Ab Initio Calculations of Possible γ-Gauche Effects in the 13C-NMR for Methine and Carbonyl Carbons in Precise Polyethylene Acrylic Acid Copolymers

The impacts of local polymer chain conformations on the methine and carbonyl 13C-NMR chemical shifts for polyethylene acrylic acid p(E-AA) copolymers were predicted using ab initio methods. Using small molecular cluster models, the magnitude and sign of the γ-gauche torsional angle effect, along with the impact of local tetrahedral structure distortions near the carbonyl group, on the 13C-NMR chemical shifts were determined. These 13C-NMR chemical shift variations were compared to the experimental trends observed for precise p(E-AA) copolymers as a function acid group spacing and degree of zinc-neutralization in the corresponding p(E-AA) ionomers. These ab initio calculations address the future ability of 13C-NMR chemical shift variations to provide information about the local chain conformations in p(E-AA) copolymer materials.


Introduction
In polyethylene (PE) the 13 C-NMR chemical shift difference between the crystalline and amorphous phases are commonly attributed to the γ-gauche effect (trans-gauche effect) arising from differences in the local polymer chain conformation [1][2][3][4]. In polymers, intermolecular effects that include chain packing, polar interactions, and hydrogen bonding, can also impact the observed 13 C-NMR chemical shifts. The γ-gauche effect has recently been employed in measuring the backbone and side chain conformations of proteins [5][6][7][8]. Ab initio calculations of NMR chemical shifts in amorphous and crystalline polymers and proteins have proven indispensable in determining the local chain conformations and microstructure for these materials. The γ-gauche effect has been documented for a variety of OPEN ACCESS substituted PE polymers. For example, the 13 C-NMR chemical shifts of methylene (CH 2 ) segments in substituted vinyl polymers such as polypropylene (PP), polyisobutylene (PIB), poly(vinyl chloride) (PVC) and polybutadiene (PBD) have been thoroughly investigated using ab initio methods [9][10][11][12][13]. Previous studies have concentrated on understanding the 13 C-NMR chemical shift and resonance band shape of the methylene (CH 2 ) carbon segment in terms of polymer chain conformations, and the role different functional groups have on the micro-structure. In PVC and PP there have also been limited evaluations of the methine (CH) 13 C-NMR chemical shifts that demonstrate a γ-gauche effect for this carbon [1,2,9,12].
Recently our group reported the 13 C-MAS-NMR for a series of precise polyethylene acrylic acid p(E-AA) copolymers and associated Zn-substituted ionomers [14]. These polymers have a carboxylic acid substituent located at precise locations along the PE backbone chain, and were used to produce the corresponding Zn-neutralized ionomers (Scheme 1).

Scheme 1. Precise p(E-AA) Copolymers and Ionomers.
The methylene (CH 2 ) 13 C-NMR chemical shifts were used to identify and quantify the crystalline and amorphous phases present in these p(E-AA) copolymers, with the spectral identification based on the well-established γ-gauche effect in PE type polymers. The broad resonance corresponding to the amorphous trans/gauche phase is shifted approximately −2 ppm with respect to the all trans conformation in the crystalline phase [14]. For these same precise p(E-AA) copolymers distinct trends in the 13 C-NMR chemical shifts were also observed for the methine (CH) carbon as a function of the chain length spacing between the carboxylic acid groups, and as a function of the extent of Zn-neutralization. For example, the chemical shift of the CH resonance increases by almost +3 ppm when the chain space length varies from p9AA to p21AA. In addition, there are distinct multiple CH resonances observed in the p9AA copolymer (δ = +44.5 and +46.5 ppm) and the corresponding Zn-neutralized ionomer (δ = +47.0 and 49.6 ppm). The carbonyl (CO) 13 C-NMR chemical shift revealed only a minor chemical shift variation, increasing +0.7 ppm between the p9AA and p21AA copolymers. Zn neutralization produced a larger variation, increasing the shift between +2 and +4 ppm [14]. Since these precise copolymers have isolated functional groups, investigation of the NMR chemical shift for the CH carbon is proposed to provide a direct probe for the local chain conformation at the point of chain functionalization.
In our original analysis of these 13 C-NMR chemical shift trends in the p(E-AA) copolymers and ionomers, we argued that the CH variations resulted from a γ-gauche effect, and that Zn-neutralization did not have a large influence on the observed shift. In contrast, we suggested that the observed CO 13 C-NMR chemical shift variations were dominated by the zinc neutralization of the acid group, with the γ-gauche effect playing a minor role. Unfortunately, there are a limited number of studies establishing the magnitude of the γ-gauche effect for CH and CO carbons in amorphous polymer systems, in particular, systems involving carboxylic acid pendant groups. In addition, the small 2-ethyl-butanoic acid model cluster employed to initially model the 13 C-NMR chemical shifts using ab initio methods was too small to eliminate end-group effects from impacting the calculated shifts of the central CH, and only a very limited number of chain conformations were considered in that analysis [14].
This paper addresses these issues by presenting a series of ab initio NMR 13 C-NMR chemical shift calculations using a larger molecular cluster to model the acrylic acid polymer fragment as a function of multiple torsional angles describing the chain conformation. The magnitude and sign of the γ-gauche effect were evaluated for both the CH and CO carbons, and are compared to the 13 C-NMR experimental results observed in the p(E-AA) copolymers and Zn-neutralized ionomers [14].

Results and Discussion
The 2-butylhexanoic acid molecular fragment ( Figure 1) was chosen as the model cluster for evaluation of the CH and CO 13 C-NMR chemical shifts within p(E-AA) copolymers. The methine (CH) carbon, C5, is sufficiently removed from the end of this molecular cluster to minimize end-group effects [9]. The dihedral angles (θ 1 , θ 2 , θ 3 , θ 4 ) describing the local polymer chain conformations around the CH carbon ranging from −180° < θ i ≤ 180° are shown in Figure 1a. The rotational sense of the individual dihedral angles was always defined looking down the bond originating at the carbon closest to the central COOH pendant group (i.e., along the C 4 -C 3 for θ 1 or the C 5 -C 4 bond for θ 2 ) as depicted in Figure 1b. The rotations of dihedral angles near the ends of the chains were not considered to be important in describing the local conformation for the central carbon, such that these dihedral angles were not constrained and allowed to adopt a minimum energy conformation.
The potential energy surface as a function of the dihedral rotation angles was calculated in the gas phase for the model cluster.
Step-wise rotation of a single dihedral angle (while allowing all other structural constraints to optimize) gives rise to the classic potential energy surface with three local energy minima designated as trans (t), along with two gauche (g and g') conformations. The potential energy surface as a function of rotation about both the θ 1 and θ 2 dihedral angles reveals 9 local minima as revealed in Figure 2. The angles and relative energies for these 9 minima are given in Table 1.
The potential energy surface as a function of the θ 3 and θ 4 dihedral angles is identical to the θ 1 and θ 2 surface (Figure 2) except for a change in the sense of rotation in defining the g and g' conformations. The potential energy surface involving all four dihedral angles (θ 1 , θ 2 , θ 3 , θ 4 ) is therefore expected to contain 81 different local minima. The structures of the model clusters corresponding to these local energy minima were extracted and used to model the dominant polymer chain conformations. It should be noted that these are gas phase calculations for a short model fragment, and that the minima and relative populations obtained are unlikely to fully reflect the conformations present within polymers. It has previously been demonstrated that the size of the molecular model fragment can be restricted to γ-and δ-neighbors for the carbon in question, with only minor changes in the predicted chemical shifts resulting from the longer range end-effects [9].

Methine 13 C-NMR Chemical Shifts
The 13 C-NMR chemical shifts for the CH carbons in these 81 optimized clusters were evaluated and are shown as a function of the |θ| 1 + |θ| 4 angle summation in Figure 3a. Variations of these dihedral angles describe the relative γ orientation of the CH carbon with respect to the remainder of the polymer chain on each side. For conformations where these two dihedral angles are both in a trans conformation the |θ| 1 + |θ| 4 summation will be 360°, while those conformations involving both dihedrals in a gauche orientation will have a summation of ~120°. It is immediately apparent that the calculated 13 C-NMR chemical shifts show significant variations, and are not a simple function of the dihedral angles. For the θ 1 , θ 4 gauche-gauche conformations (g**g, g**g', g'**g, g'**g', where * denotes the unspecified θ 2 and θ 3 dihedral angles) there is an almost 10 ppm chemical shift variation observed (vertical distribution). Not all of these chain conformations are energetically favorable, and would not be necessarily observable. For example, the relative energy of the gggg chain conformation is nearly 9 kcal higher than the tttt conformation. The all-gauss conformations (gggg, gggg', ggg'g' etc.) where found to have very high relative energies, and are not expected to contribute significantly to the observed 13 C-NMR spectra.
Obviously the γ-gauche effect for the CH carbon, as controlled by the conformation of the θ 1 and θ 4 dihedral angles, is not the dominating (or only) factor dictating the 13 C-NMR chemical shift, and that other structural components of the polymer chain are also influencing the chemical variations. These results are very different than those predicted for the CH 2 carbon in PE, where the γ-gauche effect is distinct and on the order of Δδ ~ −4 ppm in going from all-trans to all-gauche in the γ carbon position [12]. In addition, the broad distribution of predicted CH 13 C-NMR chemical shifts was not observed experimentally, where relatively narrow line widths were observed for the CH carbon in all p(E-AA) copolymer materials [14]. It is important to note that values of the θ 1 and θ 4 dihedral angles reveal a relatively narrow angle distribution (Figure 3a, horizontal spread) demonstrating that the potential energy surface for this dihedral angle pair was not significantly perturbed by the presence of the COOH group attached to C 5 .
The predicted CH 13 C-NMR chemical shift as a function of the |θ| 2 + |θ| 3 dihedral angle summation is shown in Figure 3b, and again reveals no correlation with these dihedral angles. It is clear that distinct β-orientational effects are minimal for the CH carbon. There is a significant variation (horizontal spread) in the θ 2 and θ 3 angles observed (Figure 3b), demonstrating that the potential energy surface of these dihedral angles was perturbed from classic trans-gauche values (180°, 60°) observed in un-substituted PE. The 13 C NMR chemical shift as a function of all four dihedral angles is shown in Figure 3c. No clear correlations are observed, but it can be argued that there is a decrease in the average chemical shift, and an increase in the distribution of chemical shifts with respect to the all trans conformation (tttt) when higher percentages of the gauche conformation are introduced into the polymer chain. These results demonstrate that while there is some influence of the γ-gauche on the CH 13 C NMR chemical shift, there are clearly other structural perturbations influencing the shift. Figure 3. Predicted 13 C-NMR chemical shift variation of the methine (CH) carbon as a function of (a) the |θ| 1 + |θ| 4 dihedral angle summation, (b) the |θ| 2 + |θ| 3 dihedral angle summation, and (c) the |θ| 1 + |θ| 2 + |θ| 3 + |θ| 4 dihedral angle summation.
In previous ab initio investigation of 13 C-NMR chemical shifts in "crowded" polymer systems such as PIB, there were deviations from the ideal tetrahedral coordination underlying the geometric assumptions in formulating the γ-gauche effect. As the size of the substituent was increased, the distortions of the chain tetrahedral angles near the substituent were perturbed. In the current study, the tetrahedral angles were not constrained such that these types of structural perturbations could occur in the selected clusters. To explore if tetrahedral distortions might account for the observed chemical shift distribution, the CH 13 C-NMR chemical shift as a function of the C 4 -C 5 -C 6 bond angle Ψ are shown in Figure 4. While there is a range of dihedrals observed, the distribution of the chemical shifts is again significant (Δδ ~ 10 ppm), but does not appear to have a simple correlation with bond angle distortions. Based on these ab initio results there does not appear to be a single structural parameter that can be extracted from the experimentally determined 13 C-NMR chemical shifts in the p(E-AA) copolymers. These CH chemical shift correlations ( Figure 3) along with the experimentally narrow CH resonance argue that only a limited number of local chain conformations occur around the central CH species in these p(E-AA) copolymers and corresponding Zn-modified ionomers. Other possibilities include the partitioning of CH defect between the crystalline and amorphous phases within the p(E-AA) copolymers, or that local dynamics are averaging the 13 C chemical shift over multiple conformations. The p21AA sample contained ~ 44% crystalline trans conformation phase, and the remainder of the material was in the amorphous trans/gauche or gauche defect phase, while the p9AA sample is entirely in an amorphous trans/gauche phase. For the p21AA polymer only a single CH resonance was observed, inconsistent with the CH defect being present in both the crystalline and amorphous phases (i.e., partitioned into a single phase). On the other hand, the reduction in the CH 13 C-NMR chemical shift between the p21AA and p9AA copolymer is consistent with the increase in gauche conformations with decreasing chain length. As noted above, the narrow CH line width experimentally observed is at odds with the large chemical shift distributions predicted in Figure 3. The reduced line width may reflect chemical shift averaging due to chain dynamics. A Boltzmann average of the predicted CH 13 C-NMR chemical shift for the 81 different structures investigated is δ = +50.4 ppm, which is lower than the δ = +51.9 ppm predicted for the all-trans conformation (Δδ ~ −1.5 ppm). This chemical shift difference is similar to the change observed experimentally. Chain dynamics have been observed within these p(E-AA) copolymers [14] and in precise methyl-substituted PE [15], based on experimental observation of CH 2 chemical shift anisotropy (CSA) tensor averaging.

Carbonyl 13 C-NMR Chemical Shifts
To explore the role of chain conformations on the carbonyl 13 C-NMR spectra, the COOH chemical shifts for the 81 different clusters were calculated and are shown in Figure 5 as a function of the |θ 1 | + |θ 4 | and |θ 2 | + |θ 3 | dihedral angle summations. In this case the dihedral angles θ 2 and θ 3 would control the γ-gauche effect. Similar to the methine chemical shifts there is a very large variation of ~15 ppm observed for the COOH chemical shift. This is significantly larger than the experimentally observed variation in the copolymer (~+0.7 ppm), or the Zn-neutralized ionomer (~ +2 to +4 ppm). These results ( Figure 5) demonstrate that COOH NMR chemical shift correlations with dihedral angles describing the different chain conformations were not beneficial in determining the local chain conformation. In addition, there was no correlation between the COOH chemical shift and the tetrahedral angle Ψ (data not shown). It was noted that the absolute value of the COOH chemical shifts were very sensitive to the basis set used in structural optimization, but the overall Δδ range as a function of angle was consistent between different levels of theory. This basis set effect reflects the need to adequately model the electron density in the carbonyl region, and required both diffuse and polarized basis functions. These results support the earlier conclusion that for the experimentally observed carbonyl 13 C-NMR chemical shifts, the γ-gauche effects was not observed. It is proposed that the experimental COOH NMR chemical shifts are governed by intermolecular effects such as hydrogen bonding and neutralization. Based on these correlations ( Figure 5), the lack of large dispersions in the COOH experimental line width (broadened line shape) argues against multiple different chain conformations in both the p(E-AA) copolymers and the Z,-modified ionomers. Figure 5. Predicted 13 C-NMR chemical shift variation of the carbonyl (COOH) carbon as a function of (a) the |θ| 1 + |θ| 4 dihedral angle summation and, (b) the |θ| 2 + |θ| 3 dihedral angle summation.

Computational Details
The potential energy surfaces for the 2-butylhexanoic acid model clusters as a function of the two dihedral angles θ 1 and θ 2 (see Figure 1 for definition) were obtained using the Gaussian W09 software (Gaussian Inc., Wallingford, CT, USA) with a 6-311G basis set, density functional theory (DFT) utilizing Becke's three parameter exchange functional, and the LYP correlation function (B3LYP). The two torsional angles were allowed to vary in 3 degree increments, while the remaining