Downloaded by guest on September 27, 2021 a Lei Qun-Li vitrimers linker-mediated in cross-linking Entropy-controlled www.pnas.org/cgi/doi/10.1073/pnas.2015672117 interest- More (22). experiment recent a in transition, gel–sol observed nontrivial reentrant was link- a a which free undergoes plays of system concentration vitrimer linkers the the free increasing ers, with our of that, find and entropy We be system, the role. vitrimer that can linker-mediated show and a the results sim- propose study computer catalyst, coarse-grained we to with a work, ulations combined this theory of In field need (21). mean linker- the like The without processed and (20). stability, dimen- and network sional resistance chemical the the superior have vitrimers cross-links the mediated and both changes (cross-linkers), here dynamically bis-dioxaborolanes reaction in with dioxaborolanes, metathesis dioxaborolane react of pendant with reaction units polymers metathesis functionalized the the which on based and oped equipment same the (9). which thermoplastics using processing stable, of for oxidatively conditions detrimental or conventional particularly thermally the the not is of However, usually reactions are in applications. pro- vitrimers involved the different catalysts in for or used reactants vitrimers been have of (19), (18), duction metathesis exchange olefin thiol–disulfide 17), 15), and (16, (14, reaction reaction exchange transamination trans- alkoxyamine the 13), as (2, such trig- reactions, reaction chemical esterification thermally (3–12). various liquids decade, the viscoelastic past as by the behave In reversibly they that swap so cross-linked reactions network gered as polymer behave bonds the exchangeable vitrimers the in temperatures, malleable temperatures, high at and low while, recyclable thermosets, At also 2). while insoluble (1, mechan- and are they robust specific, ically be To thermoplastics. and thermosets V cross-linking entropy-driven vitrimer this system. of vitrimer properties developed offer the newly controlling and predic- and simulations, understanding theoretical computer in guidelines Our with linkers. quantitatively polymers of agree of entropy tions translational entropy the the from conformational and originates the the which between vitrimers, determines competition the still of concentration degree Chen, linker cross-linking Q. the Huang, limit, S. temperature Yang, a H. explains Wu, which molecules vit- [S. transition, concentration, gel–sol experiment linker reentrant recent governing the a increasing a undergo the with plays rimers that, of entropy find behavior We which phase role. in the vitrimers, elucidate linker-mediated to [M. framework unknown remains theoretical principle design the R resis- polymers which cross-linked of outstanding chemical processing industry, of and and in ways thermal, new processability suggesting mechanical, good tance, as both such poly- shown performance, available commercially have various 2020) 24, mers with July review dioxaborolanes metathe- for of (received on 2020 based sis 19, vitrimers September approved linker-mediated and developed Kingdom, Recently United Cambridge, Cambridge, of University Frenkel, Daan by Edited and Singapore; 637371 University, Singapore Technological 117546 Nanyang Sciences, Mathematical and Physical colo hmcladBoeia niern,NnagTcnlgclUiest,675 Singapore; 637459 University, Technological Nanyang Engineering, Biomedical and Chemical of School ottger ¨ eety e yeo ikrmdae irmrwsdevel- was vitrimer linker-mediated of type new a Recently, xii nqepoete htcmieteavnae of to advantages known the are combine material, that polymeric properties of unique type exhibit new a itrimers, | tal., et eahssreaction metathesis 3 1019 22).Mr nrgigy ttelow the at intriguingly, More (2020)]. 1180–1190 53, a,1 Science iyn Xia Xiuyang , 5,6–5(07] eew omlt a formulate we Here (2017)]. 62–65 356, | enrn e–o transition gel–sol reentrant a,b,1 unYang Juan , c asm iaCiamarra Pica Massimo , | Macro- doi:10.1073/pnas.2015672117/-/DCSupplemental at online information supporting contains article This 2 1 the under Published Submission.y Direct PNAS a is article This interest.y competing no declare authors The deep infinitely An system. hard-sphere-chain coarse-grained this diameter of spheres hard is ulse coe 1 2020. 21, October published First R.N. and research; M.P.C., J.Y., performed X.X., Q.-L.L., X.X. and data; and paper. analyzed the Q.-L.L. R.N. wrote and research; M.P.C., J.Y., designed X.X., Q.-L.L., R.N. contributions: Author ume Vitrimer. of Model Coarse-Grained Results materials. linkers, resulting the free controlling of in of property freedom mechanical of concentration the degree extra the an on offers essentially depends which degree cross-linking still the vitrimers limit, temperature of low the at even ingly, steaeaenme fpeusr e oye htremain that polymer type per of bonds precursors of and number intact, and average cross-linkers the dangling is poten- with bonds chemical PC the cross-linking of by numbers controlled average are which system, tials molecule the byproduct the in and B C molecule reactions cross-linker two of metathesis most, numbers The at precursors. form, and reversible, different can are two cross-linker each with and bonds molecule, P B precursor intact free cross-linking another a with byproduct form react a further to dan- producing can a bond and Moreover, PC reactions. bond gling metathesis PC through B dangling molecule molecule a free cross-linker forming a by with react C can which distributed, formly are there polymer, each comprises fplmr is polymers of owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom work. To this to equally contributed X.X. and Q.-L.L. o h ainldsg ftelne-eitdvtieswith vitrimers linker-mediated the properties. of desired quan- design agrees rational guidelines theory the provides field for and mean simulations, found Our computer is with role. titatively entropy dictating sys- which a the play in to describing vitrimers, linker-mediated for of framework Here tem theoretical unknown. a remains propose principle in we processing design performance the of mechanical while high industry, possibilities with new polymers on cross-linked offer based vitrimers reactions linker-mediated metathesis developed recently The Significance eepo ot al M)smltost investigate to simulations (MC) Carlo Monte employ We V µ C ossigof consisting b and PNAS n n a Ni Ran and , N y adshrso diameter of spheres hard c P µ eateto hmsr,Ntoa nvriyo Singapore, of University National Chemistry, of Department i i φ B 2 = NSlicense.y PNAS ,P,P PC, P, = | p od e oye,rsetvl.Similarly, respectively. polymer, per bonds C epciey edefine We respectively. , = N oebr3 2020 3, November ∆G poly π 6 N b iiino hsc n ple hsc,Sho of School Physics, Applied and Physics of Division N poly n poly sterato energy. reaction the is i P m a,2 σ 2 stettlnme fpeusr or precursors of number total the is n oye his nwihec polymer each which in chains, polymer 2 C  . C σ 3 odadpouiga additional an producing and bond ntesse.Tepcigfraction packing The system. the in 1 n n r ohmdldas modeled both are C and B and , rcros(eciests uni- P sites) (reactive precursors | ecnie ytmo vol- of system a consider We . y o.117 vol. y σ https://www.pnas.org/lookup/suppl/ on 1A, Fig. in shown As . n | PC o 44 no. N C and and | n P N 27111–27115 2 B C r the are sthe as n P

APPLIED PHYSICAL SCIENCES correction per cross-linking bond for any inaccuracy arising from AB ex the ideal gas approximation. FHS is the excess free energy based on Carnahan–Starling hard-sphere equation of state (24) aris- ing 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 C site becomes homogeneous. Thus, Vp can be replaced by the available volume per polymer, that is, Vp = V /Npoly.

We define the cross-linking degree of system as fP2C =

2NP2C/(Npolym), that is, the fraction of reactive sites that are D cross-linked, and fi = Ni /(Npolym), (i = PC, B, C). Using the saddle-point approximation, ∂F /∂fi = 0, (i = PC, P2C, B, C),

one can obtain the equilibrium fPC and fP2C as h i2 −(1 + a) + p(1 + a)2 + 4c f = , [3] P2C 4c −(1 + a) + p(1 + a)2 + 4c f = , [4] PC 2c with Fig. 1. Vitrimer model. (A) Illustration of the two-step metathesis reactions a = eβ(∆G+µB−µC), [5] in the vitrimer system. (B) K2/K1 as a function of φp for different m and µC at n = 100 and βµ = −3. (C and D) Illustration of the (C) heterogeneous 3 B 2 mΛ −βµ +k−1∆S+βµex dilute and (D) homogeneous dense systems of the vitrimers. c = e C B HS , [6] Vp where µex is the excess chemical potential originating from F ex . square well-tethered bond potential (23), below, is used to mimic HS HS Here we note that the effect of µB is the same as ∆G. Since the connectivity among covalently bonded polymer beads and µex is a function of the packing fraction of the system, which cross-linkers, HS also depends on fi , self-consistent iterations are needed to obtain fPC, fP C, and the equilibrium packing fraction of the system (see ( 0 2 0 0 |r − r | < rcut SI Appendix for details). Vbonds(r, r ) = , [1] ∞ else Reaction Equilibrium. As shown in Fig. 1A, the formation a cross- P C K where rcut is the cutoff distance of the tethered bond, and we linking 2 bond requires two metathesis reactions, with 1 and K use rcut = 1.5σ throughout all simulations. See Materials and 2 the corresponding reaction constants, which satisfy Methods for the simulation details. K1ρPρC = ρPCρB, [7]

Mean Field Theory. In the dilute limit (Fig. 1C), polymer chains K2ρPρPC = ρP2CρB, [8] are isolated. The distribution of reactive sites in the system is heterogeneous, and the reactive sites can be seen as ideal gases where ρi = Ni /V is the density of specie i in the system. In refs. 3 20 and 22, the metathesis reactions in the first and second steps confined in individual polymer blobs of volume Vp = R , with g are of the same type, and one might think that the two reaction Rg as the radius of gyration of the polymer. Treating the polymer backbones implicitly as the crowding background, the free energy constants K2 and K1 should be equal. However, a recent exper- of the system can be written as iment indicates that K2/K1 calculated from Eqs. 7 and 8 can be significantly larger than 1 (22). To understand this, we first use the fact that, in chemical equilibrium, the chemical potentials for   3   X Ni Λ all species satisfy βF = N ln − 1 i V i=poly,B,C µP + µC = µPC + µB, [9]   3   X ni Λ ex µP + µPC = µP2C + µB, [10] + Ni ln − 1 + βFHS Vp i=P,PC,P C 2 where µi (i = P, PC, P2C) can be obtained by taking a direct −1 derivative of Eq. 2 with respect to Ni (SI Appendix). Combining −NP2CkB ∆S − β (NPC + NP2C + NC)µC Eqs. 7 and 8 with Eqs. 9 and 10, we have +β(NPC + 2NP2C) (∆G + µB)− βNBµB, [2] K2 V k−1∆S where the first summation is on the ideal gas terms of poly- = e B mer chains, free cross-linkers, and byproduct molecules, and the K1 NpolyVp −1 second summation is on the ideal gas terms of different reac- ( nv0 kB ∆S 3 3 e (R  V /Npoly) φp R g tive sites in polymer blobs. Here β = 1/kB T , with kB and T = g , k−1∆S 3 as the Boltzmann constant and the temperature of the system, e B (Rg  V /Npoly) respectively, and Λ is the de Broglie wavelength. This ideal gas [11] approximation of reactive sites offers a simple estimation of the 3 conformational entropy change during the formation of cross- where v0 = π/6σ is the volume of a single bead on the poly- −1 linking P2C bonds, for which we introduce ∆S as the entropy mer chain. Eq. 11 predicts a powerlaw scaling K2/K1 ∼ φp in

27112 | www.pnas.org/cgi/doi/10.1073/pnas.2015672117 Lei et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 e tal. et Lei oye akoebnsaei le e,gen n ry epciey reBadCmlclsaentdan nalsimulations, all In drawn. not are molecules C and B Free respectively. gray, and green, red, blue, and in are bonds backbone polymer m various Eqs. of systems for with simulations the from obtained like parameters fitting ∆ any invoking without simulations, requires of prediction Eqs. 2. Fig. Appendix, robust (SI is framework results theoretical our simulation that the suggests This with S1). Table well match also tions system the of simula- of the fractions range with wide a quantitatively in agree results which tion lines, solid as shown plot we 1B, Fig. to In regime. saturates dense which homogeneous the regime, dilute heterogeneous the nti regime, this In 1B. Fig in confirmed regime, R also dilute is heterogeneous scaling the powerlaw In the quantita- regime. is this sites in reactive correct the tively for approximation gas ideal the that to ing of function a with as simulations in sured hrfr,teosrainof observation the raises Therefore, also which blob, polymer each eutfo h nooeeu itiuino eciestsin sites reactive 26). 25, of (22, distribution system the inhomogeneous the from result g = S hc osntepiil eedon depend explicitly not does which ihteitouto of introduction the With erae ihincreasing with decreases nFg 2 Fig. In . n 0and 10 3, n 3, = n and 4, ∆S n a e that see can One 100. = (A–I 100. 100 = and 4, K → enrn e–o transition gel–sol Reentrant ) φ 2 G D ABC p /K 12 = 0 f and P using 12 ntehmgnosdnergm,wihsuggests which regime, dense homogeneous the in c 1 t(J at 0.1 A–F 2 = hc a emaue ohi xeiet and experiments in both measured be can which C sn h measured the using  βµ and eplot we , K K K 2 ) B /K 1 2 βµ f =  φ 1 PC C eq bandfo iuain,wietedse ie in lines dashed the while simulations, from obtained h hoeia eut ae on based results theoretical The −3. 2N = K rmorter (Eqs. theory our from bandfo efcnitn itera- self-consistent from obtained ,(K −8.0, m poly 2 φ f V /K p P K K u omr rs-ikn within cross-linking more to due , m 2 and 2 2 C 1 /K /K Λ Eq. , and 3 ) f m PC K 1 1 e ,ad(L) and −5.0, φ K β , prahs1correspond- 1 approaches > V 2 h qiiru packing equilibrium The . p ( /K 6 f 2 µ p f P 1 o different for PC /K 2 HS ex a erwitnas rewritten be can HI EF or C 1 neprmnscould experiments in , −µ 1 ssoni i.1B. Fig. in shown as x ∆S sfntosof functions as p nsmltosare simulations in C safnto of function a as ) hrfr,the Therefore, . , hti,P,P PC, is, that 1A, Fig. on based bonds of type different represent colors Different −2.0. 3 K and 2 e m m /K k B −1 and and only 4) 1 V ∆S mea- [12] p µ µ µ φ or in C C C p o different for atidctro h o–e rniino h ytm epo the plot impor- we system, an the is of As transition cluster sol–gel vitrimers. polymer the of of connected indicator the properties tant of mechanical percolation the the high controlling at in zero axis toward of degree dependence cross-linking nonmonotonic cross-linking forming the than drives favored ically polymer dangling of forming entropy Therefore, conformational chains. the increases but system dangling the two to bond either forming linkers high At of decrease µ counterintuitive the theory our that Here shows conservations. based mass explained and was equilibrium which reaction (22), on experiment recent a in observed of the 2 Fig. dence increasing with in agrees shown with as quantitatively decreases simulations which then cross-linkers, of and concentration increases first degree that prove µ also can One behavior Transition. Gel–Sol Reentrant homogeneous and dilute of heterogeneous behavior regimes. both phase dense in the vitrimers predicting quantitatively the of capable and C B ∆G + thg rs-ikrcnetaini ueyetoi effect. entropic purely a is concentration cross-linker high at G–I r udsfrteee (J eye. the for guides are m µ and C f ,adti mle httecross-linking the that implies this and Appendix), (SI P 2 ii,teratv ie r ul oddwt cross- with bonded fully are sites reactive the limit, C φ p PNAS h oi ie in lines solid The . ntecnetaino rs-ikr a also was cross-linkers of concentration the on f f P P 2 2 C C | (µ (µ PC oebr3 2020 3, November PC C C f → −∞) → od adycagsteeeg of energy the changes hardly bonds P or iuainsasosfrsseswith systems for snapshots Simulation –L) 2 oevr Eq. Moreover, +∞) C f D–F P P A–F ece h aiu at maximum the reaches 2 2 C C L K J ' ' on hsnnoooi depen- nonmonotonic This . r h hoeia rdcin of predictions theoretical the are od.Cagn one Changing bonds. PC c a c µ | → 2 C → o.117 vol. od smr entrop- more is bonds βµ 0. rvdsa additional an provides f 0, B P P 3 = 2 2 C C −3.0, a h limiting the has | ihincreasing with od,which bonds, o 44 no. β ∆G µ 2 C ,P and P, C, | = This . µ −2.0, 27113 P [13] [14] C 2 = C

APPLIED PHYSICAL SCIENCES fraction of polymers in the largest cluster xp as a function of µC lational entropy of the free cross-linker drops by ∆Strans ' 3 for various m and φp in Fig. 2 G–I. Generally, gelation occurs −kB ln ρCΛ ' −µC/T . With increasing µC, ∆Strans decreases, at xp ' 0.5 (27) and, as shown in Fig. 2 G–I, xp changes non- which favors more P2C bonds breaking into dangling PC bonds. monotonically with increasing µC, for which the typical snapshots Similarly, at the fixed polymer length n and φp , increasing m of the system are shown in Fig. 2 J–L. This implies a reentrant decreases ∆Sconf , which drives the formation of more cross- gel–sol transition in this linker-mediated vitrimer system. linking P2C bonds. Similar entropic effects in linker-mediated self-assembly were recently observed in linker-mediated DNA- Entropic Effects at Low Temperature. Another interesting theo- coated colloids (28). In Fig. 3D, we plot the resulting saturated ∞ retical prediction is that, in the limit of β∆G → −∞, one has value of percolation parameter xp at β∆G → −∞ as a func- ∞ fP2C + fPC ' 1 and tion of µC for various m. One can see that xp also increases ∞ √ with decreasing µC, but changes more sharply than fP2C. This 1 + 4c − 1 suggests a practical way to control the mechanical property f ' 1 − < 1, [15] P2C 2c of the vitrimer at low temperature by tuning the cross-linker concentration. which implies that, at the low temperature limit (β∆G → −∞) or the dilute limit of byproduct molecules (βµB → −∞), the Discussion and Conclusion system is not fully cross-linked. Since c depends on µC, m, In conclusion, we have proposed a theoretical framework and Vp (Eq. 6), this suggests that, even at the low tempera- to understand the cross-linking in a newly developed linker- ture limit, the cross-linking degree of the system can still be mediated vitrimer system. Quantitative agreements have been tuned by the concentration of free linkers or the precursor den- found between our theoretical prediction and coarse-grained sity on polymer chains. In Fig. 3 A and B, we plot fP2C and computer simulations, while the purpose of this work is to xp as functions of β∆G measured in simulations of systems reveal the essential physics in this system, rather than seeking with various µC at m = 5, n = 100, and βµB = −3, which agree a quantitative prediction for real vitrimer systems. Importantly, quantitatively with the theoretical prediction. One can see that we find that entropy plays a determining role in these systems both fP2C and xp increase and reach a plateau at low β∆G. which is reflected in the following three aspects. First, we find In Fig. 3C, we plot the cross-linking degree at the low tem- that the density heterogeneity of reactive sites can result in the f ∞ βµ f ∞ perature limit P2C as a function of C. We find that P2C mismatch of reaction constants of the same metathesis reac- decreases with increasing βµC or decreasing m, which quanti- tion in the first and second cross-linking steps, which offers tatively agrees with Eq. 15. The physical explanation is that, at a possible mechanism to explain the recent experiment (22). β∆G → −∞, all reactive sites form either dangling PC bonds This mechanism of heterogeneity-enhanced reaction rates also or cross-linking P2C bonds. Breaking a cross-linking P2C bond has implications in biochemical reactions in molecular crowd- into two dangling PC bonds does not change the energy of the ing conditions, which has been utilized by cells in the form of a system but increases the conformational entropy of the poly- membraneless compartment to enhance RNA transcription and mers by ∆Sconf . The breaking of a P2C bond also simultaneously protein translations (29). A similar effect of self-concentration absorbs one free cross-linker from the solution, and the trans- was also reported in polymer (30). Second, we find that increasing the cross-linker concentration can induce a reentrant gel–sol transition of the vitrimer. This offers a way to control AB over the reshaping or recycling of vitrimers without heating up and cooling down the system. Lastly, in the low temperature limit, we find that the cross-linking degree of the system can still be effectively tuned by the concentration of cross-linkers, which suggests that, in experiments, one should be careful with the cross-linker concentration, and lowering the temperature does not always produce highly cross-linked polymer networks. Therefore, our theory provides important guidelines in control- ling the mechanical properties of the novel functional materials. It should be noted that, in real vitrimer systems as studied in refs. 20 and 22, other factors may also influence the cross- linking in the system, for example, the polydisperse distribution CD of the reactive sites on the backbone chains, the polydisper- sity and the rigidity of the chains themselves, and the van der Waals attraction between different chemical species. Incorporat- ing these effects within our theoretical framework is necessary to have a closer comparison between the theoretical prediction and experimental data.

Materials and Methods Simulation Details. Simulations are initialized with 100 polymer chains in random positions. Periodic boundary conditions are applied in all three

directions. We perform grand canonical (µBµCNpoly VT) MC simulations with the straight event-chain algorithm (31, 32) to simulate the coarse-grained 1/3 system, in which the length of each event chain is fixed at Lc = 0.2V for Entropy-driven cross-linking at low temperature. ( and ) The Fig. 3. A B fP2C the translational moves of particles. Molecules B and C are added or deleted and xp as a function of reaction energy ∆G for different µC at m = 5, where by using the conventional grand canonical MC method. Besides, we devise the solid lines in A are the theoretical predictions of Eq. 3. (C and D) Satu- a bond swap algorithm to simulate reversible bond swap metathesis reac- ∞ ∞ rated f and x as a function of µC for different m at β∆G = −8.0, where P2C p tions in the vitrimer system (see below). The ratio of the trial moves, that is, the solid lines in C are the theoretical predictions of Eq. 15. In B and D, the translational move, adding/removing molecules and bond swap, is 4 : 1 : 5. 5 dashed lines are guides for the eye. In all simulations, βµB = −3.0, n = 100, In each simulation, we use 10 MC moves per particle for equilibration and 5 and φp = 0.1. 10 MC moves per particle for sampling, where the sampling frequency is

27114 | www.pnas.org/cgi/doi/10.1073/pnas.2015672117 Lei et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 6 .J otci .A edr .J oa,Uigtednmcbn oacs macroscop- access to bond dynamic the Using Rowan, J. S. Meador, A. M. Wojtecki, J. R. 16. 5 .Taynton P. 15. 4 .Denissen W. vitrimers. Polylactide. 14. Hillmyer, A. M. Delgado, A. P. Brutman, P. J. cova- reversible 13. on based engineering Polymer polymer Zhang, Q. in M. Rong, Z. chemistry M. Zhang, P. covalent Z. Dynamic 12. Nicola Prez, Du R. Zee, Van E. F. J. N. Leibler, 11. L. Winne, M. J. 10. e tal. et Lei o h akadfis tprato is swap reaction bond step proposed first a backward of the probability for acceptance The Similarly, energy. tion where is metathesis reaction step the first the that 1A; Fig. assume in shown we as steps systems, two vitrimer of consist in reactions reactions Simulations. swap MC Event-Chain bond in Algorithm Swap size. Bond cluster maximum the sampling for particle per moves rpsdbn wpfrtefradfis tprato is a reaction step of first forward probability the for acceptance swap bond The proposed respectively. precursor–cross-linker, PC dangling cross- and precursor, a precursor, with intact bonded an point with grafted bonded linker, point grafted the of types where sampling for particle per moves MC 10 .W eisn .M in,F .D rz irmr:Praetogncntok with networks organic Permanent Vitrimers: Prez, Du coarse- E. F. of Winne, kinetics M. reaction J. and Denissen, Dynamics W. Lu, Y. 9. Z. Qian, J. H. Liu, enhance H. Li, to J. S. entropy Wu, Harnessing B. J. Ellenbroek, 8. G. W. Storm, C. Creton, C. Tito, B. N. 7. .F eg .O ad .M eete,Eatct n eaaini uladpartial and full in relaxation and Elasticity Terentjev, M. E. Saed, O. M. the Meng, tuning and F. predicting, Understanding, 6. Janssen, M. L. Biezemans, A. highway R. a Ciarella, as S. Defects vitrimers: 5. of Dynamics Ellenbroek, G. W. Sciortino, F. 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We computational and granting (A1784C0018). for Manufacturing Singapore Grant Centre, Advanced Supercomputing Research (NTU-SUG: the Individual Grant by Young Start-Up and University Technological MOE2019- M4081781.120), Fund Nanyang Research by Academic the T2-2-010, through Education of Ministry pore ACKNOWLEDGMENTS. Availability. Data of types the on depending move trial mentioned, particles. the distance above selected perform the to is within reaction type simulation, specific particle the the our reactive as which potential In a of reaction. select particle, step between then a second and flux select a the randomly after equalize first and to we before flux states reaction two forward the the double we where backward the for move proposed the is reaction of step probability second acceptance the and is reaction step second P cross-linking a P with bonded point grafted of where 2 od h cetnepoaiiyo h rpsdmv o h forward the for move proposed the of probability acceptance The bond. C iufietilecag reactions. exchange disulfide–thiol metathesis. olefin via malleable . et sn vn hi algorithms. chain event using melts blends. systems. sphere polymer in (2000). temperatures sition Science colloids. coated system. /dioxaborolane Gelation a versible of case The vitrimers: Macromolecules self-assembled of polyethylene/dioxaborolane semicrystalline systems. and maleimide molten of morphology Hierarchical chains. bead (2020). metathesis. 1190 dioxaborolane on based vitrimers of dynamics (2017). metathesis. dioxaborolane through a/bcd ottger ¨ .Mtr Chem. Mater. J. af32(2017). eaaf4382 357, oye hsc:Apiain oMlclrAscainadThermore- and Association Molecular to Applications Physics: Polymer otMatter Soft ihpromnevtiesfo omdt thermoplastics commodity from vitrimers High-performance al., et hs e.E Rev. Phys. r pcfi atce ntesse n G and system the in particles specific are c.Adv. 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