1Hâ2H Cross-Relaxation Study in a Partially Deuterated Nematic

Article pubs.acs.org/JPCB

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H−2H Cross-Relaxation Study in a Partially Deuterated Nematic Liquid Crystal A. Gradišek,*,† P. J. Sebastiaõ ,*,‡,§ S. N. Fernandes,∥ T. Apih,† M. H. Godinho,∥ and J. Seliger⊥ †

Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia Centro de Física da Matéria Condensada, Av. Prof. Gama Pinto 2, 1649-003 Lisbon, Portugal § Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal ∥ CENIMAT/I3N, Departamento de Ciência dos Materiais, Faculdade de Ciências e Tecnologia, UNL, 2829-516 Caparica, Portugal ⊥ Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia ‡

S Supporting Information *

ABSTRACT: A detailed study of the cross-relaxation effects between the 1H and 2 H spins systems is presented in the nematic phase of a 5-cyanobiphenyl (5CB) liquid crystal, partially deuterated at α position (5CB-αd2). The proton spin− lattice relaxation time was measured at a frequency range from 5 kHz to 100 MHz at a temperature 5 K below the nematic−isotropic phase transition. In the low frequency domain, the spin−lattice relaxation rate (T1−1) dispersion clearly differs from that of the fully protonated 5CB homologue. At two distinct frequencies, T1−1 presents two distinct local maxima and for low frequencies T1−1 presents a stronger frequency dependence when compared with what is observed for 5CB. The T1−1 dispersion obtained for 5CB-αd2 for frequencies above 60 kHz was interpreted in terms of the relaxation mechanisms usually accepted to interpret the spin−lattice relaxation in nematic phases in general and 5CB in particular. For lower frequencies it was necessary to consider cross-relaxation contributions between the proton and deuterium reservoirs. A detailed model interpretation of the deuterium quadrupolar dips with respect to the proton-spin relaxation is presented. The analysis of the quadrupolar relaxation independently confirms that the order director fluctuations is the dominant mechanism of proton relaxation in the low frequency domain.

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dispersion plots), such as in the cases of 14N7−12 or 35Cl.13,14 Studies of dip frequencies and shapes often reveal information about local structure, for example, in the case of HpAB, which was revealed to have SmC structure composed of bimolecular unit cells.7 Interesting enough, no systematic study has been reported so far, to the best of our knowledge, on proton−deuteron crossrelaxation in ordered liquid crystalline phases. Partially deuterated liquid crystals are often used for selective studies of molecular dynamics, phase transitions, or molecular ordering.1,15,16 Deuterium NMR spectrum is mainly influenced by the coupling between the nuclear electric quadrupolar moment of deuterium nucleus and the nonsymmetric electric field gradient at the nucleus. Therefore, the deuterium spectrum is sensitive to site-specific chemical environments, as opposed to the proton spectra, where the shape of the spectra is mostly dominated by dipolar coupling between intra- or inter-molecular proton spins. If proton spin−lattice relaxation dispersion is measured on partially deuterated molecules, proton−deuteron cross-relaxation is expected to affect the profile of the dispersion curve, by

INTRODUCTION Nuclear magnetic resonance relaxometry is a powerful tool to study molecular dynamics in liquid crystalline systems.1−5 Proton spin−lattice relaxation is sensitive to fluctuations of local fields and by measuring relaxation over a broad range of resonance frequencies (relaxation dispersion), it is possible to access various dynamic processes that take place at different time scales. In the case of liquid crystalline phases the relaxation processes most often found can be intramolecular or intermolecular. The former are associated with local molecular rotations/reorientations or associated with collective motions such as order director fluctuations. The latter are associated with translational molecular displacements, usually as the result of self-diffusion.1 In addition to these spin−lattice relaxation processes, cross-relaxation between proton and quadrupole nuclear spin reservoirs in the liquid crystal system can be observed. This mechanism results in shortening of the relaxation time, T1, at magnetic fields where the splitting of Zeeman energy levels of proton spins matches one of the quadrupole transition frequencies of a quadrupole nuclei spin.6 Many liquid crystals contain quadrupole nuclei that can contribute to cross-relaxation of proton spins. The so-called cross-relaxation “dips” in T1 relaxation dispersions were observed in log−log scale plots (or “cusps” in T1−1 log−log scale © 2014 American Chemical Society

Received: March 13, 2014 Revised: May 2, 2014 Published: May 2, 2014 5600

dx.doi.org/10.1021/jp502542q | J. Phys. Chem. B 2014, 118, 5600−5607

The Journal of Physical Chemistry B

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producing, at specific resonance frequencies, sharp decreases of the spin−lattice relaxation time, usually referred to as “quadupolar dips”. Relaxometric studies have been carried out on partially or fully deuterated liquid crystals.2,17,18 However, in neither study were quadrupolar dips observed, probably because of a coarse frequency sampling or an inadequate frequency range. Here, we present a thorough analysis of 1H−2H crossrelaxation effects in a well-known liquid crystalline model system, 5-cyanobiphenyl (5CB).2,3 We compare the dispersions in a nondeuterated system and in a system, deuterated at the αposition. Analysis of dips frequencies and shape provides information such as the orientation of the long molecular axis for rotation and independently confirm which dynamic process dominates the relaxation in the frequency range of the dips.

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EXPERIMENTAL SECTION System Description. The α-deuterated liquid crystal 4′pentylbiphenyl-4-carbonitrile-α-d2 (5CB-αd2) was obtained as described by Kundu et al.19 using a three-step reaction route, based on a modified Gray et al.’s procedure.20 In the first step a Friedel−Crafts’ reaction of pentanoyl chloride with 4-bromobipheny occurred, yielding the subsequent ketone. In the second step, the reduction of the ketone with LiAlD4 occurred to afford 4-bromo-4′-pentylbiphenyl-α-d2, the later underwent aromatic substitution of the bromine for the cyanide group to obtain 5CBαd2. This last step was performed in a sealed tube at 240 °C with a short reaction time and high yield (90%). The compound structure of the 5CB-αd2 was confirmed by 1H and 13C NMR and by FTIR spectroscopies. The N−I transition temperature was confirmed by performing polarizing optical microscopy with varying the temperature. The N−I bulk transition temperature is 308 K. Further experimental details and characterization data are available in the Supporting Information. Experimental Details. The sample was heated 15 K above the N−I transition temperature and gradually cooled to the temperature 5 K below the I−N transition temperature, TIN, with a cooling rate of approximately 1 K/min. The relaxation dispersion data were acquired using a fast field-cycling NMR relaxometer SPINMASTER FFC-2000 (Stelar s.l.r.). Around 300 mg of sample was used for the measurements. The longitudinal spin−lattice relaxation time (T1) of protons was measured in the frequency range from 18 MHz down to 5 kHz. Above 4 MHz, the nonpolarized sequence was used to measure the relaxation times. Below this frequency, the prepolarized sequence was used, with the polarizing field of 0.42 T (1H 18 MHz). In both cases, the acquisition frequency was 9.25 MHz. In addition, T1 was measured at 100 MHz using an Oxford superconducting magnet and a homemade NMR spectrometer. The inversion recovery pulse sequence was used for these measurements. Experimental Results. The experimental results for the 1H spin−lattice relaxation rate as a function of Larmor frequency, νH = γHB/(2π), where γH is the proton’s gyromagnetic ratio and B is the Zeeman magnetic field, are presented in Figure 1, for both 5CB3 and 5CB-αd2 systems at the temperature (TIN − 5 K). The general behavior is similar for both systems, but some differences can be detected, in particular, in the low frequency region. The spin−lattice relaxation rate for the deuterated sample is slightly larger than for the nondeuterated one for frequencies above MHz and between 10 and 40 kHz, the relaxation in the partially deuterated sample expresses two cusps (dips in the T1 dispersion presentation) that are related to the cross-relaxation mechanism.

Figure 1. Spin−lattice relaxation time as a function of Larmor frequency at T = TNI − 5 K for 5CB3 (red squares) and for 5CB-αd2 (black circles).

At low fields (below 10 kHz), the relaxation in deuterated sample is faster than in the nondeuterated one. A detailed theoretical analysis of the relaxation dispersion will be presented in the following.

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ANALYSIS OF EXPERIMENTAL RESULTS AND DISCUSSION Relaxation Mechanisms. The proton spin−lattice relaxation rate reflects the modulation of the interproton dipolar interaction by the molecular motions. When molecules move, the interproton distances vary, together with the angles between the interproton vectors and the magnetic field. Proton spin−lattice relaxation is affected by fluctuations of both intramolecular as well as intermolecular interactions. In liquid crystalline phases, it is possible to identify molecular motions that either are statistically independent or have distinct characteristic correlation times. Usually, these motions correspond to characteristic mechanisms that contribute simultaneously to the spin−lattice relaxation.1,21 In the analysis of proton relaxation in the nematic phase of 5CB-αd2, we will consider the following mechanisms: •Molecular translational self-dif f usion (SD) af fected by the phase structure and local molecular organization. The relaxation model for the nematic phase was developed by Ž umer and Vilfan.22,23 It is based on the Torrey model for the isotropic phase,24 which depends on the spin density n, the self-diffusion constant, the mean square jump distance ⟨r2⟩, and the width of the molecules d. The model for the nematic phase considers two diffusion constants, D⊥ and D∥, to reflect the anisotropy of the dynamic process. nτ (T1−1)SD = KD 3D Q (νHτD ,D/D⊥) (1) d 2 2 with KD = (9/8)(μ0γH ℏ/(4π)) . τD is the correlation time associated with the molecular jump displacements. Q(vHτD,D∥/D⊥) is a dimensionless function that has to be calculated numerically. In the extreme narrowing limit Q(vHτD,D∥/D⊥) reaches a constant value and in the high frequency regime Q(vHτD,D∥/D⊥) ∼ vH−2. •Local molecular rotations/reorientaions (R) of liquid crystalline molecules. Models usually consider the rotations along the long molecular axis and reorientations around the short axis, with 5601



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correlation times τL and τS, respectively. In the isotropic phase, the Woessner model25 is usually used to describe the contributions to relaxation. In the nematic phase, this model can be extended by taking into consideration the rotational motions’ restrictions that result from the increased orientational order in this phase.3 (T1−1)R = KD[J1(νH) + J2 (2νH)]

where −1

T1̃ = (T1−1)SD + (T1−1)R + (T1−1)ODF

In the low frequency domain, around 5 kHz, it is necessary to take into account the effect of local dipolar fields which can be, in most cases, included in the mode as a low cutoff frequency affecting all relaxation mechanisms and vH → (vH2 + vloc2)1/2. The T1−1 model given by eq 6 was fitted to the experimental results using a home-written nonlinear fitting software package,30 considering a global minimum least-squares target. Model parameters obtained from the best fit to the experimental data are listed in Table 1, the best fit is presented in Figure 2. Some of

(2)

where JkR (νL)

2 |dm ,0 2(αij)|2 τm 4 2 = k ∑ ⟨|Dk , m|2 ⟩ 6 3 m =−2 1 + 4π 2νH 2τm 2 rij

(3) −1

−1

−1

Table 1. Model Parameters Corresponding to the Best Fits Obtained with Model Eq 6 As Explained in the Text (d = 5 × 10−10 m, vch ∼ 96 MHz, vcl ≃ 1 kHz, D = 3.9 ×1011 m2 s−1, vloc ∼ 4.4 ± 0.5 kHz)

−1

with the correlation times τ0 = τS, τ±1 = τS + τL , and τ±2 = τS−1 + 4τL−1. The average second rank Wigner rotation matrix ⟨|Dk,±m|2⟩ can be expressed in terms of the nematic order parameter S and of ⟨P4⟩.26 The factors |dm ,0 2(αij)|2 /rij 6 can be estimated for an average conformation of the molecule.26 •Order director f luctuations (ODF). This mechanism corresponds to orientational fluctuations of the nematic director with respect to its time-average orientation. The spin−lattice relaxation contribution has the dispersion profile T1−1 ∼ vH−1/2, well established in NMR relaxometry of liquid crystals.1,27,28 This characteristic frequency dependence is modified for frequencies above νch and below νcl, where νch and νcl correspond to the high and low cutoff frequencies associated with the high and low fluctuation wave modes, respectively.1 ⎡ ⎛ν ⎞ ⎛ ν ⎞⎤ (T1−1)ODF = A ODFνH−1/2⎢fc ⎜ ch ⎟ − fc ⎜ cl ⎟⎥ ⎢⎣ ⎝ νH ⎠ ⎝ νH ⎠⎥⎦ −6

5CB τS (10−9s) τL (10−10s) n (1028spins/m) A±0 (10−3s−2) A±1 (10−3s−2) A±2 (10−3s−2) A AODF (103s−2) ACR1 (105s−2) νH1 (kHz) τCR1 (10−5s) ACR2 (105s−2) νH2 (kHz) τCR2 (10−5s) ACR3 (105s−2) νH0 (kHz) τCR3 (10−5s)

(4)

−6

where AODF = KDKBTS ζ /(2π K aeff ). aeff depends on the interproton distances and interproton vectors’ angles with respect to the long molecular axis. fc(x) is a dimensionless function that defines the T1−1 behavior in the high and low frequency asymptotic limits. K is the elastic constant, considering an one-constant approximation,29 and ζ is an average viscosity. In addition to the above mechanisms, the cross-relaxation (CR) between 1H and 2H plays an important role as well. To describe its effect on the proton relaxation, we assumed the quadrupolar dips take shape of Lorentzian curves: τCRi (T1−1)CRi = A CRi 2 1 + 4π (νH − νHi)2 τCRi 2 (5) 2 1/2

3/2 3/2

∑ (T1−1)CRi i

5CB-αd2 0.99 ± 0.04

1.4 7.1 4.6 6 3 9

4.1 5.5 2.5 8.5 0.57

5.12 ± 0.09

5.21 ± 0.05 2.9 ± 0.5 36 ± 0.5 4.2 ± 0.7 3.5 ± 0.5 25.8 ± 0.3 4.7 ± 0.8 7.3 ± 0.7 ≤7 ± 0.3 5±1

the parameters are slightly different than in the case of nondeuterated 5CB.3 The reason behind this is the reduced number of protons in the deuterated system, resulting in a

where νHi are the cross-relaxation frequencies, whereas ACRi and τCRi are parameters that describe the amplitude and the width of the cross-relaxation dips. Two Lorenzians were modeled in the region between 10 and 40 kHz and an additional one was used to model the behavior below 10 kHz. The model used to analyze the experimental spin−lattice relaxation results considers the sum of the above contributions to T1−1. It is usually assumed that the contributions are statistically independent and/or have distinct characteristic time scales. Therefore, any cross-term contribution to the total relaxation rate is assumed to be neglibible.1,21 For the sake of clarity, the relaxation can be expressed as a sum of a relaxation background T̃ 1−1 and the cross-relaxation contributions in the following form: −1 T1−1 = T1̃ +

(7)

Figure 2. Experimental T1−1 dispersion results for 5CB-αd2 and the best model fitting curves obtained with the relaxation model discussed in the text.

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the nematic director aligns along the magnetic field and that the alignment persists during magnetic field cycling. We consider two molecular motions relevant to the quadrupole nuclei: reorientation around the long molecular axis and the order director fluctuations. Translational self-diffusion does not affect the cross-relaxation because the angles between the bonds and the field remain the same in the time scale of the correlation times associated with the cross-relaxation processes. 2 H has spin I = 1. In the absence of an external magnetic field, the nucleus in general has three quadrupole energy levels with the energies36

different spin density. Furthermore, the protons at the α-position can be viewed as a part of the molecular core. When they are replaced with deuterons, the contribution of the core movements to relaxation becomes smaller than the contribution of the alkyl tail of the molecule. τS therefore becomes shorter whereas no change is expected in τL. Although the analysis was performed at a single temperature, the temperature-dependent model was used, to keep the analysis consistent with the nondeuterated system. Due to different interproton average distances, the factors A±m = |d±m ,0 2(αij)|2 /rij 6 change slightly with respect to the values obtained for the fully protonated 5CB. Comparison of some relevant parameters is given in Table 1. The values of the correlation times τL and τL are compatible with those obtained for 5CB in previous proton and deuteron NMR studies.2,3,31,32 The ratio between the two correlation times is of the same order of magnitude as the aspect ratio of the 5CB molecule, in agreement with the rational diffusion model of elongated molecules in viscous media.33 Larger τS/τL ratios have been reported in deuteron NMR studies of other liquid crystals.34,35 However, it is likely that the discrepancy in the τS/τL ratios is the consequence of using deuterium relaxometry instead of proton relaxometry, because the former provides sitespecific information about the rotational motions including fast internal rotational motions whereas proton relaxometry is, instead, more sensitive to the average behavior of all protons in the molecule. The cross-relaxation mechanism produces two Lorentzianshaped dips with frequencies νH1 = 36 kHz and νH2 = 25.8 kHz. The parameters for the Lorentzian curve below 10 kHz cannot be obtained with high accuracy because the maximum of the relaxation curve appears at frequencies below 5 kHz. Measurements of T1 for frequencies below 5 kHz were not possible because they would require a compensation of all components of the Earth’s and laboratory fields. The region with the dips is shown better in Figure 3, where T̃ 1−1 and the cross-relaxation contributions are plotted separately. NQR Frequencies of Dips. In the analysis of the positions of the quadrupolar dips, we assume that in the high magnetic field

eQVZZ (1 + η) 4 eQVZZ (1 − η) E2 = 4 eQVZZ E3 = − 2 E1 = −

(8)

Here, eQ is the nuclear electric quadrupole moment, VZZ is the largest (by magnitude) principal value of the electric field gradient (EFG) tensor at the position of the nucleus, and η is the asymmetry parameter of the EFG tensor. The EFG tensor is a traceless symmetric second rank tensor composed of the second derivatives of the electrostatic potential V with respect to the coordinates, Vik = ∂2V/∂xi∂xk. Its principal values are labeled as VZZ, VYY, and VXX, where |VZZ| ≥ |VYY| ≥ |VXX|. The asymmetry parameter of the EFG tensor is defined as η = (VXX − VYY)/VZZ. It ranges between 0 and 1. The deuterium nuclear quadrupole resonance (NQR) frequencies are expressed as E1 − E3 eQVZZ = (3 + η) h 4h E − E3 eQVZZ (3 − η) ν− = 2 = h 4h eQVZZ E − E2 ν0 = 1 = η 2h h ν+ =

(9) 2

Here, h is the Planck constant and eQVZZ/h = e qQ/h is called the quadrupole coupling constant. The electric field gradient (EFG) tensor at the deuterium position in a static C−D bond is nearly axially symmetric (η ≈ 0), with the symmetry axis nearly parallel to the C−D bond. Free reorientation of a molecule along the long molecular axis partially averages the EFG tensor at the deuterium position and reduces the quadrupole coupling constant by the factor (3 cos2 ϕ − 1)/2, where ϕ is the angle between the long molecular axis and the C−D bond. The asymmetry parameter η of the timeaveraged EFG tensor is zero, and the symmetry axis is parallel to the long molecular axis. The two deuterons in a 5CB molecule are chemically equivalent and form the same angle ϕ with the long molecular axis; thus eQVZZ/h is equal at both deuterium positions. It should be mentioned that the sign of the quadrupole coupling constant is not observed in NMR and NQR. We can roughly estimate the angle ϕ from the geometry of the molecule. We assume that the carbon atom with two deuterium atoms attached forms chemical bonds in the tetrahedral directions. If the long molecular axis pointed along the long axis of the biphenyl part of the molecule, that would mean that ϕ = 109.4° (or 70.6°) and (3 cos2 ϕ − 1)/2 = −1/3. The reorientation would thus reduce the quadrupole coupling constant to about 1/3 of its original value.

Figure 3. Low-frequency experimental T1−1 results. The black dashed line represents T̃ 1−1, the sum of relaxation mechanisms, R, SD, and ODF (i.e., excluding cross-relaxation effects). The solid black like represents the total model fit. 5603



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(2π) matches the deuterium resonance frequency. This situation is illustrated in Figure 5. The resonant interaction of deuterons and protons (quadrupole dips) occurs at B = B1, when νH1 = νQ + νD, and at B = B2, for νH2 = νQ − νD. The corresponding proton Larmor frequencies are nH1 = vQ /(1 − γD/γH) nH2 = vQ /(1 + γD/γH)

In the present case, νH1 = 36 kHz and νH2 = 25.8 kHz. The ratio γD/γH equals 0.1535. The deuterium NQR frequency νQ, as calculated from νH1, is 30.5 kHz whereas from νH2 we obtain νQ = 29.8 kHz. The difference between the two values is within the experimental resolution. Their average value is 30.1 kHz, giving |⟨eQVZZ/h⟩| = (4/3)νQ = 40.2 kHz. The value of the nematic order parameter, S, 5 K below the I−N transition temperature, as determined using Dvinskikh et al.,37 is 0.52. The quadrupole coupling constant for deuteron site (eQVZZ/h) is about 170 kHz.38 Because we cannot determine the sign of the quadrupole coupling constants, there are two possible solutions to eq 10: (a) if the constants are of the same sign, we obtain ϕ = 37° (or 143°); (b) if the constants are of the opposite signs, we obtain ϕ = 80° (or 100°). Because the axis through the biphenyl part (which would give 70.6°) is expected to be close to the long molecular axis, the value ϕ = 80° is the value consistent with the experimental observations. In fact, it is reasonable to assume that the long molecular axis is not perfectly aligned with the biphenyl axis due to the alkyl “tail” of the molecule. Cross-Relaxation Effect on Relaxation Rates. We treat the influence of the spin−lattice relaxation of the deuterium spin system on the spin−lattice relaxation of the proton spin system in the approximation of infinite temperature. This approximation is justified when (in a field-cycling experiment) the polarizing magnetic field is much higher than the relaxing magnetic field. In our case, the polarizing magnetic field (18 MHz) is over 2 orders of magnitude larger than the relaxing magnetic field in the range of the deuterium quadrupole dips (in the 10 kHz range). Therefore, the proton magnetization in the low magnetic field relaxes toward less than 1/100 of its initial value, so the final proton magnetization can be safely approximated as zero. In the approximation of infinite temperature, the deuteron spin−lattice relaxation is governed by three transition probabilities per unit time: W1,−1, W1,0, and W−1,0 (Figure 6). The transition probabilities per unit time depend on the spectral density J(ν) of the fluctuations of the EFG tensor at different frequencies ν: W1,−1 ∝ J(2νD), W1,0 ∝ J(νQ−νD), W−1,0 ∝ J(νQ+νD). The spin−lattice relaxation rate T1−1 of the coupled proton− deuterium system in the centers of the quadrupole dips can be calculated in the same way as the spin−lattice relaxation rate of the resonantly coupled proton−14N system.39 As the result, we obtain

Figure 4. Deuterium energy levels and eigenstates in zero and nonzero magnetic fields.

Figure 5. Proton and deuteron resonance frequencies with dependence on the magnetic field B. The resonance frequencies match at B = B1 and B = B2.

Figure 6. Transition probabilities per unit time between deuteron energy levels.

ODF makes a further reduction of the eQVZZ/h by a factor S = ⟨3 cos2 θ − 1⟩/2. S is the nematic order parameter and θ is the angle between the instantaneous orientation of the local nematic director and its macroscopic orientation. The time-averaged deuterium quadrupole coupling constant in the nematic phase is thus equal to ⟨eQVZZ /h⟩ = (eQVZZ /h)S(3 cos2 ϕ − 1)/2

(11)

(10)

The asymmetry parameter η of the EFG tensor equals zero in the uniaxial nematic phase. In the absence of an external magnetic field in the nematic phase, a deuterium nucleus exhibits two energy levels: a doubly degenerated energy level with the energy E1,2 = eQVZZ/4 and a nondegenerate energy level with the energy E3 = −eQVZZ/2. The NQR frequency νQ is equal to νQ = |E1,2 − E3|/h = 3|⟨eQVZZ/h⟩|/4. In a nonzero magnetic field B directed along the symmetry axis of the EFG tensor, the doubly degenerated energy level symmetrically splits into two energy levels, whereas the energy of the nondegenerated energy level does not change. The splitting of the upper energy level equals 2νD, where νD = γDB/(2π) is the deuterium Larmor frequency. Both cases are illustrated in Figure 4. In a field cycling experiment, deuterium increases the proton spin−lattice relaxation rate when the proton Larmor frequency νH = γHB/

−1 T1−1(νH1) = (1 − ε)T1̃ (νH1) ⎛ W1, −1W1,0 ⎞ ⎟⎟ + ε⎜⎜W −1,0 + W1, −1 + W1,0 ⎠ ⎝ ν

H1

T1−1(νH2)

−1 ε)T1̃ (νH2)

= (1 − ⎛ W1, −1W −1,0 ⎞ ⎟⎟ + ε⎜⎜W1,0 + W1, −1 + W −1,0 ⎠ ⎝ ν

H2

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Here, T̃ 1−1 is the proton spin−lattice relaxation rate just outside the dip and ε = 2ND/3NH, with ND = 2 being the number of deuterons and NH = 17 the number of protons in a molecule. Thus, ε = 4/51. In the above expression we assumed that the quantities W1,−1, W1,0, W−1,0, and T̃ 1−1 are frequency-dependent (predominantly due to the ODF, because the contributions of rotations/reorientations and SD are frequency-independent in this range), and that their values at νH1 differ from their values at νH2. The transition probabilities per unit time between the deuterium energy levels are dominated by the fluctuations of the EFG tensor produced by molecular reorientations and the ODF. We consider the two processes that occur at different time scales as being independent. The transition probability per unit time between the energy levels m and n can be calculated according to the expression:6 Wmn =

1 ℏ2

∫0

From the experimental points, we can easily estimate the transition probabilities between the deuterium energy levels, if we take an average of the relaxation rates inside the two quadrupole dips and the values in their near vicinity. In our case, we obtain ⟨T̃ 1−1⟩ ≈ 38 s−1 and ⟨T1−1 − T̃ 1−1⟩ ≈ 19 s−1 (Figure 3). From these values, we get W1,0 + W1,0W1,−1 + W1,−1/(W1,0 + W1,−1) ∼ 270 s−1. On the other hand, we can estimate the transition probabilities using eq 17 and eq 18. Due to fast reorientations in the frequency region of the dips, we may take JR(ω) ≈ JR(0) = τR ≈ 7 × 10−10 s. If we further take ϕ ≃ 80° and eQVZZ/h ≃ 170 kHz, we obtain WR±1,0 ≈ 11 s−1 and WR±1,−1 ≈ 210 s−1. Because WR±1,0 ≪ 270 s−1, it is clear that the deuteron spin−lattice relaxation within the quadrupole dips is dominated by the ODF. This result is further confirmed because values −1 WODF and W±1,0 + W±1,0W1,−1/(W±1,0 + W1,−1) ∼ ±1,0 ∼ 140 s −1 280 s are obtained using eq 18 and detailed estimates of the ODF spectral densities (Supporting Information). Low Frequency Behavior. As seen in Figure 1, the relaxation at low Larmor frequencies (under 10 kHz) in the deuterated system is faster than in the nondeuterated one. This difference can also be attributed to the cross-relaxation effect. At low frequencies, the splitting 2νD of the deuterium energy levels decreases within the dipolarly broadened proton NMR line. The coupling of deuterons to the proton dipolar reservoir can be observed when the proton dipolar reservoir strongly interacts with the proton Zeeman reservoir, i.e., when the external low magnetic field in the field cycling experiment is comparable to the local dipolar magnetic fields. In this case, the thermodynamic order of the spin system is described by a single spin temperature. The density matrix ρ of the proton spin system can be, in the high-temperature approximation, written as

∞

⟨(δHQ (0))mn (δHQ (− t ))mn exp(iωmnt ) + cc⟩ dt (13) 36

where HQ(t) is expressed as δHQ (t ) =

eQ 6I(2I − 1)

⎡3 2 ⎤ (IkIl + IlIk) − I ⃗ δkl ⎥ ⎦ 2

∑ δVkl(t )⎢⎣ kl

(14)

If we further assume that the autocorrelation function ⟨δVkl(0) δVkl(−t)⟩ can be expressed as ⟨δVkl(0) δkl( −t )⟩ = ⟨|δkl − tkl|2 ⟩g (t )

(15)

and define the spectral density J(ω) of the fluctuations as

∫0

J(ω) =

∞

g (t ) cos(ωt ) dt

ρ = (1 − β(HZ + HD))/Tr(1)

(16)

Here, HZ and HD are the Zeeman and the dipolar Hamiltonian terms, respectively, and β = 1/kBT is the inverse spin temperature. The energy EH of the proton spin system is

we obtain the contribution of the molecular reorientations to the deuterium spin−lattice relaxation as W∓R1,0 =

2 9π 2 ⎛ eQVZZ ⎞ 2 2 ⎜ ⎟ sin ϕ cos ϕJ (ω Q ± ωD) R 4 ⎝ h ⎠

W1,R−1 =

9π ⎛ eQVZZ ⎞ 4 ⎜ ⎟ sin ϕJ (2ω ) D R 8 ⎝ h ⎠ 2

(19)

E H = Tr((HZ + HD)ρ) = −

2

NH 2 2 h (νH + νloc 2)β 4

(20)

νloc is the proton local frequency, defined as (17)

νloc 2 =

Here, JR(ω) is the spectral density of the reorientations. For the ODF, which are slow as compared to the reorientations, we obtain W ∓ODF 1,0 =

2 2 9π 2 ⎛ eQVZZ ⎞ ⎜⎛ 3 1⎞ ⎜ ⎟ cos2 ϕ − ⎟ ⟨sin 2 θ cos2 θ⟩JODF (ωQ ± ωD) 4 ⎝ h ⎠ ⎝2 2⎠

W1,ODF −1 =

2 2 9π 2 ⎛ eQVZZ ⎞ ⎜⎛ 3 1⎞ ⎜ ⎟ cos2 ϕ − ⎟ ⟨sin 4 θ⟩JODF (2ωD) 8 ⎝ h ⎠ ⎝2 2⎠

4 TrHD2 NHh2

(21)

The proton spin−lattice relaxation increases the energy of the proton spin system to the value corresponding to β = βL= (kBTL)−1, where TL is the temperature of the sample. In the approximation of infinite temperature, the energy of the proton spin system relaxes toward zero. The time dependence of β is, in the same approximation, described by a simple rate equation dβ −1 = −T1̃ β (22) dt −1 Here, T̃ 1 is the pure proton spin−lattice relaxation rate, as defined in eq 7. The time derivative (dEH/dt)SL of the energy of the proton spin system due to the proton spin−lattice (SL) relaxation is

(18)

Here, JODF(ω) is the spectral density of the order director fluctuations. The experimentally observed transition probabilities per unit time are the sum of the ODF and reorientation 2 2 R contributions: Wm,n = Wm,n + WODF m,n . The averages ⟨sin θ cos θ⟩ and ⟨sin4 θ⟩ can be expressed in terms of the nematic order parameter and of ⟨P4⟩ and estimated using polynomial expansions calculated with a Mayer−Saupe orientational potential for uniaxial phases.40 The magnitudes of the transition probabilities can be estimated using eq 12, if we assume that W1,0 ≈ W−1,0. The value of the deuterium term W1,0 + W1,0W1,−1/(W1,0 + W1,−1) is then between W1,0 (for W1,−1 → 0) and 2W1,0 (for W1,−1 → ∞).

⎛ dE H ⎞ N −1 ⎜ ⎟ = H h2(νH 2 + νloc 2)T1̃ β ⎝ dt ⎠SL 4

(23)

We further assume that the proton dipolar system strongly interacts with deuterons at the frequency 2νD. In this case the ratio of the population of the two deuteron energy levels 5605



Article

5CB-αd2 molecule produces some changes in both translational self-diffusion and local rotations relaxation contributions. The major difference between the T1−1 dispersions of the two 5CB systems is observed below 40 kHz, where the characteristic crossrelaxation cusps (quadrupolar dips in the T1 profiles) are observed. These cross-relaxation contributions were fitted using Lorentzian curves, and from the values of the cross-relaxation frequencies and amplitudes, it was possible to analyze in detail the quadrupolar relaxation effects in the proton spin−lattice relaxation. We determined the angle between the long molecular axis and the C−D bond to be ∼80°. The analysis of the quadrupolar dips confirms that order director fluctuations are the dominant relaxation contribution in the low frequency domain. The differences between the T1−1 dispersion profiles of 5CB and 5CB-αd2 below 10 kHz were assigned to an additional relaxation mechanism that originates from the coupling between the proton Zeeman and the proton dipolar reservoirs through the deuterium dipolar reservoir.

separated by the frequency 2νD is Boltzmann with the inverse spin temperature β: N−1/N1 = 1−2hνDβ. The energy of the part of deuterium system that is coupled to protons (spins at |−1⟩ and |+1⟩) is ED = −

2ND 2 2 h νD β 3

(24)

The energy gain per unit time (dED/dt)SL of the part of the deuterium spin system that interacts with the proton spin system due to the deuterium spin−lattice relaxation is ⎛ dE D ⎞ 4ND 2 ⎜ ⎟ = 2hνDWQ (N1 − N −1) = h νDWQ β ⎝ dt ⎠SL 3

(25)

Here WQ = W1, −1 +

W1,0W −1,0 W1,0 + W −1,0

≅ W1, −1 +

1 W1,0 2

(26)

The time derivative of the energy of the coupled proton− deuteron system, d(EH + ED)/dt is ⎛ dE ⎞ d(E H + E D) ⎛ dE H ⎞ ⎟ + ⎜ D⎟ =⎜ ⎝ dt ⎠SL ⎝ dt ⎠SL dt

■

S Supporting Information *

(27)

Synthesis of 5CB-αd2 and detailed calculation of transition probabilities. This material is available free of charge via the Internet at http://pubs.acs.org.

From this expression, we derive the equation for β, dβ/dt = −T1−1β. Here, T1−1

=

■

−1 (νH 2 + νloc 2)T1̃ + 8ενD2WQ 2

2

νH + νloc + 4ενD

2

−1

⎛γ ⎞ ν 2 + 8ε⎜⎜ D ⎟⎟ WQ 2 H νH + νloc 2 ⎝ γH ⎠

AUTHOR INFORMATION

Corresponding Authors

*A. Gradišek: e-mail, [email protected]. *P. J. Sebastião: e-mail, [email protected].

2

≅ T1̃

ASSOCIATED CONTENT

(28)

Notes

The authors declare no competing financial interest.

This expression holds when νH is equal to a few νloc or lower. At higher values of νH, the proton Zeeman reservoir decouples from the proton dipolar reservoir and the relaxation of the proton dipolar reservoir caused by deuterons is no more observed on the proton Zeeman reservoir, which is in fact observed in a fieldcycling experiment. To include the decoupling between the two spin reservoirs for νH > νloc, a Lorentzian expression (eq 5) was also used to fit the experimental spin−lattice relaxation in the low frequency range with a characteristic frequency close to the νloc. Although the low asymptotic limit of νH2/(νH2 + νloc2) differs from that of eq 5, this difference is not relevant for the fit because T1−1 was measured only for values above 5 kHz. The value of WQ (basically W1,−1), obtained using eq 28 and the value of T1−1 at νH0, is estimated to be in the range 2000 s−1 to 5500 s−1. The values in this range are larger that those that can be estimated using eq 26 but are compatible with the expected increase of the ODF contribution at frequencies lower than νH1 and νH2. A more detailed estimate would require the measure of the spin−lattice relaxation rate at lower frequencies which was not possible with our fast field-cycling relaxometer.

■

ACKNOWLEDGMENTS This research was supported by the Portuguese Science and Technology Foundation (FCT) through contracts PTDC/ CTM-POL/1484/2012 and PEst-C/CTM/LA0025/2013- 14. S. N. Fernandes acknowledges FCT for grant SFRH/BPD/ 78430/2011. The authors also thank the Slovenian Research Agency (ARRS) for support through a series of bilateral Slovenian-Portuguese projects.

■

REFERENCES

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■

CONCLUSIONS The proton spin−lattice relaxation dispersion obtained for the 5cyanobiphenyl (5CB) liquid crystal with molecules deuterated at the α position presents clear differences with respect to the relaxation dispersion reported for the fully protonated 5CB counterpart. In the intermediate and high frequency domains the spin−lattice relaxation rate is very similar to that obtained for the fully protonated 5CB. The differences in these frequency regions are related to the fact that the distribution of proton spins in the 5606



Article

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