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

Article Cite This: J. Phys. Chem. A 2018, 122, 1496−1503

pubs.acs.org/JPCA

DFT Modeling of Cross-Linked Polyethylene: Role of Gold Atoms and Dispersion Interactions Published as part of The Journal of Physical Chemistry virtual special issue “Manuel Yáñez and Otilia Mó Festschrift”. Martin Blaško,† Pavel Mach,‡ Andrej Antušek,§ and Miroslav Urban*,† †

Department of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University, Mlynská Dolina, Ilkovičova 6, 842 15 Bratislava, Slovakia ‡ Department of Nuclear Physics and Biophysics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská Dolina, 84248 Bratislava, Slovakia § Advanced Technologies Research Institute, Faculty of Materials Science and Technology in Trnava, Slovak University of Technology in Bratislava, Bottova 25, 917 24 Trnava, Slovakia ABSTRACT: Using DFT modeling, we analyze the concerted action of gold atoms and dispersion interactions in cross-linked polyethylene. Our model consists of two oligomer chains (PEn) with 7, 11, 15, 19, or 23 carbon atoms in each oligomer cross-linked with one to three Au atoms through C−Au−C bonds. In structures with a single gold atom the C−Au−C bond is located in the central position of the oligomer. Binding energies (BEs) with respect to two oligomer radical fragments and Au are as high as 362−489 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 of the number of carbon atoms, lying between 293 and 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 to the bond distance between saturated closed shell chains in the polyethylene crystal. BEs of pure saturated closed shell PEn−PEn oligomers are 51−187 kJ/mol. Both Au atoms and dispersion interactions contribute considerably to the creation of nearly parallel chains of oligomers with reasonably high binding energies.

1. INTRODUCTION Cross-linking 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 the same or different polyethylene chains are joined together to form the three-dimensional highly oriented chainextended network.1−5 Chemical reactions leading to crosslinking of polymers are mostly initiated by free radicals with a general scheme that consists of the formation of macroradicals and their subsequent recombination in which a polymer chain with an abstractable hydrogen participates. So most of the cross-linking 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 early example is intramolecular complexation of poly(N-methacryloyl-L-lysine) with Cu(II).9 Another example is cross-linking of polydiens by group 10 metals.10,11 One interesting feature of cross-linking by metal atoms is the possibility of reversible cross-linking, e,g, as in ref 12. We also © 2018 American Chemical Society

should mention that cross-linking can be mediated by metal atoms incorporated into the polymer backbone.13−16 The advantage of this method is the intrinsic introduction of a metal to the organic structure acting as, for example, an ultraflexible, yet strong magnetic actuator or generally incorporating some of the electric, magnetic, or optical properties of the metal into the polymer. Very promising is also application of this concept for intrachain cross-linking in preparation of single-chain nanoparticles.17 Another example is gold atoms and gold nanoparticles acting as cross-linkers in polymers or on its surface.18−20 In this paper we will focus on three aspects of the PE−gold cross-linked structures and energetics. Throughout the paper we use oligomers containing 7−23 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. Reference 21 is particularly inspiring in this context since it deals with nanostrips of Received: December 12, 2017 Revised: January 10, 2018 Published: January 10, 2018 1496

DOI: 10.1021/acs.jpca.7b12232 J. Phys. Chem. A 2018, 122, 1496−1503

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The Journal of Physical Chemistry A

Figure 1. PE23−Au−PE23 cross-linked oligomers with 23 carbon atoms in each “linear” oligomer chain neglecting the dispersion interaction (pure PBE0) (A) and oligomers with the D3 contribution included (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 inclusion of the dispersion interaction.

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 cross-linked PE structures with high enough interchain binding energies.

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 cross-linking energetics24 in relation to the length of oligomers cross-linked with or without participating gold atoms. Dispersion interactions between the two oriented parallel chains can be particularly important and responsible for high tensile strength, in contrast to low molecular weight polymers.1,25 Finally, we will analyze structural aspects of crosslinked species focusing on the C−C bond distances in the C− Au−C moiety. In most cross-linking 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 article,1 the length of a carbon−carbon bond can vary considerably depending on the structure of cross-linked 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 modeling 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, respectively.22 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 cross-linked by a gold atom the dispersion contribution is much larger. Therefore, we have decided to study in detail the importance of gold as a cross-linking agent and, at the same

2. COMPUTATIONAL DETAILS Geometry optimizations and energies of our model oligomers cross-linked 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 nanoclusters.27,33 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 were 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 ΔE = 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 ΔE = Ecomplex − EPEAu − EPE*. For structures optimized by the PBE0 + D3 method, influence of the following dispersion models on BEs were tested: (1) PBE0 with no dispersion included; (2) PBE0 with Grimme D3 dispersion model,30−32 denoted as D3; (3) PBE0 with dispersioncorrection with Becke−Johnson damping,37 D3(BJ); (4) PBE0 with D3 and additional three-body dispersion term;38 (5) PBE0 with dispersion-correction with Becke−Johnson damping and additional three-body dispersion term39 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 package.40 Results with 1497


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Figure 2. Oligomers cross-linked with two and three gold atoms: PE15−Au2−PE15 (triplet) (A) and PE15−Au3−PE15 (quartet) (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 inclusion of two or three gold atoms and the dispersion interaction.

Table 1. Binding Energies (BE, kJ/mol) for PEn−Au−PEn Cross-Linked Oligomersa with Respect to Fragments 2PEn* + Aub,c structure

PBE0

PE7−Au−PE7 PE11−Au−PE11 PE15−Au−PE15 PE19−Au−PE19 PE23−Au−PE23

292.6 292.9 293.3 294.2 295.5

PBE0 + D3 362.1 396.1 427.5 458.2 489.0

(258.4) (279.2) (298.3) (309.5) (333.6)

PBE0 + D3(BJ)

PBE0 + D3 + 3body

PBE0 + D3(BJ) + 3body

364.9 397.8 427.6 456.9 486.3

360.9 393.5 423.9 453.6 483.4

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. bPEn−Au−PEn, cross-linked complex, separated PEn* oligomer radicals and Au are all doublets. cPBE0 energies correspond to structures optimized with dispersion contribution neglected. D3 and alternative dispersion contributions (see section 2) are for structures optimized at the PBE0 D3 level. ΔG values at ambient conditions obtained with the PBE0 + D3 method are in parentheses.

PE15 (triplet) and PE15−Au3−PE15 (quartet), respectively. The PBE0 + D3 optimized structures as presented in Figure 2A and Figure 2B, respectively, show similar tendency toward the parallel orientation of the two oligomers like with the single Au atom. The PE15−Au2−PE15 cross-linked oligomer was also considered as an example of the closed shell singlet. More detailed analysis based on binding energies of PEn−Aux−PEn cross-linked oligomers is presented in the next three parts. 3.1. Oligomers Cross-Linked by a Single Au Atom. Binding energies of the PEn−Au−PEn cross-linked 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 of 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, 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

dispersion models 3, 4, and 5 were calculated using Grimme’s Web page.41

3. RESULTS AND DISCUSSION Polyethylene structures cross-linked with 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 cross-linked 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 to each other (note that polymers have a zigzag structure so that interoligomer 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 the tendency toward 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 cross-linking effect of two and three gold atoms is demonstrated by results for PE15 oligomers, PE15−Au2− 1498


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3.2. Cross-Linking of PEn by Two and Three Gold Atoms. The investigation of PEn oligomers cross-linked 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 cross-linking the carbon atoms chains in which the energetically favorable connection is found for cross-linked atoms separated by eight carbon atoms.42 Furthermore, PEn−Aux−PEn complexes with several gold atoms may exist in several spin states. In this section we focus on results that can indicate how additional cross-linking 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 cross-linked with two gold atoms optimized by the PBE0 + D3 method is shown in Figure 2A. C−Au−C cross-links are separated by five 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 cross-links. 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 cross-linked by a single Au atom. Next, we report PE15 chains cross-linked 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 cross-linked by three Au atoms. The C−C distance in the C−Au−C moiety remains similar as in other gold cross-linked 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 cross-link. This value is almost the same as that for PE15 oligomers cross-linked 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.

Figure 3. 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.

any conclusions arising from D3 corrections, so there is no need to discuss them separately. Alternative BEs for PEn−Au−PEn cross-linked oligomers are calculated with respect to fragments representing an oligomergold 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 crosslinking. 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 with the largest PEn−Au−PEn cross-linked oligomers having 7 and 23 carbon atoms, respectively. For assessing thermodynamically allowed ways of making cross-linked 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.

Table 2. Binding Energies (BE, kJ/mol) for PEn−Au−PEn Cross-Linked Oligomers with Respect to Fragments PEnAu(singlet)−PEn* (doublet)a structure

PBE0

PE7−Au−PE7 PE11−Au−PE11 PE15−Au−PE15 PE19−Au−PE19 PE23−Au−PE23

83.43 83.89 84.24 85.15 86.45

PBE0 + D3 128.9 161.8 193.0 223.7 254.5

(75.4) (94.9) (114.1) (125.9) (150.3)

PBE0 + D3(BJ)

PBE0 + D3 + 3body

PBE0 + D3(BJ) + 3body

129.6 161.4 191.1 220.3 249.7

127.9 159.3 189.6 219.2 249.1

128.8 159.5 188.4 216.8 245.4

Structures are optimized without (PBE0) and with the D3 contribution included. ΔG values at ambient conditions obtained with the PBE0 + D3 method are in parentheses.

a

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Figure 4. Closed shell PE15 oligomer cross-linked with two gold atoms, PE15−Au2−PE15 (singlet), optimized with the PBE + D3 method.

The spin state of the PE15−Au3−PE15 complex with three carbon atoms participating in C−Au−C 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 cross-linking 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 cross-linking 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 Au−Au bond is affected by the carbon network quite significantly. The C−Au bond length in PE15−Au2−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−Au2−PE15 closed shell singlet presented in Figure 4 is 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 150 kJ/mol higher than 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 cross-linking 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 cross-linked 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 cross-linked with a single gold atom can be applied (see section 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.

3.3. PEn−PEn Cross-Linking in Crystals and in Pure PEn Oligomers and the Role of Gold as a Cross-Linking Agent. To elucidate the role of gold as a cross-linking 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 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 interoligomer C−C distance in parallel

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.

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 Table 3. PBE0 + D3 Binding Energies (kJ/mol) for Two Linear PEn−PEn Oligomer with Respect to Two PEn fragments structure

PBE0 + D3

PE7···PE7 PE11···PE11 PE15···PE15 PE19···PE19 PE23···PE23

50.9 85.0 119.1 153.5 186.6

structure of pure closed shell PEn−PEn oligomers are much lower (by about 300 kJ/mol) than BEs for structures in which gold atoms participate. 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 cross-linked PEn−Au−PEn oligomers and in the crystal-like closed shell cross-linked PEn−PEn oligomers. In fact, for the same number of C atoms, D3 contributions are very similar. For our example of PE15n−Au2,3−PE15 with two or three Au atoms, dispersion contributions are 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 1500


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organized parallel structures. Both oligomers adopt the sp3 hybridization on each cross-linking 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.

4. 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 crystallike structures. With the PBE0 + D3 method our models for interchain interactions represented by closed shell oligomers containing 7−23 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 cross-linking elements. Already single Au atom crosslinking oligomers having 7−23 carbon atoms (PEn−Au−PEn) elevate BEs to 362−489 kJ/mol depending on the length of the oligomer chain. With the dispersion contribution neglected, BEs are lower, 293−296 kJ/mol, and are almost independent of the oligomer length. The interchain dispersion contribution depends linearly on the number of carbon atoms in the PEn oligomer in the crystal-like PE structures and similarly in gold PEn−Au−PEn cross-linked oligomers. This explains high binding energies for extended cross-linked 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 cross-linked oligomer chains. Closed shell crystal-like 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 enhanced. When the dispersion interaction is neglected, interchain C−C distances rise to more than 5 Å at the oligomer part apart from the C−Au−C bond. With Au acting as a cross-linking agent and dispersion interactions included, BEs are reasonably high, the well-organized parallel structure of oligomers is reconstructed, and the parallelism extends far from the C−Au−C cross-linking bond. In contrast, for oligomers cross-linked via the common radical mechanism with short and strong interchain C−C bonds our attempts in obtaining parallel oligomer structures failed.

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 cross-linked structure including the geometry change (see Table 1) with geometries optimized using PBE0 and PBE0 + D3 methods, respectively.

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 cross-linked 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 cross-linked 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 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 task.46,47 Finally, we discuss an example of a cross-linked oligomer created by the common radical mechanism. In Figure 7 we present the PBE0 + D3 optimized structure of the cross-linked PE15−PE15 oligomer with gold atoms omitted. Both oligomers are cross-linked 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

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +412902 219 988. ORCID

Miroslav Urban: 0000-0003-0379-6414 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. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Slovak Research and Development Agency, Contracts APVV-15-0105 and APVV16-0600, and the Slovak Grant Agency VEGA under Contract

Figure 7. PBE0 + D3 optimized structure of two PE15 oligomers cross-linked with the HC−CH bond according to the radical mechanism. C−C bond distances in angstrom are denoted by dashed lines. 1501


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1/0279/16. Research and Development Operational Programme under projects “University Scientific Park Campus MTF STU-CAMBO” ITMS: 26220220179 and CGreen-II 26240120025 and Project 003STU-2-3/2016 are also acknowledged. We thank Professor Ivan Chodák for his valuable remarks and suggestions.

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

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