DFT Modeling of Cross-Linked Polyethylene: Role of Gold Atoms and

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DFT Modeling of Crosslinked Polyethylene: Role of Gold Atoms and Dispersion Interactions Martin Blaško, Pavel Mach, Andrej Antušek, and Miroslav Urban J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b12232 • Publication Date (Web): 10 Jan 2018 Downloaded from http://pubs.acs.org on January 10, 2018

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The Journal of Physical Chemistry

DFT Modeling of Crosslinked Polyethylene: Role of Gold Atoms and Dispersion Interactions Martin Blaškoa, Pavel Machb, Andrej Antušekc and Miroslav Urbana,* a

Comenius University, Faculty of Natural Sciences, Department of Physical and Theoretical Chemistry, Mlynská dolina, Ilkovičova 6, 841 04 Bratislava, Slovakia.

b

Comenius University, Faculty of Matematics, Physics and Informatics, Department of Nuclear Physics and Biophysics, Mlynská dolina, 84248 Bratislava, Slovakia

c

Slovak University of Technology in Bratislava, Faculty of Materials Science and Technology in Trnava, Advanced Technologies Research Institute, Bottova 25, 917 24 Trnava, Slovakia.

ABSTRACT: Using DFT modeling we analyze the concerted action of gold atoms and dispersion interactions in crosslinked polyethylene. Our model consists of two oligomer chains (PEn) with 7, 11, 15, 19 or 23 carbon atoms in each oligomer crosslinked with one to three Au atoms through C-Au-C bonds. In structures with a single gold atom is the C-Au-C bond located in the central position of the oligomer. Binding energies (BEs) with respect to two oligomer radical fragments and Au are as high as 362 to 489 kJ/mol depending on the length of the oligomer chain. When the dispersion contribution in PEn-Au-PEn oligomers is omitted, BE is almost independent on the number of carbon atoms, lying between 293 to 296 kJ/mol. The dispersion energy contributions to BEs in PEn-Au-PEn rise nearly linearly with the number of carbon atoms in the PEn chain. The carbon-carbon distance in the C-Au-C moiety is around 4.1 Å, similar as the bond distance between saturated closed shell chains in the polyethylene crystal. BEs of pure saturated closed shell PEn-PEn oligomers are 51 to 187 kJ/mol. Both Au atoms and dispersion interactions contribute considerably to the creation of nearly parallel chains of oligomers with reasonably high binding energies.

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1. INTRODUCTION

Crosslinking in polyethylene (PE) and polyolefins in general is a common process used to change their properties, like creep behavior, thermal resistance, surface tension, adhesion, miscibility, microhardness etc.1. It is a process in which carbon atoms of same or different polyethylene chains are joined together to form the three dimensional highly oriented chainextended network1–5. Chemical reactions leading to crosslinking of polymers are mostly initiated by free radicals with a general scheme which consists of the formation of macroradicals and their subsequent recombination in which participates a polymer chain with an abstractable hydrogen. So, most of the crosslinking is based on the formation of covalent (mostly) carbon-carbon bonds. It is typically accomplished by ionizing radiation, peroxide chemistry, or silane chemistry6–8, and involves the recombination of a radical produced by hydrogen abstraction onto PE, with another radical. Less frequent but still viable approach is through complexation of macromolecules with metal ions. One of early example is intramolecular complexation of poly(N- methacryloyl- LLysine) with Cu(II)9. Another example is crosslinking of polydiens by group 10 metals10,11. One of interesting features of crosslinking by metal atoms is possibility of reversible crosslinking, as e.g in Ref.12.We also should mention that crosslinking can be mediated by metal atoms incorporated into the polymer backbone13–16. The advantage of this method is the intrinsic introduction of a metal to the organic structure acting as, e.g., an ultra-flexible, yet strong magnetic actuator or generally, incorporate some of the electric, magnetic or optical properties of the metal in to the polymer. Very promising is also application of this concept for intra-chain crosslinking in preparation of single-chain nanoparticles17. Another example are gold atoms and gold nanoparticles acting as cross-linkers in polymers or on its surface18–20. In this paper we will focus on three aspects of the PE – gold crosslinked structures and energetics. Throughout the paper we use oligomers containing 7 to 23 carbon atoms (n) as models of real polyethylene structures (PEn). First, we suggest that gold atom may play an important role in creating C-Au-C bonds acting this way as an intermolecular “glue”21–23 connecting two polymer chains. Ref. 21 is particularly inspiring in this context since it deals with nanostrips of quinoline-type monomers bound together through C-Au-C and C-Au-N bonds, analogous to those considered in this work. Second, we will investigate dispersion interactions in the polymer crosslinking energetics24 in relation to the length of oligomers crosslinked with or without participating gold atoms. Dispersion interactions between the two oriented parallel


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chains can be particularly important and responsible for high tensile strength, in contrast to low molecular weight polymers1,25. Finally, we will analyse structural aspects of crosslinked species focusing on the C-C bond distances in the C-Au-C moiety. In most crosslinking processes the polymer chain links are formed through short C-C bonds between two PE macromolecules with the secondary radical -CH*- as a part of the main chain. In relation to structural aspects of highly organized polymers it is interesting to calculate the C-C distance in the C-Au-C moiety and the distance between oligomer chains. As summarized by Chodák in his review article1 the length of a carbon-carbon bond can vary considerably depending on the structure of crosslinked chains. The distance between polymers chains in crystals26 with saturated -CH2- groups is typically 4.1 Å, while in the case of the macroradical chains in amorphous regions and other structural motives it can be quite different. In our recent papers22,27 we presented a computational modelling of several structures containing C-Au bonds, their binding energies, bonding characteristics and infrared spectra. Typical are PE structures in which a single gold atom substitutes a hydrogen atom. We have suggested that such structures can be prepared by ion implantation22,28 with bonding mechanism in which the most important role is played by the fragmentation of the carbon chain and the creation of the R1-CH*-R2 radicals in which R1 and R2 are saturated oligomer fragments. Binding energies for a terminal and central C-Au bonds were found to be 227 and 209 kJ/mol, respectively22. Dispersion contributions are small, 5 and 13 kJ/mol, respectively. Preliminary calculations revealed that in structures like PE-Au-PE in which the oligomer fragments are crosslinked by a gold atom the dispersion contribution is much larger. Therefore, we have decided to study in detail the importance of gold as a crosslinking agent and, at the same time, the role of dispersion interaction contributions in relation to the length of the oligomer fragment. Our goal is to suggest by computational methods that there may exist well organized linearized gold crosslinked PE structures with high enough inter-chain binding energies.


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2. COMPUTATIONAL DETAILS

Geometry optimizations and energies of our model oligomers crosslinked by one to three gold atoms were performed using DFT/PBE0 functional29 with Grimme dispersion correction term (D3)30–32. Using of PBE0 is based on benchmark calculations of interactions of selected ligands with gold atom and gold nanoclusters27,33. For hydrogen and carbon atoms the def2-TZVP basis set34 was used. For gold atoms the def2-TZVP basis set was combined with def2 scalar relativistic effective core potential (RECP)34 describing 60 core electrons. For all calculations ultrafine grid for numerical integration35 and resolution of identity36 was used. To confirm energy minima of the optimized structures, vibrational analysis was performed. The binding energies (BE) were calculated as the energy difference of the complex with respect to two separated polyethylene radicals PE* and n separated gold atoms ∆E = Ecomplex –2EPE* – nEAu. For estimates of a single C-Au binding energy we have used an alternative fragmentation of the complex in which BE is calculated with respect to PE* radical oligomer and PE-Au in which a hydrogen atom is substituted by a gold atom in a PE oligomer and the BE is defined as: ∆E = Ecomplex – EPEAu – EPE*. For structures optimized by the PBE0+D3 method, influence of following dispersion models on BEs were tested: 1) PBE0 with no dispersion included; 2) PBE0 with Grimme D3 dispersion model30–32 – denoted as D3; 3) PBE0 with dispersion-correction with Becke-Johnson damping37 – D3(BJ); 4) PBE0 with D3 and additional 3-body dispersion term38 5) PBE0 with dispersion-correction with Becke-Johnson

damping and additional 3-body

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dispersion term denoted as D3(BJ) + 3body. These calculations were performed as single point calculations without structure reoptimization in the respective dispersion model. All calculations were performed with the Turbomole 6.5 software package40. Results with dispersion models 3, 4, and 5 were calculated using Grimme’s webpage41.


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3. RESULTS AND DISCUSSION Polyethylene structures crosslinked with a single, two or three Au atoms are represented by oligomer chains with 7, 11, 15, 19, or 23 carbon atoms (oligomers are denoted as PEn), respectively. When crosslinked with a single gold atom, Au is located at the central position of PEn-Au-PEn oligomers. Optimized structures shown in Figure 1 for PE23-Au-PE23 represent a typical example of two oligomers linked with a single gold atom. These structures were optimized using the pure PBE0 functional (Figure 1A) and PBE0 supplemented by the D3 dispersion contribution (Figure 1B). Selected C-C distances demonstrate that the dispersion pushes the two oligomers closer each to the other (note that polymers have a zig-zag structure so that inter-oligomer C-C distances vary depending on their position). The same shows the side view and the C-Au-C bond angle which is smaller for structures optimized with the D3 contribution. In short, the dispersion interaction supports tendency towards the parallel orientation and closer distance between the two oligomers. This effect extends to -CH2- groups of both oligomers far away from the central carbon atom bound to Au. The PEn-Au-PEn oligomers are open shell doublets. The crosslinking effect of two and three gold atoms is demonstrated by results for PE15 oligomers, PE15-Au2-PE15 (triplet) and PE15Au3-PE15 (quartet), respectively. The PBE0 + D3 optimized structures as presented in Figures 2A and 2B, respectively, show similar tendency towards the parallel orientation of the two oligomers like with the single Au atom. The PE15-Au2-PE15 crosslinked oligomer was also considered as an example of the closed shell singlet. More detailed analysis based on binding energies of PEn-Aux-PEn crosslinked oligomers is presented in next three parts.

3.1. Oligomers crosslinked by a single Au atom. Binding energies of the PEn-Au-PEn crosslinked complexes with respect to products - the gold atom and two PEn radical fragments are presented in Table 1. These complexes are doublets with the spin density distributed over the whole C-Au-C moiety. First, we note that at the PBE0 level (i.e. dispersion terms omitted) BE is almost constant, independent on the number of C atoms in the PEn oligomer. For 23 carbon atoms in each oligomer it is 296 kJ/mol, while with the Grimme dispersion-correction term D3 it is much larger, 489 kJ/mol. Both values are calculated with geometries optimized with the PBE0 or with the PBE0 + D3 method,


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respectively. This means that for our largest oligomer the dispersion contribution (194 kJ/mol including the geometry change) represents 40% of the total binding energy. Figure 3 clearly demonstrates nearly linear dependence of binding energies on the number of carbon atoms in the PEn oligomer. Numerically, the relevance of our findings is partly supported by presenting alternative dispersion terms in Table 1, which do not modify any conclusions arising from D3 corrections so there is no need to discuss them separately. Alternative BEs for PEn-Au-PEn crosslinked oligomers are calculated with respect to fragments representing an oligomer-gold closed shell PEn-Au complex and the oligomer radical R1–CH*–R2 in its doublet state. This is considered as a natural extension of our previous calculations of PEn-Au species. In our recent paper22 we analyzed C–Au binding energies in PEn-Au complexes which can be as high as 227 kJ/mol. A possible way leading to PEn-Au species is the gold implantation of polyethylene through the creation of R1-CH*-R2 radicals, which are so important in the PE cross linking. The mechanism of such reactions was studied using the molecular dynamics simulations. Binding energies of PEn-Au-PEn with respect to the PEn-Au complex and R1-CH*-R2 radicals are presented in Table 2. Since these BEs represent breaking of just a single C-Au bond, they are lower than those presented in Table 1 calculated with respect to all doublet fragments. Again, at the PBE0 level BEs increase just slightly with the size of the oligomer, but due to dispersion interactions BEs are quite high, ranging from 129 to 255 kJ/mol for the smallest and the largest PEn-Au-PEn crosslinked oligomers having 7 and 23 carbon atoms, respectively. For assessing thermodynamically allowed ways of making crosslinked PEn-Au-PEn structures we present Gibbs energies with respect to all doublet products (single gold atom and two PEn radical fragments), Table 1. Gibbs energies with respect to the PEn-Au complex and the oligomer radical R1–CH*–R2 are presented in Table 2. Both ways are thermodynamically accessible once having oligomer radicals, as described in the previous paragraph.

3.2. Crosslinking of PEn by two and three gold atoms The investigation of PEn oligomers crosslinked by more gold atoms is not straightforward since there are many possibilities of location of Au atoms within the oligomer. Results may be different for C-Au-C bonds separated by one, two or more carbon atoms in the oligomer chain. This resembles the situation in crosslinking the carbon atoms chains in which the energetically


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favorable connection is found for crosslinked atoms separated by 8 carbon atoms42. Furthermore, PEn-Aux-PEn complexes with several gold atoms may exist in several spin states. In this Part we focus on results which can indicate how additional crosslinking Au atoms affect the structure, binding energies and the importance of the dispersion term in PEn-Aux-PEn complexes. Some insight into this problem can be obtained considering results for oligomers with 15 carbon atoms and with two or three Au atoms. The structure of two PE15 chains crosslinked with two gold atoms optimized by the PBE0 + D3 method is shown in Figure 2A. C-Au-C crosslinks are separated by 5 carbon atoms. This is sufficient for keeping the spin density fully located at the C-Au-C part of the complex. The binding energy with respect to two PE15 triplet radical fragments and the two Au atoms (doublets) is 708 kJ/mol, i.e. 354 kJ/mol per single C-Au-C crosslinks. BE with respect to the PE15 fragment (triplet) and the PE15-Au complex (singlet) is 240/kJ/mol. The C-C distance in the C-Au-C moiety, 4.2 Å, is slightly larger than in structures crosslinked by a single Au atom. Next, we report PE15 chains crosslinked with three gold atoms. We optimized several structures with various multiplicities. The structure presented in Figure 2B is for the quartet state of PE15-Au3-PE15 with C-Au-C moiety separated by four C atoms (their distance is 6.8 Å) in each PE15 oligomer optimized by the PBE0 + D3 method. Oligomer chains are nearly parallel, even though terminal CH3 groups are a bit more open than in structures with one Au atom. This is probably caused by too short CH2 chain in PE15 oligomers crosslinked by three Au atoms. The C-C distance in the C-Au-C moiety remains similar as in other gold crosslinked structures, about 4.0 Å. The C-Au bond length is short, 2.14 Å. The binding energy of PE15-Au3-PE15 (quartet) with respect to two PE15 quartet radical fragments and three Au atoms (doublets) is 1072 kJ/mol, i.e. 357 kJ/mol per a single C-Au-C crosslink. This value is almost the same as that for PE15 oligomers crosslinked with two Au atoms. The quartet PE15 is the radical (with the S2 eigenvalue 3.77) with three hydrogen atoms removed from the oligomer at the location of the binding site of Au. We use it for assessing the binding energy in the high spin PE15-Au3-PE15 complex. The spin state of the PE15-Au3-PE15 complex with three carbon atoms participating in C-AuC bonds separated by four C atoms needs some comment. First, the S2 eigenvalue is 3.78, i.e. it is almost a pure quartet. Calculated spin densities show that a single unpaired electron is located at each of well separated C-Au-C groups. The possibility of creating high spin species by


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crosslinking polymers by well separated Au atoms needs careful investigation. Finally, we note that we have also calculated PE15-Au3-PE15 in its doublet state. However, this state was heavily spin contaminated, so it could not be studied by methods used in the present paper. Finally, we have checked crosslinking of two PE15 oligomers with two gold atoms which resemble the isolated closed shell Au2 molecule, i.e. both gold atoms are in the adjacent location in the oligomer chain (Figure 4). The idea was having an example of a closed shell PEn-Aux-PEn cross linked species. Optimized Au-Au distance in this structure is 2.99 Å, larger than in the isolated Au2 molecule (2.53 Å, optimized with the PBE0 + D3 method). This means that the AuAu bond is affected by the carbon network quite significantly. The C-Au bond length in PE15Au2-PE15 is around 2.2 Å, the C-C distance in the C-Au-C part of the complex is 4.00 Å. The whole complex exhibits nearly parallel orientation of both carbon chains. With respect to the Au2 molecule and two closed shell alkenes having the C7=C8 double bond the binding energy of PE15-Au2-PE15 is 152 kJ/mol. The corresponding Gibbs energy is 33 kJ/mol. The PE15-Au2PE15 closed shell singlet presented in Figure 4 is by 101 kJ/mol energetically more favorable to the triplet state with similar location of the two adjacent C-Au-C bonds. Absolute PBE0 + D3 energy of the triplet state with two remote gold atoms as presented in Figure 2A is by 150 kJ/mol higher than is the energy of the closed shell PE15-Au2-PE15 complex with two adjacent gold atoms in Figure 4. All suggested structures with one to three crosslinking gold atoms are thermodynamically stable with respect to various products, as discussed above. The question of how to prepare these compounds remains open. The crucial step in creating the gold crosslinked structures is the availability of macroradicals resulting from the hydrogen abstraction and their subsequent reactions with gold, gold clusters or gold containing species. We believe that for high spin complexes with two or three gold atoms similar considerations as for oligomers crosslinked with a single gold atom can be applied (see Part 3.1). We have no suggestion for thermodynamically allowed

ways of making selectively the ordered structures suggested in Figures 1, 2, and 4. Further MD study focused on the optimization of processes and experimental conditions leading to ordered multiple crosslinked structures is needed.


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3.3. PEn-PEn crosslinking in crystals and in pure PEn oligomers and the role of gold as a crosslinking agent. To elucidate the role of gold as a crosslinking agent and the importance of dispersion contributions we compare BEs and the dispersion contribution in PEn-Aux-PEn (with one to three gold atoms) with crosslinked closed shell oligomers PE15-PE15 (with no Au) having structures analogous to the crystal form of the polyethylene. Optimized PE15-PE15 structure is shown in Figure 5. The inter oligomer C-C distance in parallel parts is almost constant, 4.1 Å and very similar to that in PEn-Au-PEn. Table 3 demonstrates that BEs in the crystal-like structure of pure crosslinked closed shell PEn-PEn oligomers are much lower (by about 300 kJ/mol) than BEs for structures in which participate gold atoms. This confirms the crucial role of contributions to BEs arising from C-Au-C bonds combined with the dispersion term. The importance of dispersion interactions in polymer crystals was also stressed and analyzed by Liu et al.24. As we can see from Figure 6, dispersion terms represented by D3 energies depend linearly on the number of C atoms in both Au crosslinked PEn-Au-PEn oligomers and in the crystal-like closed shell crosslinked PEn-PEn oligomers. In fact, for the same number of C atoms are D3 contributions very similar. For our example of PE15n-Au2,3-PE15 with two or three Au atoms are dispersion contributions slightly larger than with a single Au atom. D3 energies in Figure 6 are taken directly as contributions for the PBE0 + D3 optimized PEn-Au1,2,3-PEn and PEn-PEn complexes, respectively. The true dispersion contribution to binding energies in PEn-Au-PEn is represented by the dashed line in Figure 6. This contribution is calculated as the difference between PBE0 binding energies for complexes optimized by the PBE0 method and PBE0 + D3 binding energies optimized including the D3 term (see Table 1). Again, the dependence on the oligomer size is linear. One can estimate the dispersion contribution to BE per single CH2 group in an oligomer. For example, in the PE23-Au-PE23 crosslinked oligomer with 22 CH2-CH2 bonds it is 8.8 kJ/mol. It is small, but due to additive contributions they affect BEs considerably. Also, the dispersion contribution supports the parallelization of the crosslinked oligomer structures. Linear dependence of dispersion contribution on the number of carbon atoms indicates that methods used in the present paper represent this term reasonably well irrespective


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of the size of the system which is not so obvious with other methods.43–45 We also note that benchmarking of noncovalent interactions for large systems is a difficult task46,47. Finally, we discuss an example of a crosslinked oligomer created by the common radical mechanism. In Figure 7 we present the PBE0 + D3 optimized structure of the crosslinked PE15PE15 oligomer with gold atoms omitted. Both oligomers are crosslinked via a short CH-CH bond between the two oligomers with the C-C bond distance 1.54 Å. With various starting geometries we were unable to obtain well organized parallel structures. Both oligomers adopt the sp3 hybridization on each crosslinking carbon atom which results in the out-of-the plane orientation of a part of the oligomer chain and very large C-C distance between some remote parts of the oligomer.

CONCLUSIONS The importance of the dispersion contribution to interaction energies between two oligomers representing the chains in polyethylene is expected. Such interactions are also essential in creating long parallel oligomer chains in saturated PE crystal-like structures. With the PBE0 + D3 method our models for interchain interactions represented by closed shell oligomers containing 7 to 23 carbon atoms (PEn) lead to binding energies of 51 – 187 kJ/mol. Binding energies between oligomer chains can be greatly enhanced by implemented gold atoms which can act as crosslinking elements. Already single Au atom crosslinking oligomers having 7 to 23 carbon atoms (PEn-Au-PEn) elevates BEs to 362 - 489 kJ/mol depending on the length of the oligomer chain. With the dispersion contribution neglected BEs are lower, 293 to 296 kJ/mol and are almost independent on the oligomer length. The interchain dispersion contribution depends linearly on the number of carbon atoms in the PEn oligomer in the crystal-like crosslinked PE structures and similarly in gold PEn-Au-PEn crosslinked oligomers. This explains high binding energies for extended crosslinked oligomers, even though contribution per a single H2C…CH2 bond (H3C…CH3 bonds for terminal oligomer groups) is small, less than 9 kJ/mol. The concerted role of dispersion interactions and the crosslinking ability of gold plays a crucial role in the structural organization of crosslinked oligomer chains. Closed shell crystallike oligomers have rather small interaction energies, but are characterized by nearly parallel oligomer chains with the interchain distance 4.1 Å. Upon implanting gold atoms, BEs are


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enhanced. When the dispersion interaction is neglected, interchain C-C distances rise to more than 5 Angstrom at the oligomer part apart from the C-Au-C bond. With Au acting as a crosslinking agent and dispersion interactions included, BEs are reasonably high, the wellorganized parallel structure of oligomers is reconstructed, and the parallelism extends far from the C-Au-C crosslinking bond. In contrast, for oligomers crosslinked via the common radical mechanism with short and strong interchain C-C bonds our attempts in obtaining parallel oligomer structures failed.


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Figure 1. PE23-Au-PE23 crosslinked oligomers with 23 carbon atoms in each “linear” oligomer chain neglecting the dispersion interaction (pure PBE0, part A), and oligomers with the D3 contribution included (part B). Selected C-C bond distances in Angstrom are denoted by dashed lines. Side view and C-Au-C bond angles demonstrate the tendency to the parallel orientation of the two oligomers upon including the dispersion interaction.


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Figure 2. Oligomers crosslinked with two and three gold atoms: PE15-Au2 -PE15 (triplet, part A) and PE15-Au3 -PE15 (quartet, part B). Selected C-C bond distances in Angstrom are denoted by dashed lines. Side view and C-Au-C bond angles indicate the tendency to the parallel orientation of the two oligomers upon including two or three gold atoms and the dispersion interaction.

Figure 3. The dependence of BEs of PEn-Au-PEn complexes on the number of carbon atoms in the oligomer. BEs are calculated with respect to fragments 2PEn + Au. Pure PBE0 results (blue line) and results with the D3 contribution (red line) are calculated at their optimized geometries.


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Figure 4. The closed shell PE15 oligomer crosslinked with two gold atoms, PE15-Au2 -PE15 (singlet) optimized with the PBE + D3 method.

Figure 5. PBE0 + D3 optimized crystal-like structures of parallel closed shell PE15-PE15 oligomers. C-C bond distances in Angstrom are denoted by dashed lines.


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-300

D3 contribution, PE-Au-PE(d) series including the geometry change

-275

Dispersion energy [kJ/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


D3 in the PE…PE(s) oligomers

-250

D3 in the PE-Au-PE(d) structure

-225

D3 in the PE15-Au2-PE15(s) structure

-200

D3 in the PE15-Au3-PE15(q) structure

-175

D3 in the PE15-Au2-PE15(t) structure

-150 -125 -100 -75 -50 3

7

11

15

19

23

Number of carbon atoms in the single oligomer Figure 6. D3 energies in PEn-Aux-PEn and in the crystal-like PEn-PEn closed shell structures at geometries optimized by the PBE0 + D3 method. Dotted line represents the D3 contribution to the binding energy in the PEn-Au-PEn crosslinked structure including the geometry change (see Table 1) with geometries optimized using PBE0 and PBE0 + D3 methods, respectively.

Figure 7. PBE0 + D3 optimized structure of two PE15 oligomers crosslinked with the HC-CH bond according to the radical mechanism. C-C bond distances in Angstrom are denoted by dashed lines.


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Table 1. Binding energies (BE) for PEn-Au-PEn crosslinked oligomersa with respect to fragments 2PEn + Aub. PBE0 energies correspond to structures optimized with dispersion contribution neglected. D3 and alternative dispersion contributionsc are for structures optimized at the PBE0 D3 level. ∆G values at ambient conditions obtained with the PBE0+D3 method are in parentheses. structures PE7-Au-PE7 PE11-Au-PE11 PE15-Au-PE15 PE19-Au-PE19 PE23-Au-PE23

PBE0 292.6 292.9 293.3 294.2 295.5

PBE0+D3 362.1 (258.4) 396.1 (279.2) 427.5 (298.3) 458.2 (309.5) 489.0 (333.6)

PBE0+D3(BJ)* 364.9 397.8 427.6 456.9 486.3

PBE0+D3+3body* 360.9 393.5 423.9 453.6 483.4

PBE0+D3(BJ) + 3body* 364.0 395.8 424.8 453.3 481.9

a

Symbols PEn represent the number of carbon atoms in each oligomer used as a model for polyethylene. b PEn-Au-PEn, crosslinked complex, separated PEn oligomers and Au are all doublets. c see Part 2. Table 2. Binding energies (BE) for PEn-Au-PEn crosslinked oligomersa with respect to fragments PEnAu(singlet) – PE(doublet). Structures are optimized without (PBE0) and with the D3 contribution includedb. ∆G values at ambient conditions obtained with the PBE0+D3 method are in parentheses. structures PE7-Au-PE7 PE11-Au-PE11 PE15-Au-PE15 PE19-Au-PE19 PE23-Au-PE23 a

PBE0 83.43 83.89 84.24 85.15 86.45

PBE0+D3 128.9 (75.4) 161.8 (94.9) 193.0 (114.1) 223.7 (125.9) 254.5 (150.3)

PBE0+D3(BJ)* 129.6 161.4 191.1 220.3 249.7

PBE0+D3+3body* 127.9 159.3 189.6 219.2 249.1

PBE0+D3(BJ) + 3body* 128.8 159.5 188.4 216.8 245.4

PEn-Au-PEn, crosslinked complex and Au are doublets, separated PEnAu complex is a singlet.

Table 3.PBE0 +D3 binding energies (kJ/mol) for two linear PEn– PEn oligomer with respect to two PEn fragments. structures PE7...PE7 PE11...PE11 PE15...PE15 PE19...PE19 PE23...PE23

PBE0+D3 50.9 85.0 119.1 153.5 186.6


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ACKNOWLEDGMENTS

This work was supported by the Slovak Research and Development Agency, contracts No. APVV-15-0105 and APVV-16-0600 and the Slovak Grant Agency VEGA under the contract No. 1/0279/16. Research and Development Operational Programme under the project "University Scientific Park Campus MTF STU - CAMBO" ITMS: 26220220179 and project No. 003STU-2-3/2016 are also acknowledged. We thank Professor Ivan Chodák for his valuable remarks and suggestions.


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AUTHOR INFORMATION Corresponding Author *[email protected], +412902 219 988 Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally. Funding Sources APVV-15-0105, APVV-16-0600 VEGA 1/0279/16 Notes No additional notes.


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