Entropy-Controlled Cross-Linking in Linker-Mediated Vitrimers

Qun-Li Leia,1, Xiuyang Xiaa,b,1, Juan Yangc, Massimo Pica Ciamarrab, and Ran Nia,2

aSchool of Chemical and Biomedical Engineering, Nanyang Technological University, 62 Nanyang Drive, 637459, Singapore; bDivision of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore; cDepartment of Chemistry, National University of Singapore, 117546 Singapore

This manuscript was compiled on October 23, 2020

Recently developed linker-mediated vitrimers based on metathesis mean field theory combined with coarse-grained computer sim- of dioxaborolanes with various commercially available ulations to study the linker-mediated vitrimer system, and our have shown both good processability and outstanding performance, results show that the entropy of free linkers plays a nontrivial such as mechanical, thermal, and chemical resistance, suggesting role. We find that with increasing the concentration of free new ways of processing cross-linked polymers in industry, of which linkers, the vitrimer system undergoes a reentrant gel-sol tran- the design principle remains unknown [M. Röttger, et al., Science sition, which was observed in a recent experiment (22). More 356, 62 (2017)]. Here we formulate a theoretical framework to eluci- interestingly, even at the low temperature limit, the cross- date the phase behaviour of the linker-mediated vitrimers, in which linking degree of vitrimers still depends on the concentration entropy plays a governing role. We find that with increasing the of free linkers, which essentially offers an extra degree of free- linker concentration, vitrimers undergo a reentrant gel-sol transition, dom in controlling the mechanical property of the resulting which explains a recent experiment [S. Wu, H. Yang, S. Huang, Q. materials. Chen, Macromolecules 53, 1180 (2020)]. More intriguingly, at the low temperature limit, the linker concentration still determines the cross- Results linking degree of the vitrimers, which originates from the competi- tion between the conformational entropy of polymers and the trans- Coarse-Grained Model of Vitrimer. We consider a system of lational entropy of linkers. Our theoretical predictions agree quantita- volume V consisting of Npoly chains, in which each tively with computer simulations, and offer guidelines in understand- polymer comprises of n hard spheres of diameter σ. As shown ing and controlling the properties of this newly developed vitrimer in Fig.1a, on each polymer there are m  1 precursors system. (reactive sites) P uniformly distributed, which can react with a cross-linker molecule C by forming a dangling PC bond and vitrimer | metathesis reaction | reentrant gel-sol transition | entropy- producing a byproduct free molecule B through metathesis driven cross-linking reactions. Moreover, a dangling PC bond can further react with another intact precursor P to form a cross-linking P2C itrimers, a new type of polymeric materials, are known bond and producing an additional free B molecule, and each Vto exhibit unique properties that combine the advan- cross-linker can form at most two bonds with two different tages of thermosets and . To be specific, they precursors. The metathesis reactions are reversible, and ∆G are mechanically robust and insoluble while also recyclable is the reaction energy. NC and NB are the numbers of cross- and malleable (1, 2). At low temperatures, vitrimers be- linker molecule C and the byproduct molecule B in the system, have as cross-linked thermosets; while at high temperatures, which are controlled by the chemical potentials µC and µB, the exchangeable bonds in the polymer network swap re- respectively. We define nPC and nP2C as the average numbers versibly by the thermally-triggered reactionsDRAFT so that they of PC bonds with dangling cross-linkers and cross-linking P2C behave as viscoelastic liquids (3–12). In the past decade, various chemical reactions, such as the transesterification reac- tion (2, 13), transamination reaction (14, 15), alkoxyamine ex- Significance Statement change reaction (16, 17), olefin metathesis (18), thiol-disulfide The recently developed linker-mediated vitrimers based on exchange (19), have been used in the production of vitrimers metathesis reactions offer new possibilities of processing cross- for different applications. However, the reactants or catalysts linked polymers with high mechanical performance in industry, involved in reactions of the conventional vitrimers are usu- arXiv:2007.06807v2 [cond-mat.soft] 22 Oct 2020 while the design principle remains unknown. Here we propose ally not thermally or oxidatively stable, which is particularly a theoretical framework for describing the system of linker- detrimental for using the same equipments and conditions of mediated vitrimers, in which entropy is found to play a dictating processing thermoplastics (9). role. Our mean field theory agrees quantitatively with computer Recently, a new type of linker-mediated vitrimers was de- simulations, and provides guidelines for the rational design of veloped based on the metathesis reaction of dioxaborolanes, in the linker-mediated vitrimers with desired properties. which the functionalized polymers with pendant dioxaborolane

units react with bis-dioxaborolanes (cross-linkers), and the Author contributions: R.N. designed and directed the research; Q.-L.L. formulated the mean field metathesis reaction here both cross-links the polymers and theory; X.X. performed the computer simulations; and all authors discussed the results and wrote dynamically changes the polymer network (20). The linker- the manuscript. mediated vitrimers have superior chemical resistance and di- The authors declare no conflict of interest. 1 mensional stability, without the need of a catalyst, and can be Q.-L.L. and X.X. contributed equally to this work. processed like thermoplastics (21). In this work, we propose a 2To whom correspondence should be addressed. E-mail: [email protected]

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS | October 23, 2020 | vol. XXX | no. XX | 1–6 m where the first summation is on the ideal gas terms of polymer (a) (b) 1 βµ φp− 5 10 20 C chains, free cross-linkers and byproduct molecules, and the 8.0 1 − 10 6.0 second summation is on the ideal gas terms of different reactive − 4.0 sites in polymer blobs. Here β = 1/kB T with kB and T − 1 2.0 − the Boltzmann constant and the temperature of the system,

/K respectively, and Λ is the de Broglie wavelength. This ideal gas 2

K approximation of reactive sites offers a simple estimation of the 100 conformational entropy change during the formation of cross- (c) linking P2C bonds, for which we introduce ∆S as the entropy correction per cross-linking bond for any inaccuracy arising ex 3 2 1 from the ideal gas approximation. FHS is the excess free energy 10− 10− 10− φp based on Carnahan-Starling hard-sphere equation of state (24) (d) arising from the excluded volume interaction, which accounts for the crowding effect in the system. The last three terms arise from the bond formation in metathesis reactions (related with ∆G), and the exchange of molecules with reservoir (related with µB, µC). As the system approaches the dense regime (Fig.1d), polymer blobs begin to overlap and the distribution of reactive site become homogeneous. Thus, Vp can be replaced by the available volume per polymer, i.e., Vp = V/Npoly.

We define the cross-linking degree of system as fP2C = 2NP C/(Npolym), i.e., the fraction of reactive sites that are Fig. 1. Vitrimer model. (a): Illustration of the two step metathesis reactions in the 2 cross-linked, and fi = Ni/(Npolym), (i = PC, B, C). Us- vitrimer system. (b): K2/K1 as a function of φp for different m and µC at n = 100 and βµB = −3. (c,d): Illustration of the heterogeneous dilute (c) and homogeneous ing the saddle-point approximation, ∂F /∂fi = 0, (i = dense (d) systems of the vitrimers. PC, P2C, B, C), one can obtain the equilibrium fPC and fP2C as h p i2 bonds per polymer, respectively. Similarly, nP is the average −(1 + a) + (1 + a)2 + 4c number of precursors per polymer that remain intact, and f = , [3] P2C 4c Ni = Npolyni is the total number of precursors or bonds of p 2 type i = P, PC, P2C in the system. The packing fraction of −(1 + a) + (1 + a) + 4c π 3 fPC = , [4] polymers is φp = 6 Npolynσ , and B and C are both modelled 2c as hard spheres of diameter σ. with We employ Monte Carlo (MC) simulations to investigate this coarse-grained hard-sphere-chain system. An infinitely a = eβ(∆G+µB−µC), [5] deep square-well tethered bond potential (23) below is used 3 2mΛ −βµ +k−1∆S+βµex to mimic the connectivity among covalently bonded polymer c = e C B HS , [6] Vp beads and cross-linkers, ex where µHS is the excess chemical potential originating from  0 ex 0 0 |r − r | < rcut FHS. Here we note that the effect of µB is the same as ∆G. Vbonds(r, r ) = , [1] ex ∞ else Since µHS is a function of the packing fraction of the system, which also depends on fi, self-consistent iterations are needed to obtain f , f and the equilibrium packing fraction of where rcut is the cutoff distance of the tetheredDRAFT bond, and we PC P2C use rcut = 1.5σ throughout all simulations. See Materials and the system (see SI Appendix for details). Methods for the simulation details. Reaction equilibrium. As shown in Fig.1a, the formation a cross-linking P C bond requires two metathesis reactions, with Mean field theory . In the dilute limit (Fig.1c), polymer chains 2 are isolated. The distribution of reactive sites in the system K1 and K2 the corresponding reaction constants, which satisfy is heterogeneous and the reactive sites can be seen as ideal K1ρPρC = ρPCρB, [7] 3 gases confined in individual polymer blobs of volume Vp = Rg K2ρPρPC = ρP2CρB, [8] with Rg the radius of gyration of the polymer. Treating the polymer backbones implicitly as the crowding background, the where ρi = Ni/V is the density of specie i in the system. In free energy of the system can be written as Ref. (20, 22), the metathesis reactions in the first and second steps are of the same type, and one might think that the two   3   X NiΛ reaction constants K2 and K1 should be equal. However, a βF = Ni ln − 1 V recent experiment indicates that K2/K1 calculated from Eqs.7 i=poly,B,C and8 can be significantly larger than 1 ( 22). To understand   3   X niΛ ex this, we first use the fact that in chemical equilibrium, the + Ni ln − 1 + βFHS Vp chemical potentials for all species satisfy i=P,PC,P2C −1 −NP2CkB ∆S − β (NPC + NP2C + NC) µC µP + µC = µPC + µB, [9]

+β(NPC + 2NP2C) (∆G + µB) − βNBµB, [2] µP + µPC = µP2C + µB, [10]

2 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Qun-Li Lei et al. m =5 m =8 m = 10 (j) 1.0 (a) (b) (c)

PC 0.5 f

0.0 1.0 (k)

C (d) (e) (f) 2 0.5 P f

0.0 1.0

φp =0.01 (l)

p (g) (h) (i) 0.5 φp =0.10 x φp =0.30

0.0 8 6 4 2 8 6 4 2 8 6 4 2 − − − − − − − − − − − − βµC

Fig. 2. Reentrant gel-sol transition. (a-i): fPC, fP2C, xp as a function of µC for different m and φp. The solid lines in (a-f) are the theoretical predictions of Eqs.3,4 and 12 using K2/K1 obtained from simulations, while the dashed lines in (g-i) are guide for eye. . (j-l): Simulation snapshots for systems with m = 10 and φp = 0.1 at βµC = −8.0 (j), −5.0 (k) and −2.0 (l). Different colors represent different type of bonds based on Fig.1a, i.e., PC, P2C, P and polymer backbone bonds are in blue, red, green and grey, respectively. Free B and C molecules are not drawn. In all simulations, βµB = −3.0, β∆G = −2.0, and n = 100.

where µi (i = P, PC, P2C) can be obtained by taking a direct which does not explicitly depend on Vp or ∆S. Therefore, the derivative of Eq.2 with respect to Ni (see SI Appendix). prediction of fP2C and fPC from our theory (Eqs.3 and4) only Combining Eqs.7 and8 with Eqs.9 and 10 we have requires K2/K1 which can be measured both in experiments and simulations, without invoking any fitting parameters like K2 V k−1∆S B = e Vp or ∆S. In Fig.2a-f, we plot fP C and fPC as functions K1 NpolyVp 2  of µC obtained from the simulations for systems of various nv0 k−1∆S 3 B m and φp with n = 100 and βµB = −3. The theoretical  3 e (Rg  V/Npoly) = φpRg [11] results based on Eqs.3,4 and 12 using the measured K2/K1 in  k−1∆S 3 simulations are shown as solid lines, which agree quantitatively  e B (Rg  V/Npoly) DRAFTwith the simulation results in a wide range of φp and m. The π 3 equilibrium packing fractions of the system φeq obtained from where v0 = 6 σ is the volume of a single bead on the polymer −1 self-consistent iterations also match well with the simulation chain. Eq. 11 predicts a powerlaw scaling K2/K1 ∼ φp in k−1∆S results (see Table S1 in the SI Appendix). This suggests that the heterogeneous dilute regime, which saturates to e B our theoretical framework is robust and capable of quantita- in the homogeneous dense regime. In Fig.1b, we plot K /K 2 1 tively predicting the phase behaviour of the vitrimers in both measured in simulations as a function of φ for different m p heterogeneous dilute and homogeneous dense regimes. and µC with n = 100. One can see that K2/K1 approaches 1 corresponding to ∆S → 0 in the homogeneous dense regime, Reentrant gel-sol transition. Moreover, Eq.3 has the limiting which suggests that the ideal gas approximation for the reactive behaviour: sites is quantitatively correct in this regime. In the heteroge- c neous dilute regime, the powerlaw scaling is also confirmed in fP2C(µC → −∞) ' → 0, [13] a2 Fig1b. In this regime, Rg decreases with increasing m due fP C(µC → +∞) ' c → 0. [14] to more cross-linking within each polymer blob, which also 2 raises K2/K1 as shown in Fig.1b. Therefore, the observation One can also prove that fP2C reaches the maximum at µC = of K2/K1 > 1 in experiments could result from the inhomoge- µB + ∆G (see SI Appendix), and this implies that the cross- neous distribution of reactive sites in the system (22, 25, 26). linking degree first increases and then decreases with increasing With the introduction of K2/K1, Eq.6 can be re-written the concentration of cross-linkers, which quantitatively agrees as with simulations as shown in Fig.2d-f. This non-monotonic 3 dependence of f on the concentration of cross-linkers was  K2  2NpolymΛ β(µex −µ ) P2C c = e HS C , [12] also observed in a recent experiment (22), which was explained K1 V

Qun-Li Lei et al. PNAS | October 23, 2020 | vol. XXX | no. XX | 3 based on reaction equilibrium and mass conservations. Here limit, the cross-linking degree of the system can still be tuned our theory shows that the counter-intuitive decrease of fP2C by the concentration of free linkers or the precursor density with increasing µC at high cross-linker concentration is a on polymer chains. In Fig.3ab, we plot fP2C and xp as purely entropic effect. At high µC limit, the reactive sites functions of β∆G measured in simulations of systems with are fully bonded with cross-linkers forming either PC or P2C various µC at m = 5, n = 100 and βµB = −3, which agree bonds. Changing one P2C bond to two dangling PC bonds quantitatively with the theoretical prediction. One can see hardly changes the energy of the system but increases the that both fP2C and xp increase and reach a plateau at low conformational entropy of polymer chains. Therefore, forming β∆G. In Fig.3c, we plot the cross-linking degree at the low ∞ dangling PC bonds is more entropically favoured than forming temperature limit fP2C as a function of βµC. We find that ∞ cross-linking P2C bonds, which drives the cross-linking degree fP2C decreases with increasing βµC or decreasing m, which towards zero at high µC. This non-monotonic dependence quantitatively agrees with Eq. 15. The physical explanation is of fP2C on µC provides an additional axis in controlling the that at β∆G → −∞, all reactive sites form either dangling PC mechanical properties of vitrimers. As the percolation of the bonds or cross-linking P2C bonds. Breaking a cross-linking connected polymer cluster is an important indicator of the sol- P2C bond into two dangling PC bonds does not change the gel transition of the system, we plot the fraction of polymers energy of the system but increases the conformational entropy in the largest cluster xp as a function of µC for various m and of the polymers by ∆Sconf . The breaking of a P2C bond also φp in Fig2g-i. Generally, gelation occurs at xp ' 0.5 (27), simultaneously absorbs one free cross-linker from the solution, and as shown in Fig2g-i, xp changes non-monotonically with and the translational entropy of the free cross-linker drops 3 increasing µC, for which the typical snapshots of the system are by ∆Strans ' −kB ln ρCΛ ' −µC/T . With increasing µC, shown in Fig.2j-l. This implies a reentrant gel-sol transition ∆Strans decreases, which favors more P2C bonds breaking into in this linker-mediated vitrimer system. dangling PC bonds. Similarly, at the fixed polymer length n and φp, increasing m decreases ∆Sconf , which drives the formation of more cross-linking P2C bonds. Similar entropic (a)1.0 βµC (b)1.0 8.0 6.0 effects in linker-mediated self-assembly were recently observed − − 7.0 5.0 0.8 − 0.8 in linker-mediated DNA-coated colloids (28). In Fig.3d, we − 4.0 − ∞ 3.0 plot the resulting saturated value of percolation parameter xp C 0.6 − 0.6 p 2 2.0

P at β∆G → −∞ as a function of µC for various m. One can see − x

f 0.4 0.4 ∞ that xp also increases with decreasing µC, but changes more sharply than f ∞ . This suggests a practical way to control 0.2 0.2 P2C the mechanical property of the vitrimer at low temperature 0.0 0.0 by tuning the cross-linker concentration. 8 6 4 2 0 8 6 4 2 0 − − β∆− G − − − β∆− G − Discussion and Conclusion (c) (d)1.0 0.8 In conclusion, we have proposed a theoretical framework to un- 0.8 derstand the cross-linking in a newly developed linker-mediated

C vitrimer system. Quantitative agreements have been found

2 0.6 0.6 ∞ p ∞ P

x between our theoretical prediction and coarse-grained com- f 0.4 m =5 0.4 puter simulations, while the purpose of this work is to reveal m =8 0.2 the essential physics in this system, rather than seeking a 0.2 m = 10 quantitative prediction for real vitrimer systems. Importantly, 8 6 4 2 8 6 4 2 we find that entropy plays a determining role in these systems − − − − − DRAFT− − − which is reflected in the following three aspects. First, we find βµC βµC that the density heterogeneity of reactive sites can result in the mismatch of reaction constants of the same metathesis reaction Fig. 3. Entropy-driven cross-linking at low temperature. (a-b): fP2C and xp as a function of reaction energy ∆G for different µC at m = 5, where the solid lines in the first and second cross-linking steps, which offers a pos- in (a) are the theoretical predictions of Eq.3. (c-d): Saturated f ∞ and x∞ as a P2C p sible mechanism to explain the recent experiment (22). This function of µC for different m at β∆G = −8.0, where the solid lines in (c) are the mechanism of heterogeneity-enhanced reactions rates also has theoretical predictions of Eq. 15. In (b,d), the dashed lines are guide for eye. In all implications in biochemical reactions in molecular crowding simulations, βµB = −3.0, n = 100, and φp = 0.1. conditions, which has been utilized by cells in form of mem- braneless compartment to enhance RNA transcription and protein translations (29). A similar effect of self-concentration Entropic effects at low temperature. Another interesting theo- was also reported in polymer (30). Second, we find retical prediction is that in the limit of β∆G → −∞, one has that increasing the cross-linker concentration can induce a f + f ' 1 and P2C PC reentrant gel-sol transition of the vitrimer. This offers a way √ 1 + 4c − 1 to control over the reshaping or recycling of vitrimers without fP2C ' 1 − < 1, [15] heating up and cooling down the system. Lastly, in the low 2c temperature limit, we find that the cross-linking degree of the which implies that at the low temperature limit (β∆G → −∞) system can still be effectively tuned by the concentration of or the dilute limit of byproduct molecules (βµB → −∞), the cross-linkers, which suggests that in experiments, one should system is not fully cross-linked. Since c depends on µC, m be careful with the cross-linker concentration, and lowering and Vp (Eq.6), this suggests that even at the low temperature the temperature does not always produce highly cross-linked

4 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Qun-Li Lei et al. polymer networks. Therefore, our theory provides important is: guidelines in controlling the mechanical properties of the novel   functional materials. It should be noted that in real vitrimer bB bP   λ(aG0 , cC) systems as studied in (20, 22), other factors may also influ- acc aG → aG  = min 1, exp (β∆G) .  1 0  λ(a , b ) ence the cross-linking in the system, e.g., the polydisperse G1 B cPC cC distribution of the reactive sites on the backbone chains, the [17] polydispersity and the rigidity of the chains themselves, the For the second bond switching reaction in Fig. 1a, van der Waals attraction between different chemical species etc. Incorporating these effects within our theoretical framework is bP bP2C necessary to have a closer comparison between the theoretical aG c aG c prediction and experimental data. 0 G1 2 G2

dPC dB

Materials and Methods where a/b/c/d are specific particles in the system and G2/P2C are types of grafted point bonded with a cross-linkng P2C bond and cross-linker in a P2C bond. The acceptance probability of the proposed move for the forward second step reaction is: Simulation details. Simulation are initialized with 100 polymer   chains in random positions. Periodic boundary conditions are bP bP2C applied in all three directions. We perform grand canonical   acc aG0 cG1 → aG2 cG2 = (µ µ N VT ) Monte Carlo simulations with the straight   B C poly [18] event-chain algorithm (31, 32) to simulate the coarse-grained dPC dB system, in which the length of each event chain is fixed at L =  2λ(c , d )  c min 1, G2 B exp (−β∆G) , 0.2V 1/3 for the translational moves of particles. Molecules B λ(cG1 , bP) and C are added or deleted by using the conventional grand canonical Monte Carlo method. Besides, we devise a bond and the acceptance probability of the proposed move for the swap algorithm to simulate reversible bond swap metathesis backward second step reaction is: reactions in the vitrimer system (see below). The ratio of  b b  the trial moves, i.e., translational move, adding/removing P2C P   molecules and bond swap, is 4 : 1 : 5. In each simulation, we acc aG2 cG2 → aG0 cG1 = 5 5   use 10 MC moves per particle for equilibration and 10 MC [19] dB dPC moves per particle for sampling, where the sampling frequency   λ(cG , bP) is 10 MC moves per particle for sampling φeq, fPC, fP2C and 1 2 min 1, exp (β∆G) , K2/K1, and 10 MC moves per particle for sampling the 2λ(cG2 , dB) maximum cluster size. where we double the forward reaction flux to equalize the flux between the two states before and after a second step reaction. Bond swap algorithm in Event-Chain Monte Carlo simula- In our simulation, we first randomly select a particle, of which tions. To simulate reversible bond swap reactions in vitrimer the type is mentioned above and then select a potential reactive systems, we assume that the metathesis reactions consist of particle within the distance rcut as well as the specific reaction two steps as shown in Fig. 1a, the first step reaction is to perform the trial move depending on the types of selected particles. bP bB DRAFT ACKNOWLEDGMENTS. This work has been supported in part aG0 aG1 by the Singapore Ministry of Education through the Academic c c Research Fund MOE2019-T2-2-010, by Nanyang Technological Uni- C PC versity Start-Up Grant (NTU-SUG: M4081781.120), by the Ad- vanced Manufacturing and Engineering Young Individual Research where a/b/c are specific particles in the system and Grant (A1784C0018). We thank NSCC for granting computational G0/P/C/G1/B/PC are types of the grafted point bonded resources. with an intact precursor, precursor, cross-linker, grafted point 1. Montarnal D, Capelot M, Tournilhac F, Leibler L (2011) Silica-like malleable materials from bonded with a dangling PC bond, byproduct molecule and permanent organic networks. Science 334(6058):965–968. precursor-cross-linker, respectively. The acceptance probabil- 2. Capelot M, Montarnal D, Tournilhac F, Leibler L (2012) Metal-catalyzed transesterification for healing and assembling of thermosets. Journal of the american chemical society ity of a proposed bond swap for the forward first step reaction 134(18):7664–7667. is: 3. Smallenburg F, Leibler L, Sciortino F (2013) Patchy particle model for vitrimers. Phys Rev Lett 111(18):188002.   4. Ciarella S, Sciortino F, Ellenbroek WG (2018) Dynamics of vitrimers: Defects as a highway to bP bB stress relaxation. Phys Rev Lett 121(5):058003.  λ(a , b )    G1 B 5. Ciarella S, Biezemans RA, Janssen LM (2019) Understanding, predicting, and tuning acc aG0 → aG1 = min 1, exp (−β∆G) , the fragility of vitrimeric polymers. Proceedings of the National Academy of Sciences   λ(aG0 , cC) c c 116(50):25013–25022. C PC 6. Meng F, Saed MO, Terentjev EM (2019) Elasticity and relaxation in full and partial vitrimer [16] networks. Macromolecules 52(19):7423–7429. where λ(x, y) is probability of finding the pair sites (x, y). ∆G 7. Tito NB, Creton C, Storm C, Ellenbroek WG (2019) Harnessing entropy to enhance tough- ness in reversibly crosslinked polymer networks. Soft matter 15(10):2190–2203. is the reaction energy. Similarly, The acceptance probability 8. Wu JB, Li SJ, Liu H, Qian HJ, Lu ZY (2019) Dynamics and reaction kinetics of coarse-grained of a proposed bond swap for the backward first step reaction bulk vitrimers: a molecular dynamics study. Phys. Chem. Chem. Phys. 21(24):13258–13267.

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6 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Qun-Li Lei et al.