)}{2{\sigma }^{2}}\right\}$$
(4)

where σ is the standard deviation, and x is the normalized diffusion coefficient (D/Dave). As shown in Fig. 4, the shape of the distribution changes with p. These results suggest that network formation influences the spatial heterogeneity in particle dynamics [28].

Fig. 4
figure 4

Probability distributions of the normalized diffusion coefficients (D/Dave) of particles embedded in Tetra-PEG slimes with different network connectivities (p): (a) p = 0; (b) p = 0.14; (c) p = 0.25; (d) p = 0.30; (e) p = 0.35; and (f) p = 0.40

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We employed quantile–quantile (Q–Q) plots to compare the predicted values (assuming a Gaussian distribution) with the experimental results. In this manner, we could assess the deviation in the experimental results from the Gaussian distribution. Figure 5 shows Q–Q plots of the diffusion coefficients D of the particles in Tetra-PEG slimes with different p values against the theoretical quantiles, thereby assuming that the distribution follows a Gaussian distribution (Dtheo). The diagonal lines crossing the origin represent the Gaussian distribution, and the red circles denote the experimental data. At low p values, the experimental data followed an approximately linear relationship. However, with increasing p, the data increasingly deviated from a linear relationship. Figure 6 shows the coefficient of determination (R2) as a function of p. In the low-p region (<0.3), R2 is almost 0.9, which indicates that the distribution obeys a Gaussian distribution. However, R2 decreases to 0.8 when p increases above 0.34. Notably, this corresponds to the percolation point of the three-dimensional diamond lattice in the percolation model [13, 29,30,31,32,33]. This is supported by the G′ curves, which exhibit plateaus for samples with p ≥ 0.34. Interestingly, this suggests that the heterogeneity in particle dynamics increases under percolation.

Fig. 5
figure 5

Q–Q plot of the diffusion coefficients (D) as a function of the theoretical quantiles calculated from the Gaussian distribution (Dtheo) of particles embedded in Tetra-PEG slimes with different network connectivities (p): (a) p = 0; (b) p = 0.14; (c) p = 0.25; (d) p = 0.30; (e) p = 0.35; and (f) p = 0.40. The black lines denote the ideal Gaussian distribution, and the red circles represent the experimental data

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Fig. 6
figure 6

Coefficient of determination (R2) as a function of network connectivity (p)

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At the percolation point, a few precursors form percolated networks that can elastically trap particles. However, many of the precursors remain isolated and form clusters that do not suppress the particle dynamics, unlike the network regions [34]. These results, accompanied by heterogeneity in particle dynamics, have been observed in gelation phenomena [35,36,37,38,39,40,41]. For example, near the sol–gel transition point, the macroscopic dynamics are significantly delayed, whereas the heterogeneity in the local dynamics, such as the probe mobility, concentration fluctuations, and cooperative diffusivity, are enhanced. Our findings suggest that heterogeneity in particle dynamics also occurs in network systems produced via reversible interactions near the percolation point. Notably, the particle size dependence of the spatial heterogeneity in particle dynamics has been reported elsewhere. [42, 43] Our findings are limited to the microscale, which is larger than the network strand length.

Conclusion

In this study, we investigated the diffusion of probe particles embedded in transient model networks with controlled network connectivity (p). The multiple particle tracking experiments revealed the following: (i) the mean square displacement is proportional to the lag time below p = 0.34, suggesting normal diffusion of the particles; (ii) the mean square displacement significantly plateaus against the lag time (subdiffusive mode) above p = 0.34, which is attributed to the elastic trapping of particles by network structures; and (iii) the non-Gaussian parameter of the mean square displacement at all lag times reaches a maximum at p = 0.34, which corresponds to the percolation threshold. These results suggest that the microscale heterogeneity in particle dynamics is enhanced due to percolation, even in transient networks. This microscale heterogeneity in particle dynamics is also observed in sol–gel transitions. Our findings establish a foundation for exploring the relationships between mechanical properties and transient network structures, thereby advancing our understanding of general soft materials.