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, 637459 Singapore; bDivision of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore; and cDepartment of Chemistry, National University of Singapore, 117546 Singapore Edited by Daan Frenkel, University of Cambridge, Cambridge, United Kingdom, and approved September 19, 2020 (received for review July 24, 2020) Recently developed linker-mediated vitrimers based on metathe- ingly, even at the low temperature limit, the cross-linking degree sis of dioxaborolanes with various commercially available poly- of vitrimers still depends on the concentration of free linkers, mers have shown both good processability and outstanding which essentially offers an extra degree of freedom in controlling performance, such as mechanical, thermal, and chemical resis- the mechanical property of the resulting materials. tance, suggesting new ways of processing cross-linked polymers in industry, of which the design principle remains unknown [M. Results Rottger¨ et al., Science 356, 62–65 (2017)]. Here we formulate a Coarse-Grained Model of Vitrimer. We consider a system of vol- theoretical framework to elucidate the phase behavior of the ume V consisting of Npoly polymer chains, in which each polymer linker-mediated vitrimers, in which entropy plays a governing comprises n hard spheres of diameter σ. As shown in Fig. 1A, on role. We find that, with increasing the linker concentration, vit- each polymer, there are m 1 precursors (reactive sites) P uni- rimers undergo a reentrant gel–sol transition, which explains a formly distributed, which can react with a cross-linker molecule recent experiment [S. Wu, H. Yang, S. Huang, Q. Chen, Macro- C by forming a dangling PC bond and producing a byproduct molecules 53, 1180–1190 (2020)]. More intriguingly, at the low free molecule B through metathesis reactions. Moreover, a dan- temperature limit, the linker concentration still determines the gling PC bond can further react with another intact precursor P cross-linking degree of the vitrimers, which originates from the to form a cross-linking P2C bond and producing an additional competition between the conformational entropy of polymers free B molecule, and each cross-linker can form, at most, two and the translational entropy of linkers. Our theoretical predic- bonds with two different precursors. The metathesis reactions APPLIED PHYSICAL SCIENCES tions agree quantitatively with computer simulations, and offer are reversible, and ∆G is the reaction energy. NC and NB are the guidelines in understanding and controlling the properties of this numbers of cross-linker molecule C and the byproduct molecule newly developed vitrimer system. B in the system, which are controlled by the chemical poten- tials µC and µB, respectively. We define nPC and nP2C as the vitrimer j metathesis reaction j reentrant gel–sol transition j average numbers of PC bonds with dangling cross-linkers and entropy-driven cross-linking cross-linking P2C bonds per polymer, respectively. Similarly, nP is the average number of precursors per polymer that remain itrimers, a new type of polymeric material, are known to intact, and Ni = Npoly ni is the total number of precursors or Vexhibit unique properties that combine the advantages of bonds of type i = P, PC, P2C in the system. The packing fraction π 3 thermosets and thermoplastics. To be specific, they are mechan- of polymers is φp = 6 Npolynσ , and B and C are both modeled as ically robust and insoluble while also recyclable and malleable hard spheres of diameter σ. (1, 2). At low temperatures, vitrimers behave as cross-linked We employ Monte Carlo (MC) simulations to investigate thermosets, while, at high temperatures, the exchangeable bonds this coarse-grained hard-sphere-chain system. An infinitely deep in the polymer network swap reversibly by the thermally trig- gered reactions so that they behave as viscoelastic liquids (3–12). In the past decade, various chemical reactions, such as the trans- Significance esterification reaction (2, 13), transamination reaction (14, 15), alkoxyamine exchange reaction (16, 17), olefin metathesis (18), The recently developed linker-mediated vitrimers based on and thiol–disulfide exchange (19), have been used in the pro- metathesis reactions offer new possibilities of processing duction of vitrimers for different applications. However, the cross-linked polymers with high mechanical performance in reactants or catalysts involved in reactions of the conventional industry, while the design principle remains unknown. Here vitrimers are usually not thermally or oxidatively stable, which we propose a theoretical framework for describing the sys- is particularly detrimental for using the same equipment and tem of linker-mediated vitrimers, in which entropy is found conditions of processing thermoplastics (9). to play a dictating role. Our mean field theory agrees quan- Recently, a new type of linker-mediated vitrimer was devel- titatively with computer simulations, and provides guidelines oped based on the metathesis reaction of dioxaborolanes, in for the rational design of the linker-mediated vitrimers with which the functionalized polymers with pendant dioxaborolane desired properties. units react with bis-dioxaborolanes (cross-linkers), and the metathesis reaction here both cross-links the polymers and Author contributions: R.N. designed research; Q.-L.L. and X.X. performed research; dynamically changes the polymer network (20). The linker- Q.-L.L., X.X., J.Y., M.P.C., and R.N. analyzed data; and Q.-L.L., X.X., J.Y., M.P.C., and R.N. mediated vitrimers have superior chemical resistance and dimen- wrote the paper.y sional stability, without the need of a catalyst, and can be The authors declare no competing interest.y processed like thermoplastics (21). In this work, we propose a This article is a PNAS Direct Submission.y mean field theory combined with coarse-grained computer sim- Published under the PNAS license.y ulations to study the linker-mediated vitrimer system, and our 1 Q.-L.L. and X.X. contributed equally to this work.y results show that the entropy of free linkers plays a nontrivial 2 To whom correspondence may be addressed. Email: [email protected] role. We find that, with increasing the concentration of free link- This article contains supporting information online at https://www.pnas.org/lookup/suppl/ ers, the vitrimer system undergoes a reentrant gel–sol transition, doi:10.1073/pnas.2015672117/-/DCSupplemental.y which was observed in a recent experiment (22). More interest- First published October 21, 2020. www.pnas.org/cgi/doi/10.1073/pnas.2015672117 PNAS j November 3, 2020 j vol. 117 j no. 44 j 27111–27115 Downloaded by guest on September 27, 2021 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 jr − r j < rcut SI Appendix for details). Vbonds(r, r ) = , [1] 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.