Theoretical Determination of the Infrared Spectra of Amorphous

Article pubs.acs.org/JPCA

Theoretical Determination of the Infrared Spectra of Amorphous Polymers Piotr Borowski,* Sylwia Pasieczna-Patkowska, Mariusz Barczak, and Karol Pilorz Faculty of Chemistry, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 3, 20-031 Lublin, Poland S Supporting Information *

ABSTRACT: The simple procedure of calculating the infrared spectra of polymers is presented. It is based on selecting the relevant, medium-size representative fragments of a polymer, for which the vibrational frequencies are computed within the harmonic approximation, in conjunction with the multiparameter scaling techniques. Scaling is necessary to predict the reliable fundamentals, which, along with the calculated intensities and properly chosen band widths, reproduce the observed band shapes with high accuracy. Applications to the three polymers: poly(methyl methacrylate), poly(vinyl acetate), and poly(isopropenyl acetate) are presented. The simulated spectra are in good agreement with the experiment. The assignment of bands is reported. The obtained results indicate strong delocalization of the vibrational modes within polymers, which is in accord with the most recent experimental finding [Macromolecules 2008, 41, 2494−2501]. Good agreement between the observed and the calculated spectra of deuterated PMMA confirms the correctness of our approach. The preliminary results obtained for the highly irregular macromolecular compound (vinyl-functionalized silica) are also shown. macromolecule. Then the approximate force field can be diagonalized to provide frequencies and the corresponding normal modes (the latter ones being used in the determination of bands intensities from the relevant tensors). CTTM was applied a number of times to study various biomolecules (see ref 7 and references therein). The other approaches, modetracking9,10 and intensity-tracking,11,12 were also proposed. In the former the iterative search for a particular frequency (frequencies) of a large system without the necessity of calculating the force constant matrix is carried out, in the latter, the most intense IR-active modes are searched. Both methods provide the same results as the full calculations for the modes in question. Alternatively, the empirical force fields, like the wellknown Urey−Bradley force field,13,14 can be used; such calculations were carried out for polymers quite recently.15 All the methods described above are well justified mathematically and offer obvious advantages. Each of them has its own limitations though. Using the empirical force fields suffers from the lack of the information on the bands intensities. When using CTTM, one has to reconcile oneself to the neglect of the long-range interactions between atoms, which cannot be accounted for by performing calculations on small fragments. In addition, the method is suited to the calculations on the systems composed of subunits that are regularly repeated along the macromolecule, but applications to highly irregular structures are not so straightforward. Moreover, it does not

1. INTRODUCTION Proper interpretation of the infrared spectrum of a molecule consists in assigning the observed bands to the vibrational modes of a system. In some cases the assignment is obvious; otherwise the ambiguities must be solved with the aid of methods of computational chemistry (see, e.g., ref 1 for the recent review). Such calculations for large or medium-size systems are typically carried out by means of the so-called Wilson−Decius−Cross (WDC) method2,3 based on the harmonic force fields. The obtained results clearly indicate the individuals present in the IR, or, in general, in the vibrational spectrum, and reveal the atomic displacements within the modes corresponding to the observed bands. The obvious limitations of the method follow from those in the computer resources: the computational effort of the force constant matrix calculations increases rapidly with the increasing size of the system. A number of approaches were developed to date to adapt the WDC method to calculations on macromolecules. Approximations shedding light on the molecular vibrations are often based on the fragmentation of a large system into small subunits the macromolecule is composed of4,5 (good review of the literature on the fragment approximation in relation to the biomolecules can be found in ref 5). Probably the most general is the so-called Cartesian tensor transfer method (CTTM). It was originally proposed by Bouř and co-workers,6 and recently revisited by Bieler et al.7 and Yamamoto et al.,8 who carefully investigated its accuracy and reliability. The method is based on transferring the Cartesian tensors (Hessian, multiple moments, etc.) obtained for small fragments, to a large part of the © 2012 American Chemical Society

Received: April 4, 2012 Revised: June 7, 2012 Published: June 11, 2012 7424

dx.doi.org/10.1021/jp303209v | J. Phys. Chem. A 2012, 116, 7424−7435

The Journal of Physical Chemistry A

Article

vibrational spectra of the properly chosen (vide infra) mediumsize polymer fragments. The optimal geometries and the Cartesian force fields for all the considered fragments were calculated at the DFT/B3LYP level18,19 with the 6-31++G** basis set.20,21 This approach was shown to provide very accurate multiparameter-scaled frequencies.22 All calculations were carried out with the parallel version of the PQS quantum chemistry package,23,24 implemented for our 16-core cluster (4 × 4-core Xeon, 2.27 GHz). The single run took approximately 3−5 days per structure (90−110 atoms each). The structures corresponding to all real harmonic frequencies were selected. The ESFF procedure25 based on the redundant primitive internal coordinates26 was used to calculate the scaled frequencies. The 9-parameter scaling frame27 with the recently reported B3LYP/6-31++G** scaling factors22 was used. The shapes of the spectral bands were simulated using the linear combination of Gaussian and Lorentz functions for individual modes, that is,

allow for the relaxation of subunits within the macromolecule; the transferred (rotated) Cartesian tensors may differ from the actual ones due to the changes in the geometries within the irregular macromolecular structure. The mode-tracking and intensity tracking approaches are free of these problems, but accurate results obtained within a reasonable time refer to a rather narrow range of wavenumbers. In this work a simple approach to simulate the IR spectra of polymers is presented. It is shown that very accurate spectra can be obtained without the necessity of transferring the calculated tensors to larger fragments of the macromolecule. The method is based on the calculations on the medium-size fragment(s), which, when chosen properly, will be referred to as the representative fragment(s). The presence of inequivalent structural motifs within the macromolecule leading to the bands broadening can be accounted for by simulating the band widths. In practice, it means that no new software (in addition to the already possessed for the QC calculations) is needed to draw far-reaching conclusions related to the bands assignment. This seems to be of particular importance for the research groups interested in accurate (and fairly quick) interpretation of the polymer spectra without going deeply into analysis of the macromolecular force fields. In addition, the approach can be applied to irregular systems and to the nearly entire spectral range, that is, apart from the lowest frequencies. In section 2 we present the details of both the experiment and QC calculations we carried out. In section 3 we outline the methodological aspects of obtaining the final spectra and present our results. In particular, we report a detailed analysis of the infrared spectrum of poly(methyl methacrylate) (PMMA). Then we show the theoretical spectra of the two additional polymers, poly(vinyl acetate) (PVAc) and poly(isopropenyl acetate) (PIPAc), and compare them with the experimental ones. At the end we present our preliminary results for the (irregular) vinyl-functionalized silica. Finally, in section 4, we state our conclusions.

⎛ (ν − ν )2 ⎞ 1 0 ⎟ exp⎜ − σ 2π 2σ 2 ⎠ ⎝ 1 + (1 − c) (ν − ν0)2 πσ 1 + 2

a(ν , ν0 , σ ) = c

(

σ

)

(1)

where ν0 is the scaled frequency, σ is the standard deviation (the parameter determining the bandwidth; note that σ is the same for both functions, for simplicity), and c (0 ≤ c ≤ 1) is the parameter determining the contribution of a given function to the band shape. Such a shape is a standard one in the case of using the curve-resolving procedures.16,28 In all simulations we used c = 0.5. The spectra were generated as all modes

A (ν ) =

∑ i

Iia(ν , νi , σi)

(2)

where Ii denotes the calculated integrated intensity corresponding to the scaled frequency νi, and σi is the same for all modes in a given band.

2. APPLIED METHODS The standard notation for the vibrations is used throughout the paper: ν, δ, ρ, and τ denote stretching, in-plane bending, rocking, and twisting vibrations, respectively; plus denotes the coupling; comma denotes the presence of two close-lying pure vibrations; X denotes a general second-row atom; the two- (e.g., XH), and three-letter (e.g., XXX) symbols refer to the bond lengths and valence angles, respectively; XX1 and XX2 are used to denote single and double bonds, respectively; TOR refers to torsional angles of any kind. 2.1. Experimental Section. The experimental spectra were obtained only for the purpose of comparison. The FTIR spectra of PMMA (Plexiglas XT 29070, RÖ HM), PVAc (Mowilith 20, KREMER), and vinyl-functionalized silica (synthesized by the authors) were collected with a Bio-Rad spectrometer at room temperature. Preparation of the polymer film was described elsewhere;16 we followed that procedure using tetrahydrofuran (POCh) as a solvent. The silica spectrum was obtained using the KBr pellet method. All spectra were measured at 4 cm−1 resolution (which is enough for our purpose). The interferograms of 32 scans were averaged for each spectrum. PIPAc was not available to us; however, its spectrum (obtained probably for an atactic or syndiotactic material) can be found in the literature.17 2.2. Calculations. The first step of the approach presented in this work involves the “gas-phase calculations” of the

3. RESULTS AND DISCUSSION Selection of the relevant representative fragment(s), the medium-size oligomer(s) of a macromolecule (denoted RF), is the first step in the overall studies. We followed the recommendations listed below. 1. Chains (or cross-linked structures) of macromolecules have to be terminated at some points. If possible, the termination should be made with the aid of small groups that are already present in the macromolecule. Otherwise, one should use groups which are responsible for rather weak bands in the range of interest. In addition, their vibrations should not be coupled (or only weakly coupled) to those of the remaining part of a molecule, in which case the intensities of the corresponding modes can be preset to zero. 2. The search for the structure corresponding to the global minimum of the energy is not recommended. First, there is an extremely large number of possible conformers even in the case of medium size fragments (e.g., 100 atoms), and plenty of them may be found within the energy range of a few kcal mol−1. Second, there is no guarantee that the obtained minimum-energy “gas-phase” structure 7425



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to be desirable to work out simple rules readily applicable to the simulation of the IR spectra of complex systems, like the typical chain polymers. The example of the band broadening due to the presence of a great variety of modes is shown in Figure 1a. The thin line

is the most frequently repeated unit in the macromolecule. The solid-phase formation is associated with the minimization of the Gibbs free energy (G) of the macro-system. The minimization of the main, enthalpic (H) contribution should be, in principle, achieved for the fragments different from those that would have been obtained in the gas-phase − the energy gain associated with the intermolecular interactions of favorably aligned chains (weak, but acting along a substantial part of a chain) may overcome a slight energy increase due to the changes in the torsional angles (determining the shape of the macromolecule). The entropic (−TS) contribution, which is mostly due to the low-frequency vibrations, is not expected to significantly affect G. Information on the relative orientation of mers is typically not available for amorphous polymers, and therefore, as long as the selected fragment is of an appreciable size, contains the most probable arrangements of the basic units (e.g., both cis and trans conformations of the CH3−Cα−CO moieties in PMMA), and its energy is relatively low, it may be considered as a possible RF of the macromolecule (or one of the RFs). We propose to optimize at most a few structures starting with the intuitively chosen geometries, which do not show significant steric effects. Then by modeling the band widths (vide infra) for the chosen fragment(s), which accounts for the presence of various inequivalent structural motifs in the bulk-phase, and comparing the resulting theoretical spectrum with the experimental one and with those of the remaining fragments, the decision can be made as to their relevance to the description of the macroscopic sample. Note that this procedure may not work well for the very lowest frequencies (below 600 cm−1), where torsions are likely to occur. 3. If the theoretical spectra obtained for various RFs differ significantly, all of them should be considered in the modeling of the final spectrum. 4. The multiparameter scaling should be preferably used to approximate the fundamentals. Indeed, uniform scaling29−32 renders the rms deviations between the scaled and the experimental frequencies that are often in the range of 30−50 cm−1. Those values are reduced even to 10−11 cm−1 when the multiparameter scaling is applied. We decided to use the ESFF scaling procedure,25,26 but we believe that the users of SQM33−35 can apply the method with equal success. Scaling could also be used in conjunction with the CTTM method. However, the transferability of the scaling factors to systems described by the approximate force constants should be carefully investigated prior to such calculations. Chains of typical polymers weakly interact with each other, and therefore, they are not expected to significantly perturb each other. However, the pendant groups in the amorphous systems exhibit various orientations relative to the macromolecular chain; thus, mers are not fully equivalent from the spectroscopic point of view. In addition, the mechanical coupling, which may propagate down the substantial part of a chain due to the (near-)degeneracy of local vibrations, may cause the additional frequency shifts of the vibrational modes. These effects lead to the additional (inhomogeneous) band broadening even if the individual bandwidths are small. Although the theory of the band shapes is known,36 it seems

Figure 1. (a) Example of the band shape simulation: the frequency gaps and the adopted values of σ parameters of the Gaussian-Lorentz function are shown. (b−g) Rules for obtaining the approximate σ values for typical patterns of vibrational modes (see text).

represents the individual modes assuming σ = 1 cm−1, the thick line, the simulated band shape. The vibrational modes of a system built of similar units are frequently grouped, and those within each group often exhibit similar distribution in terms of local vibrations. The black and red lines shown in Figure 1b−g are used to denote the pure vibrations of different origin, the mixed line, the coupled mode. Figure 1b shows the case of an isolated group of pure modes, in that the distances between the individual vibrational frequencies are small (typically 100 cm−1, cf., Figure 1g) they may be responsible for the presence of a specific pattern of bands; in this case, we propose to use 6−8 cm−1 for each mode. The approach outlined above was first tested on PMMA, and then applied to a few other macromolecular systems. We will concentrate on the range of 600−1800 cm−1, that is, stretching vibrations involving hydrogen atoms are not considered. The interpretation of the IR spectra in the high-frequency range is often straightforward. In addition, the CH stretching vibrations

account for the presence of an extremely large number of (not fully equivalent) motifs of a given kind in a macromolecule, and, in contrast to the liquid phase,37 the spectral lines are not expected to show the motional narrowing effect. We propose to use the values not lower than 6−8 cm−1. The recipes for the cases shown in Figures 1c−e can be easily deduced. A typical pattern of modes at low frequencies (usually below 1200 cm−1), is shown in Figure 1f: pure modes of different origin, as well as their combinations, are jumbled up. As long as the group is well separated from the remaining groups and the individual frequencies span a relatively narrow range of frequencies, 7427



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than 1 kcal mol−1 higher than A and B). The PMMA-A structure contains more cis, and -B contains more trans conformations of the C(c)−C−C(a)O moieties, with an overall excess of the latter ones, as suggested earlier.40 Geometry optimization reveals that the pendant ester groups tend to be located on the opposite sides of a chain. In addition, the carbon chain bends due to the presence of steric effects of the ester groups with the methyl C(c)H3 groups (in accord with the previous findings on the helical structure of PMMA43). Note that the structures A−C contain cis conformations of the OC(a)−O−C(b) moieties. However, some bands on the IR spectrum of s-PMMA (the ν4 shoulder, vide infra) were assigned to the asymmetric stretching vibrations of C(a)−O− C(b) fragments within the trans-type ester groups.40 Thus, the additional, fourth structure, in which two ester groups were reoriented to adopt the trans conformations, was found. It turned out to be more than 30 kcal mol−1 above PMMA-A and -B due to the steric effect between C(b)H3 and C(c)H3 groups (for trans C(c)−C−C(a)O), as well as C(b)H3 and CH2 groups (for cis C(c)−C−C(a)O). Although the energetics obtained in the “gas-phase calculations” is not a decisive factor determining the final structure of a macromolecule, it seems that the energy increase as large as 30 kcal mol−1 (ca. 15 kcal mol−1 per moiety) precludes the presence of trans OC(a)−O−C(b) conformations within a polymer chain, at least in a detectable number. In addition, their νas(C(a)−O−C(b)) vibrations were found to be about 50 cm−1 above the ν4 shoulder (the value refers to the scaled frequency). 3.1.2. Theoretical Spectra. Most of the vibrational modes were found to be strongly delocalized (in accord with the previous finding16). Thus, the individual PED coefficients44 are small. More informative are the sums of PEDs for a given type of internal coordinates reported in the Supporting Information. However, the contributions specified in this way may be misleading with respect to the real nuclear motions. For instance, 80% HXH, may refer to both δas(CH3) and δs(CH2) vibrations (note that δs(CH3) typically consists of 40−50% of XXH and 20−30% of HXH). Thus, visual inspection of the vibrational modes is also needed. The calculated infrared spectra of both head-to-tail fragments are shown in Figure 2a′,b′; the data the simulation is based on is given in the Supporting Information. The most striking difference is observed in the range of 1300−1100 cm−1. Four maxima, typically denoted ν1, ν2, ν3, and ν516,40 (ν4 is the shoulder between ν3 and ν5, cf., Figure 2d) are observed in both spectra, and the ratio of total integrated intensities of the (ν1,ν2) and (ν3,ν5) “doublets” is approximately the same (and in accord with the experiment). However, the intensities ratio of the components within each doublet is different for both structures, indicating that individually they are by no means representative for PMMA. The theoretical spectrum of the head-to-head structure, shown in Figure 3a′, is totally different. Although the doublets are also observed, their total integrated intensities are reversed. This contradicts the experimental observation. Because the bands in this range are among the most intense ones on the FTIR spectrum of PMMA, we conclude that the contribution from the head-to-head arrangement of mers to the final structure should not be significant. The final theoretical spectrum shown in Figure 2c was obtained by averaging the spectra of PMMA-A and -B. Equal weights were assumed for both structures. The results are summarized in Table 1. As can be seen, all major features on the experimental spectrum are well reproduced in our

may exhibit large deviations between the scaled and experimental wavenumbers (sometimes exceeding 30 cm−1), which may lead to errors in the final bands shapes. This would probably be the case also for the CTTM frequencies (even if the proper scaling was applied). 3.1. Poly(methyl methacrylate), PMMA. This polymer has been widely studied for nearly 50 years; the post-1980 literature on the PMMA infrared spectroscopy includes well above 10 contributions.16,38−41 However, the investigations related to the PMMA spectroscopy are by no means complete; in spite of the extended studies, the origin of some bands is not fully clear. 3.1.1. Representative Fragments. RFs of PMMA were chosen to be built of seven units. The following atom labeling is |

used: C(c)H3−C−C(a)(O)−O−C(b)H3. Terminating the |

chain with the hydrogen atoms introduces two additional methyl groups bound to the aliphatic carbon atoms (already present in the macromolecule). Head-to-tail and head-to-head arrangements of units were tested. Trans−trans chain conformations were considered. As suggested,40,42 trans−trans states are strongly preferred over trans−gauche states. Syndiotacticity of PMMA was assumed; note that spectra of atactic and syndiotactic PMMA are basically the same. The geometry optimization gave us three acceptable fragments, two of which were of the head-to-tail type (essentially energetically equivalent PMMA-A and PMMA-B structures, Figure 2a,b), and one head-to-head type (PMMA-C, Figure 3a, which is less

Figure 3. PMMA. Representative fragment of the head-to-head type used in the calculations (a) and its simulated spectrum (a′). 7428



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Table 1. Observed (νexpt, in cm−1) and Calculated (νtheor, in cm−1) Vibrational Wavenumbers, Their Differences (Δν, in cm−1), Relative Percentage Errors (Error, in %), Approximate Assignment of the Corresponding Bands, as Well as RMS (in cm−1), and Average Relative Percentage Error (ARPE, in %) for PMMA νexpt

νtheor

Δν

error

1 2 3 4 5 6

1731 1484 1448 1435 1386 1362

7 −7 −20 −7 −20 −16

0.40 0.47 1.38 0.49 1.44 1.17

7 8 9 10

1270 1241 1192 1149

1724 1491 1468 1442 1406 1378 1332 1257 1233 1190 1153

13 8 2 −4

1.02 0.64 0.17 0.35

11

1064

0

0.00

12 13 14 15 16

989 966 912 841 826 810 750

1064 1021 981 956 900 849 818

8 10 12 −8 8

0.81 1.04 1.32 0.95 0.97

741

9

1.20

no.

17 rms/ARPE a

10.8

assignmenta/comment ν(C(a)O) δas(C(c)H3) δas(C(c)H3) (another type), δas(C(b)H3), δs(CH2) (weak modes) δs(C(b)H3) (note, blue-shifted as compared with C(c)H3; see below) δs(C(c)H3) δs(C(c)H3) + ω(CH2) τ(CH2) ν(C(a)−O) + ν(C(a)−C) + τ(CH2); the so-called ν1 band ν(C(a)−O) + ν(C(a)−C) + τ(CH2); the so-called ν2 band ν(C(a)−O) + ρ(C(b)H3) (variable contributions); the so-called ν3 band delocalized modes (intense, no specific assignment can be made), ρ(C(c)H3) (medium and weak modes); the so-called ν5 band with ν7 tail ν(C(c)−C) + ρ(C(c)H3) + τ(CH2); the so-called ν6 band ρ(C(c)H3) ν(C(b)−O) ν(chain) + ρ(C(c)H3) ν(chain) + ρ(C(c)H3) ν(chain) (sometimes + ρ(CH2); yet another chain stretching) ν(chain) + δ(C−C−C) probably another ν(chain) + δ(C−C−C) vibration ρ(CH2)

0.81 |

The following notation is used: C(c)H3−C−C(a)(O)−O−C(b)H3. |

of strong delocalization of all the stretches among the cis- and trans-like conformations is clear. ν3 can be unambiguously assigned to the delocalized ν(C(a)−O) stretching vibration coupled to rocking of the ester C(b)H3 groups. However, no specific assignment can be made in the case of ν5 band. The normal modes correspond to stretching of the C−C bonds of a chain coupled to stretching of C−C(c) and C(a)−O, as well as rocking of C(b)H3 and, partially, C(c)H3 groups. This is probably “the most delocalized band” in the entire spectrum of PMMA. Note that the ν4 shoulder was not reproduced in our simulated spectrum; its origin can be deduced, though. Careful analysis reveals that there are two delocalized vibrations of the XX1+XXH type in the range between ν3 and ν5: at 1170 and 1166 cm−1 for PMMA-A, and at 1166 and 1165 cm−1 for PMMA-B (see the Supporting Information) that differ from the remaining ones corresponding to ν5, in that they involve larger contribution from the strongly delocalized stretching of a chain. Their intensities are significantly lower than those of the most intense neighboring modes. This is probably too low intensity that is responsible for the absence of the shoulder on our theoretical spectrum. Indeed, manual increase of the intensities corresponding to those modes by a factor of 2 leads to the ν4 shoulder (cf., Figure 4). We believe that low intensity of these bands may be due to terminating the polymer chain after 7 units. For larger fragments the simultaneous stretch of a large number of C−C bonds would probably lead to larger changes in the overall dipole moment increasing the corresponding intensities. Note that the CTTM method may partially solve that problem. The reconstruction of the force field by transferring the Hessian from a smaller (here 7-mer) to a larger fragment would obviously lead to the delocalization of the vibrational modes in spite of neglecting the long-range interactions, due to the indirect coupling (coupling

simulation. There are two very weak maxima on the theoretical spectrum (at 1332 and 1021 cm−1) that are not observed experimentally, and one very weak maximum on the experimental spectrum (at 810 cm−1), which is not found in our simulation. The rms deviation between the experimental and calculated wavenumbers corresponding to the bands maxima is 10.8 cm−1. This shows that the scaling factors optimized using a set of 30 small molecules35 are well transferable even to the polymer fragments. Good transferability of the scaling factors from the gas to condensed phase has already been observed.45,46 Although some subtle observed features were not reproduced in our simulation, the interpretation of the infrared spectrum is straightforward. In the following we will concentrate on the most interesting and controversial range of 1300−1100 cm−1. The assignment of the remaining bands is given in Table 1. As can be seen, four maxima are present in the simulated spectrum. Our observations are consistent with the most recent findings,16 according to which “... it would seem likely that all of the bands in this region of the spectrum can be assigned to individual, highly mixed normal modes of vibration ...”. Both components of the first (ν1,ν2) doublet follow mostly from the delocalized stretching vibrations of the C−C(a) and C(a)−O bonds coupled to the methylene group twisting vibrations (we avoid further classification in terms of symmetry due to the significant differences in the observed amplitudes for both bonds, though the asymmetric character of the ν(C−C(a)−O) vibration seems to dominate). As we will see shortly, coupling to twisting has a strong effect on the band shapes of the deuterated polymers. There is a slight preference for the cis configurations of the C(c)−C−C(a)O moieties to show larger amplitudes within the ν1 band (note, cis rather than trans, as suggested earlier40), and trans, within ν2, but the general trend 7429



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Figure 4. PMMA. Original and modified (by increasing the intensities of some modes, see text) simulated spectra in the range of 1000−1350 cm−1, revealing the presence of ν4 shoulder.

of F1 to F2 and F2 to F3, cf., Figure 1 of ref 7, leads to mixing of F1 and F3, even though the direct coupling of F1 and F3 is neglected). The accuracy of such an approach was shown to be satisfactory in most cases.7 However, in the case of features as subtle as that already described, it is difficult to predict the actual performance of CTTM in advance. 3.1.3. Deuterated PMMA. Earlier assignment of bands in the range of 1300−1100 cm−1 to the highly mixed vibrational modes follows from the detailed analysis of the IR spectra of deuterated polymers (cf., Figure 12 of ref 16). In the following, PMMA-HH denotes the nondeuterated polymer, PMMA-DH, the polymer with the deuterated ester methyl groups (C(b)D3), PMMA-HD, the polymer with all but the ester methyl groups deuterated (C(c)D3 and CD2), and PMMA-DD, the fully deuterated polymer. The simulated spectra and the bands assignment are shown in Figure 5. The simulated spectra closely resemble the experimental ones. Deuteration of the ester methyl groups in PMMA-DH leaves the (ν1,ν2) doublet practically unchanged since basically no local C(b)H3 vibrations contribute to the normal modes in this range. On the other hand, the (ν3,ν5) is replaced by a single strong band corresponding to the ν(C(a)−O) vibrations coupled to the chain stretching. This is obvious: the coupled ν(C(a)−O) + ρ(C(b)H3) vibrations leading to ν3 lose the rocking character (in favor of the C−C stretching) when the corresponding mass tensor elements are changed upon deuteration, and overlap with the delocalized vibrations leading to ν5. The additional band somewhat below 1100 cm−1 is unambiguously assigned to an umbrella-type vibration of the C(b)D3 groups (in fact, δs(C(b)D3) + ν(C(b)−O) with similar contributions; it looks as if the C(b) atom was “pushing in between” the deuterium atoms). PMMA-HD and PMMA-DD show only one band instead of the (ν1,ν2) doublet; the deuteration of the CH2 groups prevents coupling of the simultaneous C(a)−O and C(a)−C stretches to the CD2 twisting by changing the mass tensor, and therefore, no splitting occurs. A band below 1100 cm−1 on the PMMADD spectrum corresponds, as in the case of PMMA-DH, to the δs(C(b)D3) + ν(C(b)−O) vibrations. Broader and less intense feature on the simulated spectrum of PMMA-HD at approximately the same position corresponds to the asymmetric

Figure 5. PMMA. Simulated spectra of various deuterated forms of the polymer in the range of 1000−1350 cm−1, as well as an approximate assignment of the bands.

stretching of the C(a)−O−C(b) fragments. The assignment of the remaining bands is given in Figure 5. All the features are also observed on the experimental spectra of deuterated PMMAs.16 However, there is one upsetting disagreement. Although the spectra of PMMA-HD and PMMA-DD are qualitatively similar, it seems that our simulated spectrum of PMMA-DD more closely resembles the experimental spectrum of PMMA-HD (and vice versa). This refers to both the location of the band above 1250 cm−1 and the shape of the band close to 1100 cm−1. It is likely that this disagreement follows from the approximations in the computational procedure. Notwithstanding, the presence of a sharp band on the third (from the bottom) spectrum shown in Figure 12 of ref 16, which is attributed to an umbrella C(b)D3 vibrations, and lack thereof on the uppermost spectrum, may indicate that the experimental spectra of PMMA-HD and PMMA-DD were accidentally swapped in the above reference. 3.2. Poly(vinyl acetate), PVAc. 3.2.1. Representative Fragments. As before, RFs of PVAc were chosen to be built of 7 |

units. The following atom labeling is used: H−C−O−C(a)(O)− |

C(b)H3. The chain can be terminated by adding the hydrogen 7430



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Figure 6. PVAc. Representative fragment of atactic polymer of the head-to-tail type used in the calculations (a), its simulated spectrum (a′), and the experimental spectrum (b). Theoretical spectrum obtained using the uniform scaling procedure (the B3LYP/6-311+G(2df,p) calculations) is shown in (c).

atom either to the methylene group, or to the carbon atom with the pendant acetoxy −O−C(a)(O)−C(b)H3 group. We are inclined to the first choice, the additional methyl groups, which are not present in the polymer, are often hardly coupled to the remaining vibrations of a fragment. Note that the second choice would introduce methylene protons in the direct vicinity of the oxygen atom (known to shift some of the frequencies). The cis conformations of the C−O−C(a)O moieties corresponding to much lower energies as compared with trans were chosen. Atacticity of PVAc was assumed. Three final structures, two of the head-to-tail and one head-to-head type, were acceptable. Two additional fragments exhibiting stronger coupling between the terminal methyl groups and the rest of the molecule were rejected. In contrast to PMMA, the backbone is fairy straight due to the presence of hydrogen atoms rather than bulky methyl groups attached to the chain. This can be seen in Figures 6a (PVAc-A structure) and 7a (PVAc-B), which shows the selected head-to-tail and head-to-head structure, respectively. The second head-to-tail structure is not considered as it exhibits nearly the same simulated spectrum as the first one, in spite of some differences in the individual frequencies and intensities. It follows from much less diversified local force fields of the pendant groups as compared with PMMA, due to the lack of significant steric effects. 3.2.2. Theoretical Spectra. The calculated IR spectrum of the head-to-tail PVAc-A fragment is shown in Figure 6a′ and

compared with our experimental FTIR spectrum of atactic PVAc (essentially the same as that reported in the literature47) presented in Figure 6b. The data the simulation is based on is given in the Supporting Information (the rows marked in red with the intensities preset to zero refer to the methyl groups vibrations). The spectrum of the head-to-head PVAc-B shown in Figure 7a′ differs from that of PVAc-A in some ranges. In particular, it exhibits the doublet structure of the CO stretching vibration (due to stronger coupling of the less distant carbonyl groups). Thus, we believe that the head-to-tail arrangements dominate, though the presence of sparse headto-head arrangements in the final material cannot be excluded. The results (including our assignment) are summarized in Table 2. As can be seen, the calculated rms value between the observed and theoretical wavenumbers corresponding to the maxima (and shoulders) is less than 10 cm−1. In principle, all bands in the range of 1800−600 cm−1 are observed in the simulated spectrum (a weak band at 1065 cm−1 appears as shoulder), which shows the additional weak maximum at 884 cm−1. Most modes are usually very delocalized. The only band that does not have a theoretical counterpart of comparable intensity is that at 947 cm−1. It cannot be assigned to any overtone or a combination band; it is of medium intensity, and there are no intense fundamentals at lower frequencies that could be combined to give 947 cm−1. The reason is analogous to that already discussed for the ν4 shoulder of PMMA. The modes below 950 cm−1 involve significant contribution from 7431



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Figure 6c shows the simulated spectrum of PVAc using the uniform scaling of the frequencies. In contrast to the multiparameter scaling, the double-ζ quality basis set may not be sufficient.48 Indeed, the “convergence” in the scaling factors and the rms values is achieved when f functions on the secondrow elements are added to the valence triple-ζ basis set.32 We used 6-311+G(2df,p) basis set,21,49 the smallest one we could still handle, beyond which no improvement in the scaled frequencies was observed. The recommended scaling factor of 0.968632 was used. The simulated spectrum looks similar to that obtained with the aid of the multiparameter scaled frequencies, probably due to not very diversified character of the vibrational modes. However, the obtained rms value increased by a factor close to 2: from 8.2 cm−1 (cf., Table 2) to 16.7 cm−1. The experimental spectrum of PVAc reveals that there must be some broad feature in the range of 1000−1150 cm−1 (cf., Figure 6b), which constitutes some kind of a background for other bands in this range. This feature is not reproduced in our calculations, though the maximum at 1022 cm−1, two shoulders on its blue wing, and the maximum at 1122 cm−1 are predicted with high accuracy (cf., Table 2). It also does not appear on the simulated spectrum of our head-to-head, and the second head-to-tail conformer. We are not aware of its origin; to our best knowledge no reliable explanation has been reported to date in the literature. 3.3. Poly(isopropenyl acetate), PIPAc. The FTIR spectra of PIPAc were reported in the literature,17 but no numerical data are available to make a quantitative comparison. Using the procedure described earlier we obtained the theoretical spectrum closely resembling the experimental one, this time after finding the first syndiotactic representative fragment of the head-to-tail type built, as usually, of 7 mers (denoted simply PIPAc). Note that, as in the case of PMMA, the terminating methyl groups are not expected to perturb the final theoretical

Figure 7. PVAc. Representative fragment of the head-to-head type used in the calculations (a) and its simulated spectrum (a′).

the simultaneous stretching vibrations of the nonpolar C(a)− C(b) bonds. The overall change in the dipole moment would probably be larger if the calculations were carried out on larger fragments. For this reason, we lowered the σ value for the most intense theoretical modes of PVAc-A close to 927 cm−1 down to 4 cm−1 (cf., Figure 6a′), to show more clearly the presence of this band on the theoretical spectrum.

|

spectrum. The following atom labeling is used: C(c)H3−C−O− |

C(a)(O)−C(b)H3. The data the simulation is based on is given in the Supporting Information. The selected structure and the corresponding theoretical spectrum are shown in

Table 2. Observed (νexpt, in cm−1) and Calculated (νtheor, in cm−1) Vibrational Wavenumbers, Their Differences (Δν, in cm−1), Relative Percentage Errors (Error, in %), Approximate Assignment of the Corresponding Bands, as Well as RMS (in cm−1) and Average Relative Percentage Error (ARPE, in %) for PVAc νexpt

νtheor

Δν

error

1 2 3 4

1736 1436 1372 1239

1732 1441 1375 1237

4 −5 −3 2

0.23 0.35 0.22 0.16

5 6 7 8 9

1122 1065 1041 1022 947

−11 9 2 0 20

0.98 0.85 0.19 0.00 2.11

10 11 12

796 630 605

1133 1056 1039 1022 927 884 796 625 594

0 5 11

0.00 0.79 1.82

no.

rms/ARPE a

8.2

assignmenta/comment ν(C(a)O) δas(C(b)H3), δs(CH2); C(t)H3 vibrations excluded δs(C(b)H3), ω(CH2) (low intensity contribution); C(t)H3 vibrations excluded ν(C(a)−O) + ν(C(a)−C(b)) (νas(O−C(a)−C(b)) character; sometimes + τ(CH2) or + ω(CH2)); ρ(CH) + τ(CH2) on the blue wing (the tail) ν(chain) (various types of stretching vibrations); C(t)H3 vibrations excluded ν(chain); Sh on the theoretical spectrum ρ(C(b)H3); Sh on both theoretical and experimental spectra νas(C−O−C(a)) + ρ(C(b)H3) (the tail on the red wing follows from ρ(C(b)H3)) ν(C(a)−C(b)) delocalized vibrations νs(C−O−C(a)) + δ(C−O−C(a)) ν(chain) + delocalized torsions, ν(chain) + δ(OC(a)−O) ω(C(a)O)

0.64

|

The following notation is used: H−C−O−C(a)(O)−C(b)H3; C(t)H3 denotes the terminating methyl group. |

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Figure 8. PIPAc. Representative fragment of the head-to-tail type polymer used in the calculations (a), and its simulated spectrum, on which an approximate assignment of bands is given (a′).

by the additional methyl and hydroxyl groups (content of vinyl groups has to be approximately in line with the composition of the synthesized material). Some of the additional modes (e.g., rocking of CH3, in-plane bending of SiOH, etc.) are easy to eliminate. Those CMe−O stretching vibrations that were coupled to the stretching of the silicon framework, do not significantly perturb the theoretical spectrum since the simultaneous stretch of an overwhelming majority of more polar SiO bonds leads to much stronger bands as compared with CO. Note that we could not terminate the system using the SiH moieties since the SiH bending modes lead to strong bands somewhat below 1000 cm−1. The scaling factors reported in ref 45 were used. The experimental and simulated spectra are shown in Figure 9. The procedure of choosing the σ values was the same as that described before, except for expanding the Δν range, for which σ ≈ 1/4 Δν rather than 6−8 cm−1 should be used (cf., Figures 1f,g), up to 200 cm−1 for the vibrations of the cross-linked framework. As can be seen the experimental band

Figure 8. When comparing our theoretical spectrum with the experimental one reported in Figure 1a of ref 17 we see that all features are well reproduced in the calculations. Thus, the assignment (also included in Figure 8) can easily be made. 3.4. Vinyl-Functionalized Silica. The procedure illustrated in the previous sections can be applied, with some modifications, to simulate infrared spectra of highly irregular macromolecular compounds. As an example, we consider the vinyl-functionalized silica. Infrared spectroscopy was used to confirm the presence of vinyl groups in the final material.50 However, our interpretation at that time was intuitive (and by no means strict): twisting and wagging vibrations used to identify the group are in the range of a very strong absorption band due to the stretching of the SiO framework. The silica was synthesized according to the known procedure50 using the 1÷1 molar ratio of tetraethoxysilane and vinyltriethoxysilane. The irregular, partially cross-linked RF shown in Figure 9 containing the vinyl groups, was terminated 7433



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Figure 9. Vinyl-functionalized silica. Representative fragment used in the calculations as well as the experimental and theoretical IR spectrum.

calculations” is a less important factor). Sometimes it may be necessary to average the simulated spectra of two (or more) representative fragments. This is computationally more advantageous than combing them into a single, larger fragment, but some information related to the delocalization of modes may be lost. In such a case the CTTM method may partially solve the problem, as it accounts for the delocalization in the approximate, indirect way. Finally, we have to point out that the presented approach is new, and therefore a lot of additional work related to further testing has to be carried out to fully verify its predictive capabilities. This may involve looking for the dependence of the final spectrum on the size of the fragments, comparison with other approaches, including CTTM, or even with the (nearly) exact calculations in the narrow spectral range (which can be achieved with the aid of the mode-tracking or intensity-tracking calculations on fragments consisting of several dozen of mers) etc. We hope that this work will encourage other research groups to contribute to the development of the reported ideas.

shape is well reproduced in our simulation in spite of using a rather simplified silica structure. Thus, the low-intensity features on both slopes of the intense band are easily interpretable.

4. CONCLUSIONS The purpose of the present work has been to examine the possibility of determining the infrared spectra of typical amorphous polymers using the concept of a medium-size representative fragment(s) of a macromolecule. The procedure is based on obtaining the reliable vibrational frequencies and the IR intensities, followed by simulating the band shapes. It is shown that such calculations are capable of predicting the major and some of the minor features observed on the experimental spectra of polymers without the necessity of reconstructing the force field (and the dipole moment derivatives) of the large fragment of a macromolecule from smaller fragments. Thus, no additional software to the ordinary QC packages is needed. This simplicity, in conjunction with a reasonable computational cost, makes the approach suitable to routine calculations in typical laboratories. In addition, the method can be successfully applied to irregular systems, and to nearly entire spectral range. Another attractive feature exists in the possibility of using the multiparameter scaling (SQM, ESFF) to obtain very accurate vibrational frequencies without the additional amount of work, as the scaling factors published to date are well transferable to the polymer fragments. Applications to the PMMA, PVAc, and PIPAc polymers are presented. Nearly complete assignment of the bands on their infrared spectra is given. In particular, the bands in the most controversial range of 1300−1100 cm−1 on the PMMA infrared spectrum are assigned to highly delocalized vibrations, which is in accord with the most recent experimental observation. Conclusions related to some subtle features, which are not reproduced in the simulation, can be reached from the detailed analysis of individual vibrations. In addition, the obtained results can be used to exclude some conformations of units that may potentially be formed in the final material on the basis of the lack of the similarity between the simulated and the observed spectrum (the energetics obtained in the “gas-phase

■

ASSOCIATED CONTENT

S Supporting Information *

Harmonic and ESFF-scaled frequencies, IR intensities, σ parameters for the Gaussian-Lorentz functions, as well as the contributions in terms of types of valence internal coordinates for the selected structures (the page headers indicate the structure the data refer to). This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 48-81-537-5614. Fax: 48-81-533-33-48. Notes

The authors declare no competing financial interest.

■

REFERENCES

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Theoretical Determination of the Infrared Spectra of Amorphous

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