Effect of Molecular Size and Particle Shape on the Terahertz

Jul 18, 2013 - Figure 3 further characterizes the motion within the modes into that related to the TMA cation (hash shading) and the bromine anion (sl...
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Effect of Molecular Size and Particle Shape on the Terahertz Absorption of a Homologous Series of Tetraalkylammonium Salts Andrew D. Burnett,*,†,‡ John Kendrick,§ Christopher Russell,† Jeppe Christensen,∥ John E. Cunningham,† Arwen R. Pearson,‡ Edmund H. Linfield,† and A. Giles Davies.† †

School of Electronic and Electrical Engineering, University of Leeds, Leeds, LS2 9JT, U.K. Astbury Centre for Structural Molecular Biology, University of Leeds, Leeds LS2 9JT, U.K. § Institute of Pharmaceutical Innovation, University of Bradford, Bradford, BD7 1DP, U.K. ∥ Dynamic Structural Science, University of Bath, Research Complex at Harwell, Rutherford Appleton Laboratory, Harwell, Didcot, OX11 0EL, U.K. ‡

W Web-Enhanced Feature * S Supporting Information *

ABSTRACT: The absorption coefficient and refractive index have been measured for a homologous series of tetraalkylammonium bromides over the frequency range 0.3−5.5 THz. Spectral features are found to shift to lower frequencies as the molecular mass is increased, as expected. However, to understand the detailed structure of the observed spectral features, density functional perturbation theory calculations have been performed on the first four crystalline compounds in the series. From these calculations, we find that each spectrum is dominated by three translatory modes involving asymmetric motion of the ammonium cation and bromine counterion, although the overall number of active modes increases with increasing molecular size. The experimentally observed absorption is not completely described by the infrared active phonon modes alone. We show that it is also necessary to include the coupling of the phonon modes with the macroscopic field generated by the collective displacement of the vibrating ions, and we have applied an effective medium theory, which accounts for particle shape to allow for this effect in the calculation of the terahertz spectra.

T

In this paper, we probe the effect of systematically changing molecular size and mass on the spectral features observed in the THz region, while also investigating the effect of crystal shape on THz spectra. We use a homologous series of seven crystalline tetraalkylammonium bromide salts, in which the size of all four alkyl chains increases sequentially from methyl to heptyl. To understand the origin of the spectral modes observed, we used a quantum-mechanical-based density functional theory package, CASTEP,6 to calculate the infrared and terahertz spectra, which allowed us to assign spectral features to specific vibrations.7 Thus we can begin to understand how increasing molecular size can significantly affect the resultant THz spectra.

erahertz frequency time-domain spectroscopy (THzTDS) probes the low frequency (below 300 cm−1) infrared active vibrational modes of molecules and crystals. Considerable work over the last 10 years has concentrated on identifying the THz spectra of pharmaceutical compounds1 as well as explosives and drugs-of-abuse2 where, owing to the ability of THz radiation to penetrate a number of common packaging and clothing materials, there is the possibility for hidden sample identification.3 The high sensitivity of THz-TDS to polymorphism,1 and the large spectral differences observed in THz spectra of different salts2 and hydrates4 are a result of the many optically active normal modes exhibited in polycrystalline materials in this region of the electromagnetic spectrum. These modes are typically external or phonon modes that arise because of the confinement of the molecules within the crystalline lattice. This sensitivity to the crystalline environment has been investigated by, for example, Walther et al.,5 who demonstrated the contrast between the sharp spectral features seen in the THz spectra of crystalline glucose with the broad, largely featureless absorption of amorphous glucose. © 2013 American Chemical Society

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EXPERIMENTAL METHODS The seven tetraalkylammonium bromide compounds were purchased from BDH Chemicals. As each compound in the series is hygroscopic, approximately 2 g of each sample was first Received: June 4, 2013 Accepted: July 18, 2013 Published: July 18, 2013 7926

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Figure 1. THz absorption spectra (solid lines) and refractive index (dashed lines) of a 12.5% w/w pellet of each of the seven compounds in the homologous series with tetramethylammonium (TMA) bromide at the bottom through to tetraheptylammonium bromide at the top all recorded at room temperature.

dried in an oven at 40 °C for 2 days and measured gravimetrically to ensure all water was lost, before being ground and mixed with a THz transparent matrix material polytetrafluoroethylene (PTFE), in a number of mass ratios. Samples were formed with concentrations of 12.5% and 5% by mass and were then pressed into pellets, approximately 0.36 mm thick, within a supporting copper ring with an 8 mm diameter, for spectral measurement. These concentrations ensured that the resulting absorptions measured were not limited by the dynamic range of the system and so were a true reflection of the absorption.8 Spectral measurements were performed using a dry-airpurged broadband THz time-domain spectrometer (THzTDS).9 A 7.5 W diode laser (Millenia, Spectra Physics) was used to pump a Ti:Sapphire laser (Femtosource, Femtolasers) to produce a train of near-infrared pulses, each of duration ∼12 fs, centered at ∼800 nm. The beam was then focused onto a low-temperature-grown gallium arsenide (LT-GaAs) photoconductive switch, with a stripline geometry, which was biased at 100 V using a 10 kHz signal generator. The THz radiation emitted from the photoconductive switch was collected and collimated from the side of the emitter excited by the laser10 and focused onto the sample pellet by a set of off-axis parabolic mirrors. The THz radiation transmitted through the sample was then recollected and focused with a second pair of mirrors onto a 150-μm-thick GaP crystal for electro-optic detection, using a time-delayed probe beam (∼ 35 mW) split off from the original near-infrared laser pulse train. The coherent nature of this detection system gives a high signal-to-noise ratio at THz frequencies and allows both phase and amplitude information to be obtained, and hence the absorption coefficient and refractive index, without recourse to the Kramers−Kronig relationship. THz spectra were recorded for both concentrations of each compound, with each spectrum comprising an average of five scans to increase the signal-to-noise ratio. The absorption coefficient and refractive index were calculated using a method described previously (Fan et al.);9 a purged system

with no sample present was used as a reference. The maximum measurable absorption (αmax, the maximum absorption coefficient detectable by the spectrometer) was also calculated, as described by Jepsen et al.8 Low temperature measurements, recorded at 4.2 K and shown in Figure 2, were performed by mounting the sample pellet, within a copper ring for good thermal contact, onto a coldfinger of a continuous-flow helium cryostat (MicrostatHe, Oxford Instruments) equipped with high-density polyethylene windows. The simulated infrared spectra of the first four compounds in the series were calculated using CASTEP,6 a solid state density functional software package. Full details of these calculations are provided in the Supporting Information. As a starting point for each calculation, the crystal structures of three of the compounds were taken from the Cambridge Crystal Structural Database (CSD access codes ZZZUQO03 (tetramethylammonium bromide)11 TUDQEO (tetraethylammonium bromide)12 and RABTIX (tetrabutylammonium bromide)13). TPRAMB (tetrapropylammonium bromide)14 has an R-factor of greater than 17% so we redetermined the crystal structure (CCSD 925366) to improve the starting structure for optimization. [CCDC 925366 contains the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www. ccdc.cam.ac.uk/data_request/cif]. Crystals were converted to their primitive form and all crystal faces described throughout this article refer to the primitive version of the crystal. Each molecular structure was first optimized to find its minimum energy configuration within the constraints of fixed, experimental unit cell dimensions.7,15 After geometry optimization, a calculation of the phonon modes at the gamma point was performed using density functional perturbation theory (DFPT) to calculate the dynamical matrix. Both the phonon and Born charge sum rules were used to ensure translation invariance. The resulting phonon frequencies, ωm, correspond to the transverse optical modes of the crystal and allow the complex, frequency depedent dielectric constant of the infinite 7927

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to the spectra, and why this systematic spectral peak shift occurs. The harmonic approximation for molecular and crystal vibrations predicts transitions without vibrational or temperature broadening. To represent the shape of the experimental spectra more realistically, Lorentzian curves with a single fullwidth-half-maximum (FWHM) broadening, chosen to best represent the experimentally observed spectra, were applied to the calculated modes and intensities; a single width was used for all peaks in the spectrum of a given compound, but each compound required a different width. Figure 2 shows the experimental spectra of the four tetraalkylammonium bromide samples (5% w/w) at room temperature (black curve) and 4.2 K (blue curve), the calculated spectral positions (vertical red lines), and the calculated spectra (red curves). The experimental and calculated spectra are both normalized to a maximum height of one for comparison. The vertical lines represent the individual phonon modes, which are then each scaled by the same normalization factor as the calculated curve.7 It is interesting that we see only a very small shift with temperature for all four compounds, much smaller than that observed for a number of organic crystals measured previously,5,9 but similar in size to that observed for cocaine hydrochloride18 another organic salt. It can be seen that the calculated spectra for all four compounds show reasonable agreement with experimentally recorded spectra. Although, there are some obvious discrepancies in fine spectral shape, the main predicted features are at the correct locations. Many of the expected modes have little or no effect on the THz spectra which somewhat simplifies the analysis. This is owing to a combination of selection rules, and a large number of modes simply having a weak infrared intensity. Before analyzing the origin of these modes, it is useful to consider which modes are expected from group theory, using tetramethylammonium (TMA) bromide as the simplest example. Tetramethylammonium Bromide. TMA bromide crystallizes in space group P4/nmm (D4h) with two molecules (36 atoms) in the unit cell. The crystal will have 108 normal modes with the irreducible representation, calculated by factor group analysis19 for the crystal, shown in eq 3.

crystal to be expressed in terms of the high frequency permittivity, ε(∞), the volume of the unit cell, Ω, the oscillator strength tensors, Sm, and γ which broadens the absorption of a given mode and is introduced to avoid discontinuities. ε(ω) = ε(∞) +

4π Ω

∑ m

Sm 2

ωm − ω 2 − iγωm

(1) 16

An effective medium theory developed by Balan et al. was used to describe the absorption of radiation by particles, which are small relative to the frequency of the radiation. The method accounts for the coupling between a particle’s shape and each phonon mode, and computes the internal field, Ein(ω), within a small crystallite as function of frequency from the dielectric constant of the crystal and the dielectric constant of the supporting medium εext. E in(ω) = εext(εextI + L(ε(ω) − εextI ))−1Eext

(2)

L is a matrix of depolarization factors which provides information as to the shape of the crystallite, and Eext is the applied infrared radiation field. The final infrared absorption of a powder containing such crystallites can be obtained by averaging over the crystallite orientation and considering the coupling between the internal field and the internal polarization of the crystal. A more detailed description of the effective medium theory, the DFT calculations and parameters used can be found in the Supporting Information, as can a full structural analysis of the results of the geometry optimization for each of the four calculations.

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RESULTS AND DISCUSSION Figure 1 shows measurements of the THz absorption spectra (solid lines) and refractive index (dotted lines) of the seven compounds (12.5% w/w), all recorded at room temperature. All seven absorption spectra show a large broad feature comprising several modes, which shift to lower frequencies as a function of increased alkyl chain length. The refractive index for all seven samples shows characteristic inflections corresponding in frequency to the observed absorption peaks, as expected.9 There is, however, no systematic change in the overall refractive index with molecular size, with the refractive index of all 12.5% w/w pellets lying between 1.26 and 1.6. It should be remembered, however, that the pellet samples are mixtures of PTFE (average refractive index 1.43)17 and the tetraalkylammonium bromide, so the recorded refractive index is an average of both. The peak positions and spectral intensities of the absorption peaks are summarized in Table 1. To probe the shift in spectral absorption, calculations were performed to obtain the THz spectra of the first four compounds in the series to understand both the origin of the normal modes that contribute

Γcrystal = 8A1g + 5A 2g + 5B1g + 8B2g + 14Eg + 5A1u + 9A 2u + 7B1u + 5B2u + 14Eu

The modes we expect to see in the THz region are likely to be those of an external or phonon nature, that is, modes that arise from the confinement of the molecule within the crystalline lattice, which reduces the translational and rotational degrees of freedom.20 There are 12 normal modes related to translational motion of the molecules (the three translational degrees of freedom of each of the four ions) within the crystal. Three of these modes will not be observed in the infrared spectrum since the required translationally invariant combinations over all unit cells correspond to the translations of the entire crystal (these modes are the translational or acoustic modes (Ac) of the crystal). The remaining nine modes are referred to as translatory lattice modes (T′) and correspond to the movement of the rigid ions within the unit cell with respect to each other. The other external modes are referred to as rotary lattice modes (R′) and arise because of the lack of free rotation within the crystal. In this crystal there are two TMA ions, each with three degrees of rotational freedom (the bromine ions have no rotational degrees of freedom) meaning

Table 1. Summary of Peak Positions and Spectral Intensities for the Seven Compounds Shown in Figure 1 alkyl name

peak position (THz) and intensity (cm−1)

methyl ethyl propyl butyl pentyl hexyl heptyl

2.19 (174.34), 3.09 (120.20) 1.5 (65.38), 2.0 (242.50), 2.99 (43.18) 1.5 (124.44), 2.20 (99.275), 3.08 (38.90), 3.72 (22.91) 1.47 (122.44), 1.68 (111.21), 2.08 (69.55) 1.03 (65.60), 1.40 (142.21), 2.37 (62.129) 0.96 (44.19), 1.37 (58.41), 1.73 (47.76), 2.27 (32.15) 0.84 (54.15), 1.40 (130.25)

(3)

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Figure 2. Comparison of the experimental THz spectra (5% w/w sample, room temperature (black), 4 K (blue)) with the calculated spectra (red curves) for (a) TMA bromide, (b) tetraethylammonium (TEA) bromide, (c) tetrapropylammonium (TPA) bromide, and (d) tetrabutylammonium (TBA) bromide. Calculated spectra have been prepared by adding a Lorentzian broadening of fwhm (a) 35, (b) 15, (c) 12, and (d) 20 cm−1 to the predicted normal modes (vertical red lines). Experimental and calculated spectra were both normalized to a maximum value of 1.0 for comparison. The calculated spectral modes have been normalized to the same scale.

2.41 and 3.53 THz, with the first a strong translatory mode, and the second a very weak rotary mode, and a singlet mode (A2u) at 2.78 THz, which can be assigned to a translatory lattice mode. In previous work,7 we developed a procedure that can be used to partition the energy of a given mode into components arising from molecular center of mass motion (translatory modes) and rigid body rotation (rotary modes), with the rest of the energy within the mode assumed to be vibrational in character. Using this analysis, Figure 3 shows a graphical representation of the percentage energy contribution to all the calculated modes between 0 and 5.5 THz with molecular center of mass motion in black, rigid body rotation in red, and vibrational motion in blue. The figure shows that, although many modes have small contributions from all three types of motion, most are often dominated by a single type of motion. It is also clear, if we compare the irreducible representations from eq 4 with the Mulliken symbols in Table 2, that modes dominated by center of mass motion (black) correlate to the translatory lattice modes while the modes dominated by rigid body rotation correlate with rotary lattice modes. Figure 3 also shows that no internal modes are present between 0 and 5.5 THz in this crystal, since none of the modes have a large contribution to their energy from molecular vibrations. Figure 3 further characterizes the motion within the modes into that related

there are six of these rotary modes.19 Using factor group theory,19 we can then determine an irreducible representation for the acoustic (ΓAc), translatory (ΓT′) and rotary (ΓR′) lattice modes, which can be seen in eq 4 with “ir” signifying infrared activity. ΓAc = A 2u + Eu ΓT ′ = A1g + A 2u(ir) + B2g + 2Eg + Eu(ir) ΓR ′ = A 2g + B2u + Eg + Eu(ir)

(4)

Importantly, owing to selection rules, only the A2u (one translatory mode) and Eu (one set of rotary and one set of translatory) normal modes are infrared active. We, therefore, expect to see three distinct external modes in the THz spectrum of TMA bromide. The remaining 90 modes are all internal and although some may appear in the THz spectra, the likelihood is that they will appear at much higher frequencies. In the CASTEP calculation of TMA bromide, 18 normal modes are found between 0 and 5.5 THz (3 and 183 cm−1), as shown in Table 2, which shows the irreducible representation of each mode along with an assignment to one of the external mode types indicated in eq 4. As the factor group analysis also suggests, five of these modes are infrared active (shown in bold in Table 2): there are two pairs of degenerate modes (Eu) at 7929

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dominated by three translatory modes involving antisymmetric movement of the four ions along each of the three crystal axis directions. Animations of these three modes are available online (animation of mode 7, animation of mode 8, and animation of mode 10). From the results of these calculations, we can now understand the origin of the three modes that contribute to the THz spectra of TMA bromide. However, the experimental spectra shows two clear peaks (Figure 2a): one centered at 2.19 THz and the second at 3.09 THz, whereas the calculated spectra with a Lorentzian broadening of 35 cm−1 added to each of the five infrared active modes shows only a single peak centered at 2.6 THz. The reason for this is that the theoretical description of absorption outlined so far ignores any contribution arising from the macroscopic electric field generated in a crystal by the collective displacement of ions. This is the so-called nonanalytical term, in the expression for absorption intensity, which gives rise to an intensity contribution from shifted longitudinal optical modes. For neutral molecular crystals, this effect is small, but for salts the effect can be large. In the case of single crystals, or polycrystalline material, where the size of the crystal is large relative to the wavelength of the infrared radiation, the calculation of the absorption and transmission of radiation is complicated by the need to consider the anisotropy of the dielectric tensor.21 However, by using an effective medium theory for particles which are small relative to the wavelength of the radiation, Balan et al.16,22 developed a methodology for describing the infrared absorption properties of dielectric particles embedded in a low dielectric constant medium. This approach has been adopted here to describe the interaction of THz radiation with small crystallites of powdered salt embedded in a PTFE matrix. Because of the strongly ionic nature of the salts it is expected that the absorption intensities and frequencies will be strongly influenced by particle shape and the nature of the surfaces which may dominate any particular morphology. As discussed previously by Balan et al.,16 the presence of charges on the surfaces of a crystallite modifies

Table 2. Calculated Modes of TMA Bromide between 0 and 5.5 THz using CASTEPa mode number

frequency (cm−1)

frequency (THz)

intensity (km/mol)

irreducible representation

lattice mode type

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

−0.06 −0.04 −0.04 49.67 49.67 61.06 80.31 80.31 84.17 92.76 99.96 101.74 101.74 117.66 117.66 130.04 130.04 132.10

0.00 0.00 0.00 1.49 1.49 1.83 2.41 2.41 2.53 2.78 3.00 3.05 3.05 3.53 3.53 3.90 3.90 3.96

0.000 0.000 0.000 0.000 0.000 0.000 2.018 2.018 0.000 1.887 0.000 0.000 0.000 0.003 0.003 0.000 0.000 0.000

A2u Eu Eu Eg Eg A1g Eu Eu A2g A2u B2g Eg Eg Eu Eu Eg Eg B2u

Ac Ac Ac T′ T′ T′ T′ T′ R′ T′ T′ R′ R′ R′ R′ T′ T′ R′

Included in the table are the mode numbers, frequency in both cm−1 and THz, the infrared intensity, the Mulliken symbol of each mode, and whether the mode is acoustic, translatory, or rotary in nature with the bold text highlighting the infrared active modes. a

to the TMA cation (hash shading) and the bromine anion (slash shading). Of the modes that are not infrared active, three can be described as translatory lattice modes (modes 4−6), where virtually all motion comes from the bromine ions, while the other three only involve the motion of the alkylammonium ion (modes 11, 16, and 17). As the infrared active rotary modes (numbers 14 and 15) have an infrared intensity several orders of magnitude smaller than the three infrared active translatory modes, they make very little contribution to the calculated spectra shown in Figure 2a. The THz spectrum is therefore

Figure 3. Stacked histogram showing the percentage energy contributions to the calculated normal modes of TMA bromide between 0 and 5.5 THz. Black shows center of mass motion, red shows rigid body rotation, and blue shows vibration. Hashed shading signifies motion of the alkyl ammonium cation and slashed shading signifies motion of the bromine anion. Modes marked with an asterix are infrared active. 7930

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its internal electric field by a depolarization field. This can be calculated using an effective medium theory, from a combination of the electric field arising from the infrared radiation in the surrounding low dielectric constant medium and the complex frequency dependent permittivity tensor of the crystal ε(ω). The details of how this effect was included in our calculations are summarized in section 4 of the Supporting Information. One of the parameters that must be introduced is a damping factor, γ, which is used to avoid discontinuities at the mode frequencies, and is the basis for introducing Lorentzian broadening in Figure 2. To understand the effect of particle shape on the spectra, the γ damping factor used here is the same as the Lorentzian broadening term employed in Figure 2 and has been chosen to provide a reasonable match to the width of the experimentally observed absorption peaks. Spherical, needle, and platelike crystal shapes were investigated as they approximate the particle shapes well and are straightforward to incorporate in the theory.23 SEM images of the sample particles after grinding were also recorded and show that a large number of the particles are plates or fragments of plate crystals formed in the grinding process (Figure 4). A

Figure 5. Comparison of the experimental THz spectrum of TMA bromide (12.5% w/w sample, black) with the calculated spectra using CASTEP with no particle shape taken into account (red curve, with the vertical lines showing predicted infrared active peak positions), for four plate-shaped crystals (plate orientation indicated in parentheses), and for spherical particles.

but with the single broad peak shifted to slightly higher frequencies. All four simulated spectra of TMA bromide for crystals with a plate morphology show a large peak centered between 2.3 and 2.6 THz with a shoulder at higher frequencies although the ratio of peak heights and widths varies between plates. It is clear that the prediction for a platelike morphology associated with the {101} surfaces most closely resembles the experimentally observed spectra, and it is likely that the majority of particles show this morphology. Tetraethylammonium Bromide. The CASTEP calculations of the three larger compounds can be analyzed in a similar way to that of TMA bromide. Tetraethylammonium (TEA) bromide has twelve molecules in the Bravais lattice, so we expect 54 external modes (33 translatory and 18 rotary lattice modes). As can be seen in Figure 6a, which displays the energy contribution to all the predicted normal modes (except acoustic modes) between 0 and 5.5 THz (a total of 77 modes), there is a sharp drop in the center of mass/rigid body rotation energy contribution after mode 54 (that is, for frequencies >3.23 THz). The low frequency phonons have a much more complex energy profile than was seen for TMA bromide. All the modes show some contribution from molecular center of mass motion, rigid body rotation, and vibration, although the translatory and rotary modes are still easily identifiable in Figure 6a. Because of the increased mass of the molecule, there are also internal modes now appearing below 5.5 THz (although still higher in frequency than any external mode). A full list of all the mode frequencies, infrared intensities and corresponding Mulliken symbols is provided in Table S2 of the Supporting Information. This shows that only 28 of the 77 normal modes are infrared active, and of these, three modes

Figure 4. SEM image of the ground TMA bromide particles at 250× magnification. The image shows that the sample has a preponderance of flat surfaces, consistent with the material comprising platelike particles, rather than spherical or needlelike particles.

number of crystal morphologies can grow from any unit cell. To determine which crystal morphologies were most likely to be encountered, we calculated growth morphologies24 for each compound using Materials Studio 4.025 with the Dreiding forcefield26 and charge equilibrated (QEQ) charges.27 These calculations indicated the most likely crystal habit on the basis of the calculated attachment energies28 of a large number of low-index surfaces. For TMA bromide, these calculations suggested four families of faces {001}, {101}, {100}, and {111} to be the most important. Using effective medium theory, which is outlined in the Supporting Information, we then calculated the effect of particle shape on the THz spectra for a PTFE matrix containing small crystallites with four different platelike morphologies with these faces as the dominant face, and also for spherical particles (Figure 5). It can be seen in Figure 5 that particle shape has a dramatic influence on the calculated spectra with some modes shifting to higher frequencies by up to 800 GHz. For spherical particles, the calculated spectrum is similar to that using the transverse optical (TO) frequencies and intensities reported by CASTEP 7931

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and {211} are the most important families of faces. Effective medium theory calculations were carried out in a similar manner for TMA bromide, but this time considering particles with the two platelike morphologies, and for spherical particles. The correlation between theory and experiment based on TO frequencies and intensities (Figure 2b) is already very good without the consideration of particle shape, but its inclusion causes a broadening of the main feature with a pronounced shoulder appearing at higher frequencies (see Supporting Information Figure S5). The experimental spectrum is similar to that calculated both for spherical particles and plate morphologies involving {211} faces; however, the size of the calculated shift to higher frequencies for plate morphologies involving {11̅0} faces is too large. Tetrapropylammonium Bromide. The spectra of tetrapropylammonioum (TPA) bromide are more complex than the previous two cases, but the analysis remains relatively simple with only 9 external modes. Analysis of the TO phonon modes calculated by CASTEP shows a clear distinction between external and internal modes (see Figure 6b) although, due to the mixed energy contribution to all the modes, the differentiation between translatory and rotary modes becomes more difficult. Of the 18 modes between 0 and 5.5 THz, 11 are infrared active with 6 modes dominating the calculated spectra (Supporting Information Table S3). Modes 6, 7, and 8 are very similar to the most active modes in the previous two molecules, but, all three modes show a significant rotational and vibrational deformation of the long alkyl chain arms during the vibration because of the increased chain length, which increases the flexibility of the molecule. There are three other modes that also show strong infrared activity (modes 8 and 9 at 1.29 THz and mode 10 at 3.07 THz), which both show a different motion to the other infrared active modes. Modes 8 and 9 are degenerate modes with a large degree of rotary motion, very much like modes 14 and 15 in TMA that showed very weak infrared activity. Mode 10, on the other hand, is almost entirely vibrational in character and involves deformation of the four alkyl chains with only small changes in bromine ion position. This is the first example in this series where an internal mode provides any significant contribution to the THz spectra. It can be seen in Figure 3c that the spectrum of TPA bromide has two main features at 1.5 and 2.20 THz along with a number of weaker features at higher frequencies. This same structure is found in the simulated spectrum with mode 6 correlating with the peak at 1.5 THz and modes 7 and 8 correlating with the feature at 2.20 THz. The next strongest feature at 3 THz correlates well with mode 10, whereas modes 4 and 5 produce a small shoulder on the first peak at 1.6 THz that is not seen experimentally. SEM images showed that, as in the case of the other compounds, the ground TPA bromide powder is a complex mixture of particle shapes with a large number of particles being either spherical or platelike. Therefore, as with the previous compounds, growth morphology calculations were performed which predicted three significant families of faces ({010}, {001}, and {1̅11}). Effective medium theory calculations were performed on the three corresponding platelike particles, as well as with spherical particles. Inclusion of particle shape in the spectral calculations can be seen in Supporting Information Figure S6. As with the previous examples particle shape has a dramatic effect on the calculated spectra, plates with families of faces represented by {001} and {1̅11} showing a significant reduction in the weak shoulder produced by modes 4 and 5, while producing a

Figure 6. Stacked histograms showing the energy contribution to the normal modes calculated for (a) TEA bromide, (b) TPA bromide, and (c) TBA bromide. Black signifies center of mass motion, red rigid body rotation, and blue is vibration.

(mode numbers 23, 24, and 25) dominate the spectrum (confirmed in Figure 3b). These three modes are all antisymmetric translatory modes, as were the modes in the spectrum of TMA bromide. They have, however, been shifted to lower frequencies owing not only to the increased mass of the cation but also to the increased energy contribution from rotation and vibration. SEM images of the particles show that the mixture after grinding is made of a few large plates and a large number of small platelike fragments and spherical particles. For TEA bromide growth morphology calculations suggested that {11̅0} 7932

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shoulder at ∼2.3 THz owing to a shift in frequency of the TO phonon. The spherical particles on the other hand reduce the intensity of the first peaks (modes 4, 5, and 6) dramatically, while there appears to be no discernible shift in the TO mode of the second peak (modes 7 and 8). Neither of these spectra greatly improved the correlation between simulation and experimental results alone. Tetrabutylammonium Bromide. Finally, we discuss tetrabutylammonium (TBA) bromide, which is expected to have 33 external modes and 612 internal modes. Figure 6c shows the breakdown of the TO phonon modes predicted by CASTEP into internal and external contributions. Unlike the other molecules, there is no longer a clear distinction between external and internal modes (this is much like other small molecular systems).7,29 Of the 90 normal modes predicted between 0 and 5.5 THz, 42 show infrared activity, while the predicted spectrum itself is dominated by four modes (Supporting Information Table S4). Modes 13, 16, and 17 (1.23, 1.39, and 1.41 THz, respectively) have a large proportion of center of mass motion (over 65% in all cases), and are similar in motion to the three most active modes in the previous three calculations. Mode 22 (1.71 THz) has equal contributions from center of mass motion and vibration (40% each) with the remaining being rotation. This mode shows translation of the bromide anion along with a degree of flexing and rotation of the alkyl chains. Growth morphology calculations for TBA bromide predict three significant faces in the morphology ({101̅}, {100}, and {001}). Effective medium theory was used to calculate the THz spectra for these three platelike particles along with spherical particles. Figure 3d shows excellent agreement of the basic CASTEP calculation with experiment, even without the inclusion of particle shape. On including particle shape, however, we see a broadening of the main feature with several shoulders appearing at higher frequencies for all the analyzed particle shapes as for TEA bromide (Supporting Information Figure S7). The calculation for a particle with platelike morphology dominated by the {100} family of faces best reproduces the experimentally observed spectra with SEM images indicating that the majority of particles are indeed platelike. Origins of the THz Spectra of the Four Smallest Compounds in the Series. An overall comparison of the phonon calculations of the four smallest tetraalkylammonium bromides in the series show that the spectra are dominated by three modes in all four cases. These can be described as translatory motion of the bromine ions and alkylammonium ions in opposite directions, generating a large change in dipole moment and thus intense infrared normal modes. The atomic motion corresponding to these normal modes is therefore similar in all four compounds, even though they crystallize into different space groups, with different intermolecular interactions. The shift to lower frequencies of the main spectral feature is due to the same modes vibrating but with an increased mass as the chain length increases. Although calculations on the largest three molecules have not been performed owing to the lack of experimental crystal structures, it may be a reasonable assumption that the same is true for these molecules also. The size of this shift seems to be not only dependent on the increased size of the cation but also on the increased flexibility, and thus the greater rotational and vibrational contributions to the energies of the three dominant modes. Larger molecules also have more modes that are

infrared active and contribute to the spectrum in the THz region. This gives rise to a more complex underlying structure to the spectrum, although this is not always apparent experimentally as the resulting bands overlap. Significantly, we find that the peak intensities and positions are influenced by crystal shape, and that by modeling this, a considerable improvement in the correlation between measurement and theory was achieved in most of the calculations. The fact that all the spectra from the series are dominated by modes involving the motion of the bromine ion also suggests that the exchange with a different anion would have a dramatic effect on the modes. In contrast, a homologous series of neutral compounds may not follow the pattern seen here, since the infrared active modes would be greatly influenced by the orientation of molecules in the crystal. Neutral compounds would also be expected to have a much smaller change in spectra resulting from their particle shape. Finally, we speculate that this reliance on the motion of the counterion in the dominant spectral modes may also explain why THz spectra of a number of compounds change so dramatically upon salt formation.2

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CONCLUSION

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ASSOCIATED CONTENT

The THz spectra of a homologous series of tetraalkylammonium bromide compounds have been measured between 0.3 and 5.5 THz, and a systematic shift to lower frequencies of the main spectral feature with increasing mass of the alkylammonium cation observed. DFPT calculations have been performed on the smallest four crystals in the series and the main spectral features in all cases have been shown to be dominated by three translatory lattice modes where the bromine anion and alkylammonium cation move in opposite directions. Increasing the size and flexibility of the molecules reduces the distinction between external and internal modes and also increases the number of infrared active internal modes at these frequencies, which increases the complexity of the underlying THz absorption spectrum. DFPT calculations of the TO frequencies have been used previously in the interpretation of THz spectra. The present results clearly show that, for compounds with charge separation such as salts, where there is a large coupling between the electric field induced by the phonon vibrations and that induced by the radiation, it is essential to have an understanding of the impact of the particle shape on the absorption of radiation to understand the underlying contributions to the spectrum. In the case where the particle shape can be well characterized,16,22 it is possible to calculate the spectrum. But where the particle shape is not fully characterized it is only possible to use the calculations to estimate the range of shifts in peak positions and intensities. Further detailed understanding of the nature of the vibrational contributions to the observed spectra is only possible for extremely well characterized and consistent particles.

S Supporting Information *

Theoretical procedures, complete analysis of DFT structural calculations, and tables of calculated peak positions and intensities. This material is available free of charge via the Internet at http://pubs.acs.org. 7933

dx.doi.org/10.1021/ac401657r | Anal. Chem. 2013, 85, 7926−7934

Analytical Chemistry

Article

W Web-Enhanced Features *

(19) Nakamoto, K. Infrared and Raman Spectra of Inorganic and Coordination Compounds Part A: Theory and applications in inorganic chemistry, 6th ed.; Wiley: New York, 2009; p 432. (20) Decius, J. C.; Hexter, R. M. Molecular Vibrations in Crystals; McGraw-Hill: New York, 1977. (21) Mayerhöfer, T. G. J. Opt. A: Pure Appl. Opt. 2002, 4, 540. (22) Balan, E.; Delattre, S.; Roche, D.; Segalen, L.; Morin, G.; Guillaumet, M.; Blanchard, M.; Lazzeri, M.; Brouder, C.; Salje, E. K. H. Phys. Chem. Miner. 2011, 38, 111−122, DOI: 10.1007/s00269-0100388-x. (23) Van De Hulst, H. C. Light Scattering by Small Particles; Dover Publications: New York, 1957. (24) Docherty, R.; Clydesdale, G.; Roberts, K. J.; Bennema, P. J. Phys. D Appl. Phys. 1991, 24, 89−99, DOI: 10.1088/0022-3727/24/2/001. (25) Accelrys Material Studio, version 4.0; Accelrys Software Inc.: San Diego, CA, 2007. (26) Mayo, S. L.; Olafson, B. D.; Goddard, W. A. J. Phys. Chem. 1990, 94, 8897−8909, DOI: 10.1021/j100389a010. (27) Rappe, A. K.; Goddard, W. A. J. Phys. Chem. 1991, 95, 3358− 3363, DOI: 10.1021/j100161a070. (28) Hartman, P.; Bennema, P. J. Cryst. Growth 1980, 49, 145−156, DOI: 10.1016/0022-0248(80)90075-5. (29) Jepsen, P. U.; Clark, S. J. Chem. Phys. Lett. 2007, 442, 275−280, DOI: 10.1016/j.cplett.2007.05.112.

Animations of three translatory modes involving antisymmetric movement of the four ions along each of the three crystal axis directions are available in the HTML version of the paper.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Engineering and Physical Sciences Research Council (EPSRC, U.K.), the Leverhulme Trust, the Royal Society, and the Wolfson Foundation.

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REFERENCES

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dx.doi.org/10.1021/ac401657r | Anal. Chem. 2013, 85, 7926−7934