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Entropy-Controlled Cross-Linking in Linker-Mediated Vitrimers

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Entropy-Controlled Cross-Linking in Linker-Mediated Vitrimers 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 polymers 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 polymer 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 thermoplastics. 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.
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