Size extensivity of elastic properties of alkane fragments
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Using MP2, CCSD, and B3LYP methods of computational chemistry, we show length dependence in the intrinsic elastic properties of short alkane fragments. For isolated alkane fragments of finite length in the gas phase and zero temperature, the intrinsic elasticity constants are found to vary with the number of carbon atoms and its parity. From extrapolation of the elasticity constants calculations to infinite chain length, and by comparing with in-situ elasticity constant of single poly(ethylene) molecule obtained with atomic force microscopy, we estimate the softening effect of environment on the extension response of the polymer.
KeywordsElasticity constant Quantum chemistry Atomic force microscopy Single molecule extension response
Quantum chemistry (QC) is an established tool to calculate molecular structure and energy. Application of its methods is recently extended to the investigation of polymer molecules extension response using atomic force microscopy (AFM) [1, 2, 3, 4, 5, 6, 7], but is also applicable to other force microscopy techniques [8, 9, 10]. In the QC approach, the energy profile of a short polymer fragment is computed as a function of a constrained geometry, which is normally the end-to-end extension. Usually derivatives of this energy profile are used to calculate the elastic properties of the fragment, which are then used in interpretation of in-situ elastic response of polymer molecules [11, 12, 13]. Within these efforts, conformational transition of poly(ethylene oxide) was modeled using the QC energy profile [2, 7, 13], and extension response of various polymers were modeled using the QC elasticity constants [3, 4, 5, 6, 12].
Significant differences exist between QC fragments and the actual polymer molecules. For example, polymer molecules are at least two orders of magnitude longer than the fragments used in the computations. Experiments are performed in solvent environment while computations at gas phase and zero temperature. To circumvent these differences, it is assumed that the computed elasticity constants are intrinsic properties of the short fragments. It is also usually assumed that the gas-phase zero-temperature elasticity constants are preserved in situ.
Previous work has supported some of these assumptions. For example it was shown that the energy profiles of propane, heptane, and undecane when normalized by their lengths agree reasonably, and thereby their elasticity constants are intrinsic properties of these fragments . These computations were performed using Hartree-Fock (HF) method which is a low level of computational theory due to its exclusion of electron-electron interaction correlations. The previous work also investigated an extension range with tensile force up to about 12 nN that is beyond accessible to current AFM single molecule force microscopy methods with an upper limit of about 2 nN [3, 4, 5, 6]. We have reproduced these results using a similar method and found agreement with previous work over a similarly long extension range, see Fig. S1. However, investigation of the energy profiles in AFM-related range (up to about 2 nN), using higher levels of computational theory, shows that the normalized energy profiles of alkane fragments do not actually overlap. While length dependence in elasticity constant of DNA molecules computed using molecular dynamics (MD) simulation has been reported , we are not aware of any QC method with inclusion of correlation energy that have been used to accurately examine the elasticity constants of alkane fragments and their size dependence.
The impact of experimental condition, as compared with gas-phase zero-temperature condition of computations on molecular processes was addressed by Politzer and co-workers . These authors showed computationally that solvent may facilitate structural changes of molecules in a chemical process. In particular, forces near the equilibrium geometry are weakened in the presence of the solvent (see Fig. 2b in ref. ). The elastic properties of DNA molecules showed variations with salt concentrations of monovalent and divalent salts using MD simulations [14, 16]. The effect of temperature was addressed in MD simulations of poly(ethylene glycol) monomers . In the force-extension profiles, the structural transition observed at 250 K disappears at higher simulation temperatures where the force-extension profiles do not show significant variations (see Fig. S9a in ref. ).
We show that the computed elasticity constants of short alkane fragments are not intrinsic properties of these fragments. The elasticity constants vary with the length of the fragment or the number of carbon atoms and its parity. We extrapolate the computed elasticity constants to infinite chain length and compare with the elasticity constant of poly(ethylene) obtained with AFM-based single molecule force microscopy. This comparison gives an estimation of the contribution of the environment on the elastic response of the polymer.
Methods and materials
The computations are performed for fragments of the simplest polymer that is poly(ethylene). We employ Moller-Plesset-2 perturbation (MP2) method which adds second order corrections to HF energy from electron pair correlations from doubly excited states . We use four types of split valence basis sets including polarization functions. In Pople style, these are 6-31G*, 6-31G**, 6-311G*, and 6-311G** [19, 20]. The computations are performed for alkane fragments of various lengths starting from the shortest, propane, to the longest, dodecane. Dodecane is the longest fragment that we study as its contour length is slightly shorter than the length expected to induce bent structures [21, 22].
Our MP2 conclusions are further verified using coupled cluster with singly and doubly excitations (CCSD) method which adds infinite order corrections to HF energy from electron correlations of singly and doubly excited states and is more accurate than MP2 . We perform additional computations using density functional theory (DFT) using B3LYP exchange-correlation functional [24, 25]. With CCSD and B3LYP methods, 6-31G* basis set is used. Computations using MP2 method and B3LYP method are performed for all fragments. To reduce computational costs, simulations using CCSD method are performed only for propane, butane, heptane, octane, undecane, and dodecane. All computations are performed using Gaussian 09 program .
We start with geometry optimization in the ground state where energy of a fragment is minimized by optimizing all its degrees of freedom including bond lengths, bond angles, and dihedral angles. This procedure results in the ground state energy E0 and the contour length L which is the distance between terminal carbon atoms in the fragment in the ground state.
The energy-extension profile is obtained in a procedure which consists of varying the distance x between terminal carbon atoms from the ground state. This distance is constrained while all the remaining degrees of freedom are optimized to obtain a minimized energy.
Details of the AFM experiments are explained elsewhere  and data reprocessed here. Poly(ethylene) was purchased from Sigma Aldrich and had a molecular mass of 4500 kg mol−1. A small piece of naturally oxidized silicon wafer (Silchem, Germany) was cleaned for 20 min with piranha solution, which consisted of a mixture of H2SO4 98% and H2O2 30% in a volumetric ratio of 3:1. The piece was then rinsed with Milli-Q water and dried in a stream of nitrogen. It was then treated with oxygen-enriched UV-ozone cleaner (PSD Pro, Novascan, Ames, USA) for 20 min. The clean piece was then immediately used for polymer deposition. The substrate was coated with the polymer by placing a 10 μL drop of polymer solution of a concentration of 100 mg/L in toluene for 20 min on its top. The coated substrate was then rinsed with toluene, dried in a stream of nitrogen, and mounted in the fluid cell of the AFM (Cypher, Oxford Instruments). The interaction between an AFM-tip (BL-AC40TS, Olympus, Japan, spring constant 0.1 N/m measured by thermal method ) and this substrate was probed with repeated approach-retraction cycles with a velocity of 200 nm/s. Force–distance profiles were acquired with a sampling rate of 2 kHz, and were subsequently converted to force–extension profiles by subtraction of the tip deflection .
Results and discussion
We fit the energy-extension profile of propane in Fig. 1a with Eq. (1) and obtain K1 = 29.0 nN, K2 = − 37.9 nN, and K3 = 38.6 nN. These elasticity constants are in close agreement with computations of Hugel et al. . These authors found K1 = 28.7 nN, K2 = −42.0 nN and K3 = 16.9 nN using MP2 method and triple zeta valence (TZV) basis set.
The force-extension profile of propane is shown in Fig. 1b. One observes that the inflection point on the energy profile (Fig. 1a) corresponds to maximum in force profile with value 6 nN. This rupture force is however much larger than that is accessible to AFM single molecule force microscopy experiments where a force of about 2 nN is normally obtained [3, 4, 5, 6]. As shown in Fig. 1b force F = 2 nN corresponds to relative extension x/L = 1.1. To comply with AFM experiments, we limit our subsequent energy computations to relative extensions x/L = 1 to 1.1. Figure 1c shows that, within this range, a force profile calculated from only the first two elasticity constant terms, the quadratic and cubic terms, gives an excellent agreement with the complete force profile. Using only one elasticity constant term results in 10% deviations from the complete profile at large forces. This error is small when compared with uncertainties in AFM force measurements. Thereby, the quadratic approximation provides enough accuracy when applied to AFM results. This result shows that up to a force of about 2 nN using only the quadratic elasticity constant term still provides an adequate accuracy in the force. It is worthwhile to recall here the studies of the elastic properties of carbon nanotubes by Politzer and co-workers . In the linear polymers investigated in the present work and in the carbon nanotubes investigated in ref. , the nature of the carbon-carbon bonds is significantly affected at maximally stretched molecule (relative extension of the overall length of the molecule equal to 10%). Despite this fact, the overall shape of the energy profile is dominated by the quadratic term in either case. Nevertheless, we fit the rest of the energy-extension profiles in this work using two elasticity constant terms.
For each method, the variations of the bond length and bond angle are computed and compared in Fig. 2b and c. Once again very good agreement between the theories is observed. One observes that the bond length R12 between carbon atoms 1 and 2, and the bond angle A123 between carbons atoms 1, 2, and 3 increase monotonically with the end-to-end extension.
One observes that while variations of the bond lengths in heptane and in propane are somewhat similar, the variations in the bond angles are very different. Only the bond angle at the end of heptane, A123, varies similarly to the bond angle in propane, A123. The inner bond angles in heptane are found to vary to a much lesser extent. Since within a similar relative extension, variations of the inner bond angles in heptane are smaller, they result in a relatively stiffer extension response as compared with propane. This relatively stiffer response is evident from comparison between their elasticity constant terms.
The elasticity constant of poly(ethylene) polymer molecules is lower than that obtained from an extrapolation of the elasticity constants obtained from the computations to an infinite chain length. A possible explanation for the softer polymer elastic response is that there is a significant population of gauche structures in the polymer molecule. The gauche structures are softer than trans structures [2, 7, 32]. For example, while the elasticity constants of trans-butane are K1 = 47 nN and K2 = −101 nN, those of gauche-butane are K1 = 8 nN and K2 = −19 nN (Fig. S3).
At equilibrium, poly(ethylene) has a population of about 24% gauche structures, and 76% trans. These populations are estimated from Boltzmann distribution and the ground state energy difference between the two structures which is about 5 × 10−21 J calculated from MP2 (Fig. S4). The energy barrier between gauche to trans structural transition is about 20 × 10−21 J . The area under force-extension profiles of poly(ethylene) molecule gives work that is input by the AFM cantilever (Fig. 6b). This work amounts to about 13(±4) × 10−21 J per monomer. The small energy barrier, together with the input AFM work suggests that gauche to trans structural transition may proceed in the AFM experiments by thermal activation. Using Bell model, one may estimate a maximum transition force of about 600 pN for such structural transition [34, 35, 36]. We observe that our force-extension profiles do not show a sign of structural transition akin to what has been observed for other systems which is in the form of force plateaus or kinks [27, 37, 38, 39, 40]. Lack of such observation suggests that the transition occurs at much lower forces and at a rate that is obscured to the AFM. Moreover, our AFM measurements have a high force of more than 600 pN, where all monomers should be in the trans structure.
In contrast to AFM experiments, QC computations are in the gas phase and zero temperature. Environment effects generally reduce rupture forces of covalent bonds and rings [37, 38, 39], rupture force of physical bonds in supramoleucles , and cis-trans isomerization force of carbon-carbon double bonds [27, 40]. For example, while breakage of a covalent bond is expected to occur at around 5 nN in the gas phase , in the experimental condition it occurs at about 1–2 nN . The impact of environment on facilitating structural transitions in molecules is consistent with computations by Politzer and co-workers . We suspect that similar effects reduce the in-situ elastic response of poly(ethylene) as compared with the elastic response of an infinite alkane chain in the gas phase and zero temperature.
Using the Hartree-Fock (HF) method of computational chemistry, it was previously shown that the energy-extension profiles of short alkane fragments are similar when normalized by their length. This result implied that the elasticity constants are intrinsic properties of these fragments. As a result, the elasticity constants of propane were used in interpretation of the elasticity of various polymer molecules. Using MP2, CCSD, and B3LYP methods, which are more accurate than HF due to inclusion of electron correlation energy, we show that the normalized energy profiles of alkane fragments are not the same. More importantly that the elasticity constant terms are not intrinsic properties of the fragments. The elasticity constants depend on length and are size extensive. They also depend on even and odd number of carbon atoms in the fragment. Both length and parity dependencies are found to diminish with the length of the fragments. It is then possible to extrapolate the elasticity constant terms to infinite chain length where we find a significantly stiffer elasticity constant than the elasticity constant of the actual polymer measured with AFM. The difference between the elastic properties obtained from high-level, gas-phase, zero-temperature calculations extrapolated to infinite chain length and the corresponding experimental data provides direct information about the contribution of the environment on the elastic response in a finite temperature.
The authors gratefully acknowledge stimulating discussions with Professor Michal Borkovec. Support of this research was provided by the National Center of Competence in Research (NCCR) for Bio-Inspired Materials and University of Geneva.
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