Rheological Isotope Effects for Molecular Insight in Covalent Adaptable Networks

Abstract

Understanding how small-molecule reactivity translates to bulk properties remains a fundamental challenge in polymer science. It is widely accepted that the viscoelastic behavior of covalent adaptable networks (CANs) is related to the kinetics of dynamic covalent bond exchange, but the nature of this relationship is complicated by the network environment. However, there are a lack of molecular tools to study exchange directly in polymer networks. In this work, we establish rheological isotope effects (RIEs) as a new tool for studying dynamic covalent exchange in bulk networks. By swelling dithioalkylidene-based CANs in H₂O or D₂O, we perturb the thiol/thiolate equilibrium to favor the reactive thiolate in D₂O, accelerating exchange and thus stress relaxation rates. We found that bulk networks composed of thiol-terminated, multi-arm poly(ethylene glycol) cross-linked by dithioalkylidenes exhibit a RIE between 0.31 and 0.64 for stress relaxation, depending on the cross-linker structure, cross-link density, and polymer topology. Monte Carlo simulations of network formation and evolution indicate that differences in RIE magnitude likely arise from topological defects including bridging chains. These defects reduce effective network size, decreasing the dependence of stress relaxation on bond exchange. We believe RIEs present new opportunities to study mechanisms of exchange in bulk networks, deconvolute modes of stress relaxation in complex CANs, and provide insight into the impact of defects on viscoelasticity.

Graphical Abstract

Introduction

Covalent adaptable networks (CANs) that exchange via an associative mechanism maintain their network integrity during topological rearrangement and network self-healing.^1,2 The simultaneously solid and liquid-like behaviors of these viscoelastic materials arise from a combination of molecular exchange, polymer structure, and network architecture.^3–7 Molecular control over the bulk viscoelastic properties of CANs is central to the design and synthesis of new materials.⁸ Yet, there are few experimental mechanistic tools available to connect the molecular behavior of a dynamic network to its bulk physical properties.⁹ With CANs becoming increasingly complex in recent years, including networks with multiple dynamic bonds and complex strand architectures,^10–12 molecular tools to characterize and tune the temporal hierarchy¹³ of CANs will provide the insight needed to design tailored dynamic materials.

Numerous reports detail molecular designs based on the translation of small-molecule insights to macromolecular CAN behavior.¹⁴ For example, Smulders and coworkers reported imine CANs with stress relaxation timescales dictated by Hammett parameters of the bis-aniline crosslinks.¹⁵ Winne and Du Prez have extensively explored the impact of internal catalysis on CAN viscoelasticity.¹⁶ Our lab systematically tuned acceptor structure to modulate conjugate addition–elimination rate.^17,18 However, we and other experimentalists largely rely on ex situ studies of the molecular exchange reaction and the assumption that the mechanism remains unchanged when embedded inside a polymer matrix. This assumption can fail when the polymer matrix¹⁹ or deformations^20,21 alter exchange reactivity, or when non-exchange processes dominate the relaxation behavior.^22–25 Unfortunately, there is a dearth of methods to directly study molecular exchange within the polymer matrix. CANs have been studied with broadband dielectric spectroscopy, but assignment of the relaxation modes remains challenging, and the terminal regime may be outside the accessible frequency range.^26–29,25 Dissociative dynamic bonds exhibit distinct bound and unbound states that can be monitored in situ using FTIR,^30,31 but from a spectroscopic perspective, associative exchange is degenerate without observable intermediates. While fluorophores have been incorporated into CANs to report on the state of the network,^32,33 the addition of these large π-conjugated groups could impact relaxation pathways. To address this absence of mechanistic tools for associative CANs, we introduce rheological isotope effects as a method to probe dynamic exchange directly in bulk networks.

Isotope effects are a time-honored technique for studying reaction mechanisms with minimal structural perturbation.³⁴ Most commonly applied in small-molecule mechanistic investigations, kinetic isotope effects (KIEs) provide information about the rate-limiting step of a reaction based on kinetic differences arising from the slower vibrational frequencies of the heavier atoms. Equilibrium isotope effects (EIEs), though less frequently employed, provide mechanistic insight by perturbing equilibria prior to the rate-limiting step of a reaction. A solvent isotope effect (SIE) arises from a combination of both kinetic and equilibrium isotope effects, most often originating in the rapid transfer of an exchangeable proton with an isotopically labeled solvent such as proton/deuterium exchange in D₂O (Figure 1).³⁵ Many studies of enzyme-catalyzed reactions, including those involving cysteine residues, metal-bound water, low-barrier hydrogen bonds, and exchangeable protons, employ SIEs to glean mechanistic information.³⁶ Interestingly, reactions involving thiol nucleophiles such as cysteine are often faster in D₂O than in H₂O (i.e., an inverse solvent isotope effect). This effect originates in the thiol-thiolate equilibrium, which favors the thiolate more in D₂O than in H₂O by a factor of ~2. This difference in the equilibria is referred to as the fractionation factor, or ratio of affected species in deutero vs. protio solvents.

Figure 1.

Proposed origin of the SIE in thiol-thiolate exchange: reaction coordinate diagram depicting the rapid equilibrium of thiol and thiolate in water and deuterated water prior to a rate-limiting step. Horizontal lines represent zero-point energy (ZPE) levels.

In the context of polymer chemistry, isotopic substitution has been used to improve phase contrast in neutron scattering experiments, influence thermodynamic interactions and thermomechanical properties, and improve the stability of organic electronics.³⁷ KIEs have been sparingly employed to elucidate cross-linking in static polymer networks³⁸ and polymerization mechanisms in linear polymers.^39–41 For example, Richtering and coworkers demonstrated dramatic effects of side chain vs. vinyl deuteration in the self-cross-linking of ultra-low-cross-linked poly(N-isopropylacrylamide) (pNIPAM).³⁸ Selective deuteration on the isopropyl group of NIPAM inhibited network formation, while deuteration of the vinyl group allowed cross-linking. These results, alongside ab initio calculations and other experimental data, indicated that self-cross-linking of NIPAM proceeded primarily through hydrogen atom transfer from the tertiary carbon of the isopropyl side chains rather than from the backbone. This study illustrates the power of isotope effects to provide mechanistic insight in polymer networks.

To the best of our knowledge, isotope effects on the mechanical properties of CANs, or rheological isotope effects (RIEs), have not been explored previously. Herein, we report RIEs as a mechanistic tool to probe dynamic exchange in dithioalkylidene-based CANs. Dithioalkylidenes are an emerging class of dynamic covalent cross-linkers based on the reversible conjugate addition–elimination of thiols. Our lab previously established that the structure of the dithioalkylidene cross-linker markedly affects exchange kinetics and consequently the viscoelasticity of the corresponding polymer network¹⁷. Additionally, we determined that in aqueous media, the active nucleophile is primarily the thiolate, rather than the neutral thiol.¹⁸ We reasoned the large inverse fractionation factors of thiols in H₂O vs. D₂O, while primarily studied in enzyme kinetics, would also apply to swollen polymer networks. By perturbing the ratio of thiol/thiolate in CANs via the SIE, we hypothesized that the higher equilibrium concentration of reactive thiolate in D₂O should afford faster exchange kinetics. The faster exchange is expected to translate to accelerated stress relaxation as measured by shear rheology. Because structural and environmental factors that impact exchange are already well established for the dithioalkylidene dynamic bond, it presents an ideal model system for a proof-of-concept demonstration of RIEs in dynamic networks.

Experimental Section

General procedures.

Unless otherwise noted, reactions were performed under N₂ atmosphere in oven-dried (150 °C) glassware. Reaction progress was monitored by thin layer chromatography (Merck silica gel 60 F254 plates), visualizing with fluorescence quenching, KMnO₄, or ninhydrin stains. Automated column chromatography was performed using SiliCycle SiliaFlash F60 (40–63 μm, 60 Å) in SNAP cartridges on a Biotage Isolera One. Organic solvents were removed in vacuo using a rotary evaporator (Büchi Rotovapor R-100, ~20–300 torr) and residual solvent was removed under high vacuum (<200 mtorr).

Nuclear magnetic resonance spectroscopy.

Proton nuclear magnetic resonance (¹H NMR) spectra and carbon nuclear magnetic resonance (¹³C NMR) spectra were recorded on a Bruker AVANCE-500 spectrometer at 500 MHz and 125 MHz, respectively, and referenced to the solvent residual peaks. NMR data are represented as follows: chemical shift (δ ppm), multiplicity (s = singlet, d = doublet, t = triplet, q = quartet, m = multiplet), coupling constant in Hertz (Hz) and integration. MNova (Mestrelab research) was used to analyze NMR data.

Liquid chromatography–mass spectrometry (LCMS).

LCMS kinetics experiments was performed on a liquid chromatography mass spectrometry system (Agilent 6120 Quadrupole LC/MS) equipped with a Poroshell 120 eC-C18 2.7 μM column (2.1 × 50 mm) and a variable-wavelength detector. HPLC grade acetonitrile and water, both with 0.1% formic acid, were used as the mobile phase at a flow rate of 0.8 mL/min. A gradient of 5–100% acetonitrile in water over 5 minutes was used to separate the compounds. High-resolution mass spectra were recorded on an Agilent 6545 Q-TOF using electron spray ionization (ESI).

Rheometry.

Mechanical properties of gels were measured with a TA HR 30 Rheometer with a 12-mm parallel plate attachment. Following sample loading, excess was trimmed to prevent errors from underfilling or overfilling. Silicone oil was applied around the edge of the sample to minimize solvent evaporation during measurement. Temperature was maintained at 25 °C using the Advanced Peltier Plate attachment. All measurements were performed in triplicate on independently prepared samples.

Amplitude sweeps were conducted from 0.1–10% shear strain with an angular frequency of 25 rad/s to confirm samples lay within the linear viscoelastic regime. Frequency sweeps were performed with 5% strain at a frequency range of 100 to 0.1 rad/s. Stress relaxation profiles were collected using a 5% step strain. Characteristic relaxation times were calculated by fitting stress relaxation curves to a one-element (IND and control) or two-element (AA) Maxwell model. For all stress relaxation plots, the data before 0.3 s was trimmed and the remaining data normalized to the modulus at 0.3 s to omit noise present early in the measurement.

Materials.

Commercial reagents were purchased from Sigma-Aldrich, Alfa Aesar, or TCI and used as received. PEG-SH (4-arm/5 kDa and 8-arm/20 kDa) was purchased from JenKem Technology. The synthesis of ET, BT, and crosslinkers, and corresponding NMR spectra, are reported in the Supporting Information (SI, Sections I and VII).

Small-molecule kinetics.

Stock solutions of the cross-linker (AA_ET at 1.0 M or IND_ET at 10 mM), BT (10 mM), and umbelliferone as an internal standard (3–6 mg/mL) were prepared in acetonitrile. To an LCMS vial were added HEPES buffer in H₂O or D₂O, (500 μL, 0.1 M, pH 7.4), the cross-linker solution (50 μL), and the internal standard solution (50 μL). The mixture was diluted with acetonitrile such that the total volume would reach 1000 μL following the addition of BT in the subsequent step, and the initial ratio was analyzed by LCMS. The BT solution (6.25–50 μL of 1.0 M BT or 100 μL of 10 mM BT) was added and the samples were subjected to LCMS analysis, removing aliquots every 7 minutes. The concentration of AA_ET and IND_ET were determined by the relative ratio of the integration of the starting material absorbance peak to the integration of the internal standard at 360 nm. For more details and data, see the SI, Section III.

Network fabrication.

Stock solutions of 4-arm or 8-arm PEG-SH (20 w/v%), cross-linker (0.1 M), and eosin Y (0.4 M) were prepared in dichloromethane (DCM). Appropriate concentrations of each solution were added to a shell vial with a 12-mm internal diameter. The samples were briefly sonicated, and irradiated with a green light source (Kessil PR160L-525 nm; reported average intensity 399 mW/cm² measured from 1 cm distance) for 5–12 minutes to reach a target stiffness. Irradiation times vary due to differences in sample absorption between formulations and slight changes to irradiation conditions. The samples were then dried in vacuo at room temperature to remove residual DCM, and rehydrated with HEPES buffer in H₂O or D₂O (200 μL, 0.1 M, pH 7.4). Gels were allowed to equilibrate in capped vials for 3 h prior to mechanical characterization. For details, see the SI, Section IV. For all replicates of rheological characterization, see Section V.

Results

First, we established a small-molecule system to study the thiol exchange kinetics for model dithioalkylidene derivatives using liquid chromatography-mass spectrometry (LC-MS) (Figure 2a–b). Under pseudo-first order conditions (i.e. a large excess of thiol relative to substrate⁴²), acetylacetone-based derivative AA_ET was subjected to exchange with thiol BT in a 1:1 solution of acetonitrile to H₂O or D₂O HEPES buffer (0.1 M, pH 7.4, see SI for pD to pH conversion). We monitored the consumption of AA_ET as it reacts with thiol BT to form AA_ET/BT and AA_BT over the course of two hours by UV-vis absorbance normalized to an internal standard (umbelliferone). We have previously developed a kinetic model for this non-degenerate exchange that allows us to calculate the exchange rate, k_ex.¹⁸ The stoichiometric ratio of BT to AA_ET ranged from 12.5 to 75 equivalents. Due to the large excess of deuterated solvent, we expected free thiol BT to undergo rapid proton-deuterium exchange in D₂O buffer, generating the deuterated thiol in situ. Under these conditions, we determined the slope of k_obs vs. [BT] to obtain second-order rate constants k_ex in H₂O and D₂O solvents of 1.13 × 10⁻³ and 3.68 × 10⁻³ M⁻¹s⁻¹, respectively (Figure 2c). This difference corresponds to an exchange rate acceleration in D₂O by a factor of three, affording a measured inverse SIE (k_H/k_D) of 0.31. The observed rate acceleration is consistent with the hypothesized shift in equilibrium to favor the more reactive thiolate in deuterated solvent.

Figure 2.

Small-molecule model system to monitor exchange kinetics of dithioalkylidene AA and IND in H₂O and D₂O. (a) Representative scheme of exchange reaction with AA. (b) General structure of AA and IND derivatives. (c) Determination of k_ex for AA under pseudo-first order conditions. (d) Determination of k_ex for IND under second-order conditions.

To confirm this acceleration persists across exchange timescales, we synthesized the cyclic 1,3-indanedione-based derivative IND_ET (Figure 2b). As previously reported, cyclic dithioalkylidene derivatives are known to undergo exchange with thiols at faster rates than acyclic acceptors due to resonance stabilization and restricted conformational freedom of the tetrahedral intermediate. The faster exchange rate necessitated the use of second-order reaction conditions. While pseudo-first order conditions containing a large excess of thiol were needed to capture exchange with AA_ET at relevant timescales, the reaction with IND_ET proceeded too quickly to monitor by LCMS under the same conditions. Therefore, we employed second order conditions with a 2:1 stoichiometric ratio of thiol to IND_ET (one equivalent for each reactive site of IND_ET). Lowering the concentration of thiol BT slowed the reaction rate such that we could capture the equilibration of the system. Using the same HEPES-buffered H₂O and D₂O solutions, we employed two equivalents of thiol BT to IND_ET. We monitored the consumption of IND_ET and determined k_ex values to be 0.16 and 0.52 M⁻¹s⁻¹ for H₂O and D₂O, respectively (Figure 2d). As with dithioalkylidene AA, we observe an inverse SIE of 0.28, or a ~3.6-fold acceleration in D₂O. The magnitude of these SIEs are consistent with literature reports for systems involving thiol nucleophiles.³⁶

Confident that dithioalkylidenes demonstrate SIEs for the small-molecule model system, we next sought to determine if the effect translates to bulk hydrogels. We hypothesized that bulk hydrogels would exhibit faster stress relaxation when swollen with D₂O than with H₂O because the enriched thiolate concentration promotes faster exchange. We tested this hypothesis in a series of dynamic networks with varied cross-linking density, polymer topology, and cross-linker structure.

All networks were fabricated from thiol-terminated multi-arm star poly(ethylene glycol) (PEG-SH) cross-linked with bifunctional, norbornene-bearing AA or IND derivatives (Figure 3a). The network fabrication protocol is based on our previous studies.¹⁸ The norbornene group engages in a photoinitiated thiol-ene reaction, cross-linking the networks when irradiated with 525 nm green light in the presence of eosin Y photocatalyst. Gelation was carried out in dichloromethane (DCM) to solubilize all components. After DCM removal in vacuo, gels were rehydrated in H₂O or D₂O HEPES buffer (0.1 M) at comparable pH and pD values. Networks are named using the following convention: gel-XL-x(y), where XL is the crosslinker, x is the number of arms in the PEG star, and y is the mol % of excess thiol relative to dithioalkylidene. Thus, gel-AA-4(50) is a 4-arm PEG crosslinked with AA at a ratio that provides 50% excess thiol. Networks were prepared with excess thiol to allow associative exchange. All samples were subjected to oscillatory shear rheology to characterize their viscoelastic behavior and stress relaxation measurements to quantify their response to step strain. We ensured gels within the same dataset have similar plateau moduli, and thus comparable cross-linking densities, and that all samples were tested within the linear viscoelastic regime (see SI, Figures S7–S12). The results of stress relaxation experiments are summarized in Table 1.

Figure 3.

(a) Cartoons depicting the various polymer matrices and relative cross-link density. (b) Non-dynamic cross-linker used for control gels. (c-f) Stress relaxation curves of networks swollen in H₂O (blue) or D₂O (pink) buffer for gel-AA-4(50) (c), gel-AA-4(10) (d), gel-AA-8(50) (e), and gel-IND-4(50) (f). (g) Frequency sweep data for gel-IND-4(50) displaying the crossover point between G’ (solid symbols) and G” (open symbols). (h) Stress relaxation curves for the control gel. Each experiment was performed in triplicate on independently prepared samples; the data shown are the average and the shadows indicate the standard deviation.

Table 1.

Summary of Stress Relaxation Data¹

Network	τ2 (s)		RIE	τ2 contribution2
	H2O	D2O		H2O	D2O
gel-AA-4(50)	400 ± 9	255 ± 13	0.64	0.94	0.91
gel-AA-4(10)	1796 ± 347	864 ± 62	0.48	0.92	0.91
gel-AA-8(50)	928 ± 14	350 ± 62	0.38	0.81	0.86
	ωc (rad/s)3
gel-IND-4(50)	0.75	2.4	0.31	-	-

¹Stress relaxation was measured at 5% strain for 3 independently prepared samples; mean and standard deviation are shown. Data were fit to a two-element Maxwell model; only the slow mode is shown.

²Contribution of the slow mode to stress relaxation based on a two-element Maxwell model, out of 1.

³Reported as the crossover frequency, ω_c, the point where G’ and G” intersect in the frequency sweep. Crossover frequency is related to characteristic stress relaxation time, τ, for the terminal relaxation mode by the relationship $\tau =1/{\omega }_c$.

We began by fabricating a set of hydrogels with AA cross-linker at a 2:1 molar ratio of thiol to norbornene. After irradiation, this provides cross-linked networks with 50% excess thiol dangling ends to participate in exchange. We normalize the stress relaxation data at 0.3 seconds to remove noise from experiment initiation and simplify visual comparison between samples. The resulting curves exhibit two relaxation modes: a fast, ~1 second mode, and another mode at longer times contributing the majority of stress relaxation (Figure 3c). These data were not well described by single-element Maxwell or stretched exponential models, so to account for both modes, we employed a two-element Maxwell model to calculate the characteristic relaxation times (τ). The fast mode, τ₁, was similar for H₂O and D₂O (no statistically significant difference). However, the second mode, τ₂, exhibits statistically significant differences, on average 400 and 255 s for networks swollen with H₂O and D₂O, respectively.

While molecular isotope effects are expressed as k_H/k_D, the characteristic timescale of network relaxation provided by stress relaxation measurements should be inversely related to the rate of molecular exchange. Therefore, for the RIE, we define the ratio as ${\tau }_{\text{D}}/{\tau }_{\text{H}}$ to enable comparison with the molecular SIE. The ratio between slow relaxation modes (τ₂) for D₂O gels compared to H₂O gels corresponds to an inverse RIE of 0.64, or approximately a 1.6-fold acceleration in stress relaxation in D₂O. Both the SIE and the RIE are inverse, suggesting that τ₂ can be assigned to bond exchange at the cross-linker. However, the magnitude of the isotope effect is less (i.e., closer to 1) for the RIE compared to the SIE.

We next explored the impact of cross-link density on the RIE. Apart from slowing stress relaxation rates overall due to the lower population of free thiol, we were curious if magnitude of the SIE would scale with the population of exchangeable groups. For more highly cross-linked networks with 10% free thiol (gel-AA-4(10)), we again observed two stress relaxation modes (Figure 3d). The first mode at ~1 s did not differ significantly between H₂O and D₂O. The second mode, τ₂, exhibited characteristic stress relaxation times of 1796 and 864 s for H₂O and D₂O, respectively, corresponding to a RIE of 0.48 (~2-fold rate acceleration in D₂O).

To measure the RIE with a different polymer topology, we fabricated networks with 8-arm PEG-SH (20 kg/mol) cross-linked by AA with 50% free thiol (gel-AA-8(50)). The molar concentration of free thiol and weight percent of polymer (10 wt%) in the samples were kept consistent between networks fabricated with 4-arm and 8-arm PEG-SH. As in the previous experiments, we observed two modes of stress relaxation (Figure 3e), with only τ₂ demonstrating statistically significant differences. The relaxation time was 928 and 350 s for H₂O and D₂O, respectively, corresponding to a RIE of 0.38, or a 2.6-fold acceleration in D₂O.

As further confirmation, we sought to determine if the RIE was observed when the cross-linker identity was varied. Although changing the cross-linker can alter the rate of stress relaxation, we expected that the RIE should persist if the mechanism still relies on nucleophilic attack of the thiolate. Using the cyclic cross-linker IND, we fabricated networks with 4-arm PEG-SH (gel-IND-4(50)). Consistent with the faster exchange kinetics observed in the small-molecule study, these networks were found to relax stress significantly faster than AA-based networks (Figure 3f). With the stress relaxation data normalized at 0.3 seconds, the data could be fit to a single-element Maxwell model, and characteristic relaxation times were determined to be 1.11 and 0.41 s for H₂O and D₂O, respectively, yielding a RIE of 0.37. Given the fast timescale, we were able to corroborate these results with shear oscillatory rheology, and the frequency sweeps may be fit by a two-element Maxwell model (Figure S11). For both H₂O and D₂O gels, frequency sweeps revealed crossovers between G’ and G” at 0.75 and 2.4 rad/s, respectively (Figure 3g). Since these crossover points are reported in inverse time, like rate, the RIE is calculated from ${\omega }_{\text{c},\text{H}}/{\omega }_{\text{c},\text{D}}$ to be 0.31, representing a ~3.2-fold acceleration in D₂O. We believe the small discrepancy between the RIEs determined from stress relaxation and frequency sweep arises from insufficient data in the stress relaxation spectra below 0.3 s, limiting the accuracy of this measurement for the fast relaxation event.

Finally, we synthesized a non-exchanging control cross-linker consisting of a short ethylene glycol chain to evaluate RIEs in the absence of an exchange reaction. We used this crosslinker to prepare control networks composed of thiol-terminated 4-arm PEG with 10% free thiol. These networks relax a portion of their stress, albeit slowly, and we fit the data to a single-mode Maxwell model to allow calculation of the characteristic relaxation times, which exceeded the experimentally accessible range. We did not observe a statistically significant difference in this characteristic time (3616 ± 375 s) for control networks in H₂O vs. D₂O (Figure 3h). The lack of isotope effects for the control system indicates that the modulation in characteristic stress relaxation time for the CANs is a product of the SIE and not other factors like viscosity. We hypothesize that the observed stress relaxation in the absence of cross-link exchange arises from network defects and incomplete percolation of permanent bonds through the network.

To further analyze the stress relaxation behavior of the dynamic networks, we calculated their continuous relaxation spectra (CRS). While our previous Maxwell fitting analyses employed a two-element model, CRS, also called infinity spectra, represent an infinite parallel series of Maxwell elements. Infinity spectra often make it easier to visually identify distinct stress relaxation modes, as each mode appears as a peak. The position of the peak maximum provides the characteristic relaxation time for a given relaxation mode, and the relative area under each peak is related to each mode’s contribution to total relaxation. We used pyReSpect, a python program published by Shanbhag,⁴³ to calculate the CRS. We note that due to the nature of the pyReSpect fitting protocol, all CRS include datapoints at earlier times than those used for fitting the stress relaxation data to Maxwell models.

The CRS calculated from our stress relaxation curves largely corroborate our previous fitting with two-element Maxwell models. All AA-cross-linked samples exhibit two distinct peaks corresponding to two modes of stress relaxation. The spectra for gel-AA-4(50) and gel-AA-8(50) are displayed in Figure 4a and 4b, respectively. Much like our earlier Maxwell model analyses, the first, minor peak corresponds to a sub-1 s fast relaxation mode and the second, major peak to a slower relaxation mode. For gel-IND-4(50), infinity spectra reveal two fast, overlapping relaxation modes (Figure S16). In all cases, only the slower mode is significantly impacted by the SIE. Therefore, we can ascribe this slow relaxation mode to cross-link exchange.

Figure 4.

Continuous relaxation spectra (average of triplicate data, standard deviation shaded) for (a) gel-AA-4(50) and (b) gel-AA-8(50).

Discussion

While all the dynamic networks exhibited a RIE, the magnitude of the isotope effect varied and was closer to 1 than the small-molecule SIE. This difference between RIEs suggest that the network environment also influences the timescale of this relaxation. For the AA series, the isotope effects follow the trend small-molecule AA < gel-AA-8(50) < gel-AA-4(10) < gel-AA-4(50) < 1. To rationalize these differences and understand the evolution of network structure during gelation and dynamic exchange, we turned to modeling and simulations.

We employed a mean-field model of gelation based on an infinite Bathe lattice⁴⁴ to examine the gel topology. Within this framework, molecules are categorized into four groups depending on their network connectivity, including sol (unconnected), dangling (one-arm linked), bridging (two-arm linked), and cross-linking (≥3-arm linked) molecules (Figure S22).^45,46 Elastically active chains that are capable of storing elastic energy under deformation can form directly between two cross-linking molecules or incorporate bridging molecules to extend their contour length. Using the extent of reaction (i.e., the conversion of thiol), we quantified the fraction of molecules in each category (Figure S23). Our results reveal that gel-AA-4(10) contains the lowest fraction of elastically inactive molecules (i.e., sol, dangling, and bridging molecules), followed by gel-AA-8(50) and gel-AA-4(50). According to classic theories like the affine network model, this hierarchy implies that gel-AA-4(10) should exhibit the largest shear modulus, in agreement with our experimental data (Table S7). The fraction of bridging molecules increases from 4.9% (gel-AA-4(10)) to 11.3% (gel-AA-8(50)) and 33.4% (gel-AA-4(50)), indicating that elastic strands in gel-AA-4(50) often include series of bridging chains. Additionally, we found that 23.6% of molecules in gel-AA-4(50) exist as sol (i.e., isolated clusters), whereas this fraction decreases to 0.1 % in gel-AA-4(10). The presence of these isolated clusters in gel-AA-4(50) likely contributes to the observed fast (~1 second) relaxation mode, as small clusters can relax more readily than those integrated into the network.⁴⁷ We expect Rouse-type relaxations, with timescales of 10⁻⁶–10⁻⁴ s for related systems, to fall beyond our experimental resolution.⁴⁸ Furthermore, we do not believe reptation and contour length fluctuation are relevant relaxation modes in this system as each arm of the star polymer is below the entanglement molecular weight of PEG, and the volume fraction of the gel is significantly lower than the entanglement concentration.^44,45 By ruling out other possibilities, we tentatively assign the fast mode to defect-mediated relaxation, including dangling ends, loops, and isolated clusters.

We note that the mean-field model assumes the absence of loops. Cross-linkers that form intramolecular linkages with 2 arms of a single PEG multi-arm polymer are primary loop defects, which decrease connectivity and are elastically ineffective.⁴⁹ For static networks, Johnson and Olsen have shown that primary loop fractions can be used to determine the concentrations of higher-order loop defects.^49,50 To gain a more detailed understanding of the network topology, we adopted the Monte Carlo (MC) algorithm originally developed by Stepto⁵¹ and later widely applied to study gel topology.^52,53 In this method, the polymer network is represented as a graph, where initially isolated nodes are gradually connected by edges to simulate the gelation process. We extend this method to consider dynamic bond exchange by implementing edge swap moves. For details, equations, and parameters, readers are referred to the original work and SI.

While SIEs are based on the contributions of all chemical reactions in solution, the rheological response is based on the concentration of elastically active chains. Within the network, not all exchange reactions will occur within elastically active chains. We used the topologies obtained from MC simulations to track the populations of these defects and characterize the effective network sizes (Figure S24). We obtain the effective network sizes by three stages of postprocessing: extract the largest connected component, prune loops up to the fourth order, and subsequently merge the bridging molecules (Figure 5).

Figure 5.

(a) Simulated effective network sizes for gel-AA-4(50), gel-AA-4(10), and gel-AA-8(50). (b) Fraction of post-processed effective network size relative to the total network size. The gray section represents the unprocessed network size. The size of the largest network following the removal of free molecules (sol) component is shown in green. The size of the remaining network after the removal of free molecules, loops, and dangling ends is shown in pink, and after the merging of bridging chains is shown in purple. Error bars indicate standard deviation between five independent simulations.

Unlike networks that exchange by dissociative mechanisms, associative bond exchange imparts a constant network size, yet the size of its elastically active component fluctuates. The simulations suggest that when dynamic exchange is enabled, the number of loop defects decreases, reaching an equilibrium (Figure S24). The ability of dynamic bonds to perform topological “error correction” in CANs was recently proposed by Ricarte and coworkers.⁶ However, some proportion of elastically ineffective defects persist even after equilibration. In the simulation, we find that when free molecules (sol) and loops are taken into account, the effective network size is decreased by 18% in gel-AA-4(50), and 6–7% percent in gel-AA-4(10) and gel-AA-8(50). Sol fragments contribute to 14% the total network in gel-AA-4(50) and less than 1% in gel-AA-4(10), and gel-AA-8(50). In the case of gel-AA-4(50), merging bridging molecules also leads to significant reduction in effective network size, indicating that it contains series of bridging chains, in agreement with the mean-field estimation. Overall, when only accounting for elastically active chains, the effective network sizes are 29%, 87%, and 86% of the total network composition for gel-AA-4(50), gel-AA-4(10), and gel-AA-8(50), respectively (Figure 5b).

We propose that the trend in experimental RIEs may be rationalized by the “damping” effect of bridging chains and loop defects. As illustrated in Figure S22, a bond exchange reaction that happens at any cross-link site along a bridging chain is degenerate with respect to network elasticity. In other words, as the fraction of bridging chains increases, the slow stress relaxation mode becomes less dependent on the cross-link exchange rate and more dependent on isotope-insensitive processes, such as slow chain dynamics.²⁹ Analogous damped rheological responses due to defects also emerged in Olvera de la Cruz’s previous study of vitrimers, in which varied backbone architectures resulted in different defect levels.⁵⁴ Using hybrid molecular dynamics-Monte Carlo simulations, it was found that an elevation in exchange energy barrier leads to more pronounced dynamic constraints in the low-defect uniform and random sequenced vitrimers than in the high-defect block and gradient sequenced vitrimers. While the simulations would suggest a similar RIE for gel-AA-4(10) and gel-AA-8(50) based on effective network size alone, other factors, such as finite simulation size, crosslink heterogeneity, and swelling heterogeneity, could lead to some deviation between simulations and experiments.

We note that the interpretation of RIE damping due to bridging chains is most applicable when bond exchange is the primary and rate-limiting mechanism for stress relaxation. In other words, it assumes segmental motion occurs much faster than the exchange reaction occurs. This assumption remains valid for AA-based gels, where there exists a clear separation between the fast relaxation mode associated with defects and the slow relaxation mode corresponding to bond exchange. However, the slow and fast relaxation modes are convoluted for gel-IND-4(50), as suggested by the infinity spectra (Figure S16). We believe the similarity between SIE and RIE for gel-IND-4(50) may arise from coupling between exchange reactions and incomplete segmental or diffusive relaxation of clusters within the network because these processes are occurring on similar timescales. Based on the calculated proportions of defects in each network architecture, we might expect to observe more significant differences in the timescale and magnitude of the fast relaxation mode; however, these differences likely cannot be resolved by stress relaxation measurements.⁵⁵

In this system, we can rule out alternative explanations often used to account for differences between small-molecule and network reactivity. Du Prez and coworkers observed significantly different flow activation energies and relaxation times when varying the polymer backbone and molecular weight of a vinylogous urethane vitrimer.¹⁹ Some of those differences can be ascribed to the polarity of the polymer matrix; such a medium effect should not be relevant in our study because both small-molecule and network experiments were performed in an aqueous environment. In polystyrene vitrimers, Ricarte and coworkers proposed that deviation from the sticky Rouse model could be explained by cross-linker diffusion or local elasticity fluctuations.⁶ If cross-linker diffusion were the dominant effect on dampened RIEs, we would expect that networks with more defects would allow more facile diffusion,⁵⁶ and thus more behave more similarly to the small-molecule model system. Instead, we observe the opposite. At this stage, we cannot rule out the effect of local elasticity fluctuations on the RIE.

Conclusions

In this work, we show that isotope effects, a historically important tool for studying small-molecule and enzymatic reaction mechanisms, can be extended to CANs. We translate a large inverse SIE observed in a small-molecule system to the bulk stress relaxation and viscoelasticity of swollen associative networks. The observation of the same inverse isotope effect in the slow relaxation mode of a CAN allows us to rigorously assign this mode to chemical exchange. While this assignment may seem trivial in a simple system, with this proof of concept, RIEs could be applied to more complex systems. There is a growing interest in CANs with multiple dynamic interactions, where the relaxation modes are more challenging to assign.^22,57–65 Currently, these assignments are often based on flow activation energies. However, even systems with a single dynamic bond can exhibit discrepancies between the small-molecule and flow activation energies.²⁰ In such cases, Torkelson assigned the slow relaxation mode in aromatic disulfide and thiourethane CANs to backbone $\alpha$-relaxation rather than bond exchange.^23,24

For CANs where isotopic substitution can impact the rate of exchange, such as mechanisms involving proton transfer prior to or in the rate-determining step, RIE experiments represent a new tool to molecularly assign macroscopic rheological events. Examples of potentially relevant exchange chemistry include transesterification, vinylogous urethane exchange, silyl ether exchange, Michael addition/elimination, and urea/urethane dissociation. The chemical information RIEs provide about bulk networks is complementary to other recommended techniques like time-temperature superposition.^66–68,55 Our initial study was conducted with swollen networks; in principle, H/D exchange could also be conducted in dry networks to allow the application of RIE experiments to vitrimers. RIEs could differentiate between mechanisms in systems with multiple exchange pathways, such as boronic ester transesterification (involving protons) vs. metathesis (no protons). Moreover, trends in the RIE, coupled with Monte Carlo simulations, provide insight into the impact of network defects on the slow relaxation mode. This work motivates further investigation into network defects in associative networks.

Supplementary Material

Supporting Information. A listing of the contents of each file supplied as Supporting Information should be included. For instructions on what should be included in the Supporting Information as well as how to prepare this material for publications, refer to the journal’s Instructions for Authors.

The following files are available free of charge.

Materials and methods; synthetic procedures; NMR spectra; conversion of pD to pH; kinetic data; network fabrication procedures; rheological data; modeling and simulation details and data (PDF)