Polymer gel with a flexible and highly ordered three-dimensional network synthesized via bond percolation

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Science Advances  06 Dec 2019:
Vol. 5, no. 12, eaax8647
DOI: 10.1126/sciadv.aax8647


Gels are a soft elastic material consisting of a three-dimensional polymer network with nanometer-sized pores and are used in a variety of applications. However, gel networks typically have a substantial level of defects because the network formation reaction proceeds stochastically. In this study, we present a general scheme to fabricate gels with extremely low levels of defects by applying geometric constraints into pregel solution based on the “bond percolation” concept. In the formed gel, stationary laser speckles, which are an indicator of spatial defects, were not observed at all. In addition, we found that the concentration fluctuations of the polymer chains were ergodic across the whole gel network. In such a homogeneous gel, both the spatial and temporal correlations of polymer chains are the same before and after gelation.


Ordered three-dimensional (3D) network structures with nanometer-sized pores are important for a wide range of applications such as filtration (1), sensing (2), drug release (3), electronics (4), and optics (5). Although various technologies such as nanolithography (6), metal-organic frameworks (7), and microphase separation of block copolymers (8) enable us to construct precise structures with nanometer resolution, the overall size of each object is limited to several cubic micrometers. It is challenging to construct an ordered 3D nanostructure in a large scale. Polymer gels are a familiar soft material consisting of a giant 3D polymer network with nanometer-sized pores spanning throughout their bulk (9). The gel network can be rapidly synthesized on a large scale, in principle, unlimited, by conjugating long polymer chains with cross-linkers in a solvent (10). However, the formed gel network generally has a substantial level of defects including dangling ends, loops, entanglements, and nonuniform pore sizes because of the stochastic cross-linking reaction (11, 12). These defects result in nonuniform distribution of polymer chains, i.e., spatial defects, that are usually detected by photon or neutron scattering techniques as strong small-angle scattering (13, 14), stationary laser speckles (15), and nonergodic concentration fluctuations (16). Numerous attempts have been made to remove these defects from polymer gels, including by synthesizing them from monodisperse polymer chains (17), by using a relatively uniform cross-linking process such as a photoreaction (18), by limiting unfavorable intramolecular reactions via the A-B type cross-coupling of star polymers (19), and even by cross-linking polymer chains with movable cross-linkers (20). Nevertheless, discernible signs of spatial defects have been persistently observed in gels (18, 19, 2123). Gels have been believed to be inherently disordered as a result of the stochastic reaction. In this study, we break this preconception by presenting a general scheme to fabricate polymer gels with an extremely low level of spatial defects under a mild synthesis condition.


Our strategy to fabricate a homogeneous gel is based on the “bond percolation” model, which is one of the classical percolation models along with site percolation and site-bond percolation (24). In the site or site-bond percolation model, the system is partially filled with reactive units in the initial state, and the percolation is diffusion-controlled (25), which consequently leads to a variety of spatial configurations of the units in the developed network (fig. S1A). On the other hand, the bond percolation model assumes that the space is uniformly prepacked by mutually exclusive units in the beginning, and the percolation occurs as a consequence of cross-linking between the nearby units (fig. S1B), i.e., reaction-controlled (25). Because the units are mutually exclusive, the space is always uniformly filled by the units regardless of the extent of cross-linking and network formation, leading to a highly ordered, ideal network structure after reaction completes. In past studies, the bond percolation condition has not been fully satisfied because the units failed to fill the space either in the initial state or during the cross-linking reaction, due to an insufficient excluded volume effect of the units. To achieve the ideal bond percolation, (i) we used monodisperse star polymers, four-armed poly(ethylene glycol), as a space filling unit (Fig. 1A) because polymers with multiple arms show a strong excluded volume effect that prevents other polymers to come into the pervaded volume (26). Then, (ii) we dissolved the star polymers in a good solvent at a concentration [12 weight % (wt %)] well above its chain overlapping concentration (~6 wt %) to ensure that the star polymers uniformly and tightly fill the space (Fig. 1B) as the bond percolation model assumes. (iii) To prevent the segregation of polymer chains during cross-linking (27), we chose dehydrated acetonitrile as the solvent because it shows excellent affinity to the star polymers used in this study (fig. S2). Moreover, (iv) to remove dust and nanobubbles (28) from the solution as much as possible, we filtered the dissolved polymer solutions through ultrafine syringe filters with an average pore size of 20 nm, which is moderately larger than the size of the star polymers (~10 nm). Although all the aforementioned issues have already been pointed out separately in the past studies, they have never been resolved altogether in a single system. After addition of the stoichiometric amount of small bifunctional cross-linkers to the solution, the star polymer solution underwent a sol-gel transition in approximately 1 hour at room temperature (Fig. 1, C and D). The more accurate gel point was determined in a dynamic viscoelastic measurement as the crossover point (29) of the storage modulus G′ and loss modulus G″ (Fig. 1E). The value of G′ for the fully developed star polymer gel reached 10 kPa, which is comparable to the ideal elastic modulus expected for the phantom network model (30) (see the Supplementary Materials for details of this comparison).

Fig. 1 Schematic of the gel preparation via bond percolation.

(A) Star polymer: tetrafunctional poly(ethylene glycol) (PEG) with active ester end groups; bifunctional cross-linker: 1,14-diamino-3,6,9,12-tetraoxatetradecane (amino-PEG4-amine). (B) Stoichiometric mixture of the star polymer and the cross-linker in a good solvent. The system is uniformly prepacked with the star polymers. (C) Polymer gel formed by end-linking of the star polymers with the small cross-linkers via bond percolation. 2D schematics are shown instead of the real 3D polymer network for the sake of legibility. (D) Optical image of the fully developed transparent gel in a glass tube. Photo Credit: Xiang Li, Institute for Solid State Physics, The University of Tokyo. (E) Gelation of the star polymers is confirmed by dynamic viscoelastic measurements. G′ and G″ represent the storage and loss shear moduli, respectively.

To examine the nanometer structures in the gels, we evaluated the spatial correlations of the polymer chains in Fourier space using static laser light scattering (SLS) and small-angle x-ray scattering (SAXS), which cover a wide length scale in real space from ~2 to ~60 nm in SAXS and from ~250 to ~920 nm in SLS according to Bragg’s law. Figure 2A shows the scattering profiles of the pregel solution and the fully developed gel synthesized by the bond percolation scheme. The SLS intensity from the sol and gel samples has been scaled by the same factor so that it matches the SAXS intensity I(q) at the low-q limit, where q is the magnitude of the scattering vector. The scattering profiles of the polymer gel and the non–cross-linked pregel solution were almost identical. Namely, the spatial correlation between the polymer chains did not change by cross-linking. This result strongly suggests that gelation of the star polymers proceeded by bond percolation. Hereafter, we call this gel the bond percolation gel. Another point to note is that the abnormal strong low-q scattering, which has always been observed in past polymer gels (11, 13, 14, 18, 2123), was not observed in the bond percolation gel (Fig. 2A). The spatial defects must be largely reduced from the gel network. To further confirm the homogeneity in this bond percolation gel, we performed laser speckle tests to visualize the spatial defects in the gel. The gelling samples were irradiated with a coherent laser beam, and the 2D pattern of the scattered light was monitored with an electron-multiplying charge-coupled device (EMCCD) camera (Fig. 2B and fig. S3). In the bond percolation gel, the scattering pattern remained the same from the pregel solution up to the fully developed gel (Fig. 2, C and D). Unexpectedly, even at the gel point, there was little change in the scattering pattern. Stationary laser speckles (i.e., bright spots), which are an indication of spatial defects (16, 3133), were not observed at all (see movie S1 for the scattering patterns throughout the whole gelation process). To make this observation even clearer, we performed the same measurement on a conventional gel synthesized by copolymerizing small monomers and cross-linkers in water, using a common monomer, N-isopropylacrylamide (NIPAM) (34). This monomer/cross-linker system serves as a representative example of site-bond percolation because the small monomers and the cross-linker do not space-fill the solution. As we expected, the scattering patterns of the conventional gel markedly changed during the network formation (Fig. 2, E and F). The total scattered intensity increased with the progress of reaction and reached a maximum at the sol-gel transition point. In addition, numerous laser speckles that reflect the nonuniform distribution of the polymer chains appeared when the incipient gel network was formed (see movie S1 for the scattering patterns throughout the whole gelation process). All these static scattering results show that the spatial homogeneity in polymer gels can be significantly improved by the proposed bond percolation scheme. Note that, although the bond percolation gel was spatially homogeneous, the scattering patterns did not show Bragg diffraction peaks that are expected for spatially ordered systems such as crystals. The absence of clear interpolymer interference is likely because the cross-linked polymer chains are very flexible and fluctuate in the length scale much larger than the atoms in crystals do, which would smear the interference between the chains.

Fig. 2 Static structure of polymer solutions and gels.

(A) SLS and SAXS intensity profile [I(q)] of the star polymer solution and the corresponding gel synthesized at the bond percolation condition. I(q) from SLS has been scaled to match I(q) from SAXS at the low-q limit. (B) An illustration of the measurement system of the following 2D laser speckle images. A photo of the setup is shown in fig. S3. (C) Time-resolved 2D laser speckle patterns of the gelling star polymer solution via the bond percolation and (D) the corresponding counts versus pixels plot of each image indicated by a dashed line in the image. (E) Time-resolved 2D laser speckle patterns from a conventional gelling solution consisting of the monomers (NIPAM) and the cross-linker and (F) the corresponding counts versus pixels plot indicated by a dashed line in each image. Each laser speckle image is accumulated for 30 s.

In addition to the static correlation experiments, we also evaluated the homogeneity of the gel network from the viewpoint of temporal correlations. We first measured the time correlation functions (g2) of scattered intensity throughout the whole gelation process at a fixed sample position by time-resolved dynamic light scattering (DLS) measurements. Despite the fact that the sol-gel transition certainly occurred at approximately 60 min after the reaction was initiated, there was no noticeable change in the time correlation functions in the bond percolation gel (Fig. 3A). All g2 curves fall on a single master curve regardless of the extent of cross-linking. See movie S2 for a comparison of the time-resolved DLS and the time-resolved dynamic viscoelastic measurements. The behavior of g2 in the bond percolation gel is completely different from that in any other gels (15, 18, 19, 22, 3537), where g2 changes markedly at the sol-gel transition point and a noticeable suppression of the initial value, g2(0), is commonly observed after the gel is formed. Figure 3B shows an example of the gelation of the conventional gels, using the NIPAM monomer/cross-linker system. The suppression of g2(0) indicates the presence of local oscillators that do not go out of the observation length scale ~q−1 within the experimental time (38). The origin of the local oscillators in gels has been unclear and often attributed to the restricted dynamics of the cross-linkers and the nearby polymer chains (16, 38, 39). However, such suppression of g2(0) is not observed in the bond percolation gel (Fig. 3A), indicating that the local oscillators are not an inherent property of polymer gels. Because the bond percolation gel did not show any signs of spatial defects, the local oscillators observed in the past gels are highly likely due to the locally arrested spatial defects, which could be eliminated using the bond percolation scheme. The invariance of g2 in the pregel solution and the fully developed gel in Fig. 3A also implies that the polymer chains in the gel network travel a distance at least longer than their own size by thermal fluctuations, making the apparent local chain dynamics insensitive to cross-linking. This implication is consistent with the absence of the Bragg diffraction peak in the static scattering patterns as we discussed above.

Fig. 3 Dynamic properties of polymer solutions and gels.

(A) Intensity time correlation functions (g2) of the gelling solution via the bond percolation scheme. τ is the lag time in the correlation function. The whole gelation process is continuously monitored by time-resolved DLS every 30 s at a fixed sample position. (B) g2 of the conventional gelling solution consisting of the monomers (NIPAM) and the cross-linker. (C) g2 of the fully developed bond percolation gel at 100 different sample positions and the ensemble-averaged time correlation function (g2E) computed from g2. (D) g2 of the fully developed conventional gel at 100 different sample positions and the computed g2E.

As polymer gels are a nonergodic material, measurements at a single sample position as shown in Fig. 3A may not represent the dynamics of the whole gel network. Therefore, we performed additional DLS measurements in a fully developed gel at 100 different, randomly chosen sample positions to examine the identity of local and ensemble-averaged dynamics [see the Supplementary Materials for the computation of the ensemble-averaged time correlation function (g2E)]. All g2 curves of the bond percolation gel measured at different sample positions overlap well with each other (Fig. 3C). The ensemble-averaged function g2E also agrees with the individual local g2 curves. The relaxation time of g2E in the bond percolation gel (~0.01 ms) yields a diffusion coefficient of ~3 × 10−10 m2 s−1, which is consistent with that of thermal concentration fluctuations of typical polymer chains under semidilute conditions (40). Again, the position-independent relaxation in the bond percolation gel is in marked contrast with the behavior of any reported gels (Fig. 3D) (16, 22, 37), in which each g2 curve depends significantly on the sample position and is never consistent with the ensemble-averaged one. The perfect consistency of local and ensemble-averaged dynamics suggests the ergodic concentration fluctuation in our bond percolation gel, which is yet another evidence of the high homogeneity of the formed gel network.


Gels have often been discussed in analogy with glass because both gels and glass have a highly disordered structure (35). However, our results suggest that the bond percolation gel is different from glass because almost identical star polymers are closely packed in the space and no spatial defects were detected. Even if there are no spatial defects, gels are nonergodic in nature because the constituting polymer chains cannot exchange their position with the neighbors. However, such nonergodicity is not observed by DLS for the bond percolation gel. This is likely because the entire space is evenly filled with the polymer chains and all the polymer chains can still fluctuate over a distance longer than the mesh size of the network even after gelation, which causes an apparent disappearance of the nonergodicity. This fluctuation of the polymer chains would also lead to substantial blurring of the observable correlation between chains. Hence, the scattering experiments on the bond percolation gel did not show any interference peaks that are expected for such a regularly packed system according to Bragg’s diffraction law. Note that the absence of spatial defects does not necessarily guarantee the absence of other topological defects such as dangling ends, loops, and entanglements. However, these defects should be significantly suppressed to the level that they no longer cause detectable spatial heterogeneity. The good agreement between the obtained elastic modulus and the value predicted from the theory of rubber elasticity also suggests the low level of topological defects.

Polymer gels with a highly ordered yet flexible network may be applied for a variety of fields such as high-performance filtration and sensing, effective actuators, controlled drug release, and speckle-free optical fibers. Our strategy to fabricate ordered polymer networks does not rely on specific chemistry of the polymers and cross-linking reaction, which further expands the range of applications. It also offers an ideal experimental platform to explore the fundamental physics of polymer networks that have been unclear because of the extensive defects in the conventional gels. We anticipate that our findings will usher in a new era in the field of gels.


Preparation of the star polymer gel

Tetrafunctional poly(ethylene glycol) terminated with N-hydroxysuccinimidyl ester (tetra-PEG-NHS; Mw = 20 kg mol−1, NOF) and 1,14-diamino-3,6,9,12-tetraoxatetradecane (amino-PEG4-amine; Sigma-Aldrich) were separately dissolved in anhydrous acetonitrile containing 100 mM acetic acid. Tetra-PEG-NHS (12 mM) and amino-PEG4-amine (24 mM) solutions were mixed together using a centrifugal mixer (AR-100, THINKY). The molar concentration of amino-PEG4-amine was twice that of tetra-PEG-NHS to match the molar concentrations of the amine and NHS groups. The mixed solution was immediately filtered through a fine syringe filter with a pore size of 20 nm (FD5AXFR12, Entegris) into a clean glass tube with a diameter of 10 mm, which was then sealed tightly with parafilm. Acetic acid was added to acetonitrile to tune the rate of the reaction between the NHS and amine groups so that the gel point would be approximately 1 hour after mixing tetra-PEG-NHS and amino-PEG4-amine. All the preparation procedures were performed at room temperature (approximately 25°C).

Preparation of the conventional gel using NIPAM as the monomer

NIPAM (4 mmol) and N,N′-methylenebis(acrylamide) (BIS; 0.08 mmol) were dissolved in 10 ml of water. Argon gas was bubbled through the NIPAM/BIS solution for 20 min to remove dissolved oxygen from the solution. Subsequently, ammonium persulfate (0.008 mmol) and N,N,N′,N′-tetramethyl-ethylenediamine (0.04 mmol) were added to the solution. The mixed solution was filtered through a 20-nm syringe filter (FD5AXFR12, Entegris) into a clean glass tube with a diameter of 10 mm, which was then sealed tightly with parafilm. All the preparation procedures were performed at 25°C.

Dynamic viscoelastic measurements

Dynamic viscoelastic measurements of the star polymer gel gelation process were conducted using a rheometer (MCR 501, Anton-Paar). The temperature, shear strain, and shear frequency were 25°C, 2%, and 1 Hz, respectively.

1D static light scattering

SLS was performed on a commercial light scattering instrument (ALV-5000, ALV) with a He-Ne laser (wavelength λ = 632.8 nm) as the incident light source. Scattered light was collected by a single-pixel avalanche photodiode detector mounted on a goniometer with the scattering angle 2θ in the range of 30 to 150°, with 30-s exposure time at each angle. The scattering intensity was corrected for the scattering volume at each scattering angle and for the solvent scattering and lastly plotted as a function of the magnitude of the scattering vector q = 4πnsinθ/λ, where n is the refractive index of the solvent. Toluene was used to scale the observed photon counts to excess Rayleigh ratio. Then, the excess Rayleigh ratios of sol and gel samples were equally scaled by a factor of 1.2 × 104 to match the scattered intensity (differential scattering cross section) from SAXS at the low-q limit. All the experiments were performed at 25°C.

Small-angle x-ray scattering

SAXS measurements were carried out using a lab-source small-angle instrument (SAXSpoint 2.0, Anton-Paar). The sample contained in a sealed planar cell with two 30-μm-thick glass windows was placed in a vacuum chamber and illuminated with a microfocused x-ray from a Cu-Kα source with a wavelength λ of 1.542 Å. The beam diameter at sample is approximately 1 mm. The sample-to-detector distance was 812 mm. The scattered x-ray was collected by a 2D hybrid pixel detector (2D Eiger R 1M, Dectris). The exposure time per sample was 24 hours. The scattered intensity was circular-averaged to obtain 1D intensity profile and then corrected for incident beam flux, sample absorption and thickness, exposure time, and cell and solvent scattering using a custom-made data reduction package Red2D ( on a scientific data analysis software (Igor Pro 8, WaveMetrics). The intensity was plotted as a function of the magnitude of the scattering vector q. Glassy carbon (National Institute of Standards and Technology, USA) and silver behenate standards (Nagara Science, Japan) were used to calibrate the absolute intensity (differential scattering cross section) and the accurate sample-to-detector distance, respectively. All measurements were performed at 25°C.

2D static light scattering with a microscope

The glass tubes filled with pregel solution were placed on the stage of an inverted microscope (ECLIPSE Ti-U, Nikon). A coherent laser beam with a wavelength of 488 nm and a diameter of 0.5 mm (OBIS 488 LS, Coherent) was directed onto the glass tube. The scattered light was collected via a 10× objective lens (TU Plan Fluor EPI, Nikon) and recorded by an EMCCD camera (iXon3, Andor Technology). The measurements were performed at room temperature (approximately 25°C).

Time-resolved DLS

Time-resolved DLS was carried out during the gelation process on the same light scattering instrument (ALV-5000, ALV) described in the “1D static light scattering” section. The scattering angle was fixed at 90°. Time-resolved measurements were performed throughout the gelation process with 30-s data collection per frame. All measurements were performed at 25°C.

DLS at multiple sample positions

The same light scattering apparatus as described above was used. All the samples were measured at 100 different positions for 300 s per position using an automatic rotation and up-down motion unit. All measurements were performed at 25°C.


Supplementary material for this article is available at

Section S1. Comparison of the obtained elastic modulus with the phantom network model

Section S2. Computation of ensemble-averaged time correlation function

Fig. S1. 2D schematic of the site percolation and bond percolation models at different cross-linking extent.

Fig. S2. Time correlation function (g2) of the star polymer(tetra-PEG-NHS) sol in water and acetonitrile without the addition of the cross-linker.

Fig. S3. Photographs of the optical image capture system for 2D scattered light from the polymer solutions and gels.

Movie S1. Time-lapse optical images of the scattering intensity patterns from the star polymer solution/gel system and the conventional poly-NIPAM (PNIPAM) solution/gel system during gelation.

Movie S2. A comparison of time-resolved DLS and time-resolved dynamic viscoelasticity data of the star polymer solution and gel.

References (4144)

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: We thank T. Sakai (University of Tokyo) for the valuable advice on manuscript preparation. Funding: This work was supported by JSPS KAKENHI grant numbers JP16H02277, JP17K14536, and JP19K15628. Author contributions: X.L. conceived the idea. X.L., S.N., and M.S. designed the experiments. Y.T. prepared the samples and performed the rheological, 1D static light scattering, and DLS measurements. X.L., S.N., Y.T., and N.W. performed the 2D static light scattering experiments. S.N. contributed to data analysis and figure preparation. X.L. wrote the manuscript with extensive help from S.N. and M.S. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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