Slow dynamics in a liquid crystal: 1H and 19F NMR relaxometry
M. Rajeswari, Trivikram R. Molugu, Surajit Dhara, V. S. S. Sastry, K. Venu et al.
Citation: J. Chem. Phys. 135, 244507 (2011); doi: 10.1063/1.3672180
View online: http://dx.doi.org/10.1063/1.3672180
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i24
Published by the American Institute of Physics.

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THE JOURNAL OF CHEMICAL PHYSICS 135, 244507 (2011)

Slow dynamics in a liquid crystal: 1 H and 19 F NMR relaxometry
M. Rajeswari,1,a) Trivikram R. Molugu,1,2 Surajit Dhara,1 V. S. S. Sastry,1 K. Venu,3and R. Dabrowski4
1

School of Physics, University of Hyderabad, Hyderabad 500 046, India
Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA
3
Soctronics Technologies Pvt. Ltd, Banjara Hills, Hyderabad 500 034, India
4
Institute of Chemistry, Military University of Technology, 00-908 Warsaw, Poland
2(Received 11 February 2011; accepted 5 December 2011; published online 29 December 2011)
Spin-lattice relaxation rates (R1H and R1F ) of two nuclear species (1 H and 19 F) are measured at different temperatures in the isotropic phase of a liquid crystal (4 -butoxy-3 -fluoro-4isothiocyanatotolane–4OFTOL), over a wide range of Larmor frequency (10 kHz–50 MHz). Theirdispersion profiles are found to be qualitatively very different, and the R1F in particular shows significant dispersion (varying over two orders of magnitude) in the entire isotropic range, unlike R1H .
The proton spin-lattice relaxation, as has been established, is mediated by time modulation of magnetic dipolar interactions with other protons (case of like spins), and the discernable dispersion in
the mid-frequency range, observed as the isotropic to nematic transition is approached on cooling,is indicative of the critical slowing of the time fluctuations of the nematic order. Significant dispersion seen in the R1F extending to very low frequencies suggests a distinctly different relaxation
path which is exclusively sensitive to the ultra slow modes apparently present in the system. We
find that under the conditions of our experiment at low Zeeman fields, spin-rotation coupling of thefluorine with the molecular angular momentum is the dominant mechanism, and the observed dispersion is thus attributed to the presence of slow torques experienced by the molecules, arising clearly
from collective modes. Following the arguments advanced to explain similar slow processes inferredfrom earlier detailed ESR measurements in liquid crystals, we propose that slowly relaxing local
structures representing such dynamic processes could be the likely underlying mechanism providing the necessary slow molecular angular momentum correlations to manifest as the observed low
frequency dispersion. We also find that the effects of the onset of cross-relaxation between the twonuclear species when their resonance lines start overlapping at very low Larmor frequencies (below∼ 400 kHz), provide an additional relaxation contribution. © 2011 American Institute of Physics.
[doi:10.1063/1.3672180]
I. INTRODUCTION

The laterally fluorinated liquid crystal compounds haveattracted recent attention because of their potential applications in display technology and telecommunications.1–9 The
replacement of hydrogen with the fluorine on the aromaticcore reduces the melting point and smectogenity leading to awide nematic range. Mixtures of these singly or doubly fluorinated compounds on the aromatic cores have high figureof-merit, low viscosity, and high birefringence, which are thefavorable physical parameters for liquid crystal display devices. Detailed dielectric studies were carried out on singly
fluorinated compounds to examine the correlation times ofreorientations.6 Our present interest is to examine possible
differences in the relaxation mechanisms of 1 H and 19 F residing on the molecule of a singly fluorinated liquid crystal (bothnuclei being spin- 12 , with comparable gyromagneteic ratios
differing by about 6%) using NMR relaxometry. This has thepotential to yield perhaps different perspectives of dynamic
processes of the same molecule as reported by the two distinct
probing nuclei. With this objective we chose a member of thea) Electronic mail: raji.hcu@gmail.com.

0021-9606/2011/135(24)/244507/9/$30.00

tolane family, 4 -butoxy-3 -fluoro-4-isothiocyanatotolane–
4OFTOL, with the molecular structure and phase sequence
as indicated in Fig. 1. This family of liquid crystals was earlier investigated with detailed dielectric studies to infer on themolecular reorientational dynamics and its possible correlation to the structure.6 We present the frequency dispersion of
spin-lattice relaxation rates of the nuclei (R1H and R1F , of proton and fluorine, respectively) at different temperatures in theisotropic phase of this system, and attempt to interpret themin terms of the underlying molecular processes. Analysis of
R1H data indicates that slowest processes aiding its dispersion are the pretransitional effects arising from the nematicorder fluctuations, affecting the time correlations of the reorientational dynamics of the molecules. The dramatic enhancement of R1F at lower frequencies, and its presence even at
the highest of the temperatures in the isotropic phase where
the critical effects are not so evident in R1H dispersion, suggests the presence of another type of slow molecular processes in the system (not accessed by protons) and sensed
by fluorine apparently by a qualitatively different relaxation
path. Among the possible interactions relevant to this nucleus,
we argue that the fluorine spin-rotation interaction (its coupling constant being significant relative to proton) is primarily

135, 244507-1© 2011 American Institute of Physics

Rajeswari et al.

244507-2

J. Chem. Phys. 135, 244507 (2011)

4'-butoxy-3'-fluoro-4-isothiocyantotolane (4OFTOL)
F

C4H9O

C

C

NCS

Cr 64.2oC N 71.6oC Iso
4'-butoxy-2-fluoro-4-isothiocyantotolane (4OTOLFm)
F

C4H9O

C

C

NCS

Cr 70.3oC N 94.4oC Iso
FIG. 1. Molecular structure and phase sequence of liquid crystal 4OFTOL(molecular structure of 4OTOLFm is also given for comparison).

mediating this relaxation and thus helping fluorine to reporton time correlations of molecular angular momentum components. This then suggests that there are very slow components
in such time correlations, pointing to correspondingly slow
local torques experienced by the reorienting molecules. We
attempt to connect this finding to the earlier observations ofslowly relaxing local structures (SRLS), also referred to as
cages, based on detailed ESR measurements, and suggest its
applicability to possibly account for the present experimental
observations.
A subtle change takes place as far as the dipolar interaction between the lone fluorine and the neighbouring protons is concerned, at the very low frequency end of thisfield-cycling experiment. In NMR terms10 the H-F dipolarinteraction, which is of hetero-nuclear nature (case of unlike spins) at high fields (weak-coupling limit), acquires ahomo-nuclear character (case of like spins) at low enoughfields (strong-coupling limit). Here, the resonance lines of thetwo species (despite the differences in gyromagnetic ratios)
start overlapping, promoting additional mechanism (calledcross-relaxation T21 process11 ). In the present sample, wefind that this facilitates additional effective coupling of fluorine to the lattice via the proton spin reservoir, below about
400 kHz, and its efficacy increases as the lines progressively
overlap.
The paper is organized as follows. In Sec. II we describe
the experimental details and present data on the proton andfluorine spin-lattice relaxation rates in this system. In Sec. III,
we discuss the relaxation mechanisms appropriate to interpretthe observations. In Sec. IV, R1H and R1F dispersions at different temperatures are analyzed based on these models, leading to an estimation of the relevant molecular dynamic parameters. The last section summarizes the conclusions of thisstudy.

II. EXPERIMENTAL DETAILSThe sample used in this study was synthesized in the laboratories of the Warsaw group, and the method of synthesis
and characterization of some of the properties are discussed
earlier.12, 13 In our work we sealed the compound in a standard NMR tube after removing the dissolved oxygen throughfreeze-pump-thaw cycles. Frequency dispersions of protonand fluorine spin-lattice relaxation rates, R1H and R1F , aremeasured at four different temperatures in the isotropic phase,over a wide frequency range, with a commercial fast fieldcycling NMR relaxometer (Stelar, Italy) to cover 10 kHz–
10 MHz, supplemented by home-built variable-field pulsed
NMR spectrometer for higher end frequencies (10–50 MHz).
In all the cases, we find that the magnetization recoveries
could be fitted to a simple exponential over at least three
decades of time, permitting the determination of a unique spin
relaxation rate associated with the specific spin magnetization. This corresponds to decoupled evolution of proton andfluorine Zeeman reservoirs.10 The frequency dependence ofthe relaxation rates of both the nuclei are shown in Figs. 2 and3. These rates are also measured as a function of temperature
at chosen Larmor frequencies ν L (ω = 2π ν L ) for both the nuclei and the results are shown in Figs. 4 and 5. Temperature of
the sample was maintained by passing dry air over the sample to an accuracy of ±0.1 ◦ C. Relative errors in the relaxation
rate measurements of proton are within 2% and are within 5%
for fluorine. The frequency dispersion data of proton clearlyshow the onset of critical fluctuations even at 11 ◦ C away fromthe isotropic to nematic transition temperature (TNI ). The temperature variation of proton data (Fig. 4) further confirms thecontribution of critical fluctuations near, and as well as far
away, from the transition. In comparison, the relaxation rate
dispersions of fluorine are however qualitatively very different. The R1F values show a steady and very significant enhancement at low frequency regions, and this persists at all thetemperatures in the isotropic phase (Fig. 3). The temperature
variation of R1F (Fig. 5) is also qualitatively different from the

FIG. 2. Proton spin-lattice relaxation rate at different temperatures as a function of Larmor frequency.

244507-3

Slow molecular processes in liquid crystals

FIG. 3. Fluorine spin-lattice relaxation rate at different temperatures as afunction of Larmor frequency.

behavior of proton relaxation rates. These observations point
to possible differences in the relaxation pathways of the twonuclear species, even though they reside on the same liquid
crystal molecule.
III. RELAXATION MECHANISMS

Spin-lattice relaxation of protons in these systems
occurs typically via the time modulation of dipolarinteractions10, 14, 15 and the relevant dynamic mechanisms thatare of interest to the NMR experiment are: individual molecular reorientations modulating intra-molecular dipolar couplings; translational diffusion effecting the inter-molecular
dipolar interactions; collective dynamics arising from critical fluctuations of the nematic order; and possibly a constantbackground contribution from such other processes which are

FIG. 4. Proton spin-lattice relaxation rates plotted against temperature atdifferent Larmor frequencies. The dotted line in the figure indicates theisotropic-nematic transition temperature.

J. Chem. Phys. 135, 244507 (2011)

FIG. 5. Fluorine spin-lattice relaxation rates plotted against temperature atdifferent Larmor frequencies. The dotted line in the figure indicates theisotropic-nematic transition temperature.

too fast for the experiment. We find from the analysis of
the proton data that the translational diffusion contribution
is practically constant over the frequency range, and hencecannot be distinguished from the background. The dispersion
contribution to the data thus is primarily attributed as due tomolecular reorientations (R) about the molecular short axes
and to critical fluctuations (CF). The experimental value ofthe relaxation rate is assumed to be a sum of all such contributions, neglecting the possible statistical correlations among
them.
Analysis of the proton dispersion data is carried out by
separating the contribution from different mechanisms as
R1H = R1H0 + R1HR + R1HCF .

(1)

Here R1H0 represents a frequency independent contribution from all processes too fast on the time scale of the experiment. Dynamic models accounting for R1HR from reorientational process10, 16 and for R1HCF from order parameter criticalfluctuations are well documented earlier,14, 17 and lead to thedetermination of corresponding correlation times (τR and τCF )and their couplings to the spin system (AR and ACF ).
Analysis of the fluorine data (R1F ), qualitatively different
from proton, requires additional considerations. Apart fromthe dipolar coupling with protons (hetero-nuclear) and withfluorine nuclei on other molecules (homo-nuclear), chemical shift anisotropy (CSA), and spin-rotation interaction (SR)of fluorine with the molecular angular momentum need tobe considered. While CSA, being field-dependent, is significant in relaxing fluorine in normally applicable fields,
the present experiments focusing on very low fields makethis interaction ineffective. Thus, in the case of fluorine wefind that intra-molecular dipolar interaction via hetero-nuclear
coupling with proton, and spin-rotation interaction involving
molecular angular momentum time modulations, are more
appropriate.18–21 The contributions to R1F from the modulation of the dipolar interaction are represented by R1FR (from244507-4

Rajeswari et al.

R mechanism) and R1FCF (from CF mechanism), while theconstant background is included in R1F0 . It may be noted
that, considering the very similar nature of proton and fluorine (both spin- 12 and very nearly the same gyromagnetic
ratio), the order of magnitude of the contributions from thismechanism to the R1F data as well as the analytical dependence on frequency are expected to be comparable to the caseof proton (but for a caveat mentioned below), and hence are
not expected to be able to account for the steep increase (byabout 150 times) in the fluorine relaxation rate in the low frequency region. Thus the problem of interpretation of rather
simple looking, monotonically increasing relaxation rate offluorine, all the way to kHz region, is one of finding suitable low frequency phenomena whose combined effect is tosustain this steady increase. We find that a single mechanism
cannot reproduce this data, and our analysis indicates the needto include several qualitatively different processes to accountfor this steadily increasing strong coupling of fluorine to thelattice.
The caveat referred to earlier is concerned with theconsequences of hetero-nuclear dipolar coupling on the
spin-lattice relaxation of individual spin reservoirs.10 In
principle there should be a coupling between the two Zeeman
systems and the observed magnetization recovery could
be multi-exponential due to the coupling.10 However theexperimental observation of a single exponential decay of themagnetization rules out this coupling. But, in the presenceof hetero-nuclear dipolar interactions the spectral density J(0)at the difference-frequency can also in principle contributeto the individual spin-lattice relaxation rate according to
the following equation,10 thus leading to a low frequency(ωF − ωH ) component


1 (0)
3
3
FH
J (ωF −ωH )+ J (1) (ωF )+ J (2) (ωF +ωH ) .
= AFH
R1F
12
2
4
From an analysis based on this modified equation forthe dipolar relaxation rate including a J(0) contribution at low
frequencies, we find that it is not adequate to account for the
steep increase of the relaxation rate at low frequencies.
The second mechanism which is specifically effective forfluorine in relation to proton, is the coupling of this nuclear
spin to the (transient) fields generated by molecular rotations(spin-rotation interaction–SR), and is expressed as18
HSR = −¯S.C.J.
Here J is the angular momentum of the molecule, C is
the spin-rotation interaction tensor, and S is the nuclear spin
under consideration. Considering the case of a quantized spinin the NMR experiment, and fast molecular reorientations in
the isotropic phase, the tensorial character of the spin-rotation
constant can be approximated for simplicity to a constant, asis the case for example for a spherically symmetric molecule.
The spin-lattice relaxation rate of fluorine is then expressible
as19
τSR
R1FSR = ASR T,(2)
2
1 + ω2 τSR
ASR = kB ¯2 IM 2C⊥2 + C||2 .

J. Chem. Phys. 135, 244507 (2011)

Here, IM is the moment of inertia of the molecule, C||
and C⊥ are the elements of the spin-rotation interaction tensor of the fluorine nucleus in the principal axes system. AFSR
represents the effective coupling of the nucleus under consideration to the SR process, and is dependent on the strength ofthe coupling tensor. Proton dispersion data are not sensitive tothis mechanism because of its relatively weak coupling to themolecular angular momentum.20 For typical molecules constituting simple liquids, τSR , the correlation time associatedwith the angular momentum fluctuations of the molecule, isin the range of 10−12 to 10−13 s (too fast on NMR experimental time scale), and hence for such processes no dispersion is
to be expected (Eq. (2)). The temperature dependence of thepresent data at low frequencies (Fig. 5) indicates the signatureof SR (increase of relaxation rate with increase of temperature). Observed sensitivity of this data to temperature at low
enough Larmor frequencies and the dramatic enhancement of
the rate at such low frequencies even at elevated temperatures(Fig. 3), together with the strong pointer to SR as perhaps the
only effective mechanism in this case, motivate further discussion on possible sources of slow modulations affecting the
SR mechanism.
Earlier investigations on the line shapes of ESR spectra22
showed that the simple rotational dynamic models were notadequate in explaining the observed line width variations withdifferent spin probes.23 The time domain 2D ESR methods
also confirm this observation.24 These ESR observations wereexplained in terms of slowly relaxing local structures within
the liquid crystal medium, modulating the rotational diffusionof the ESR probes. Thus the dynamics corresponds to a rotational diffusion in the presence of an orienting potential ofmean torque, which itself is undergoing a simple rotational
diffusion on much slower time scale. This is viewed as a cage
inducing local ordering, tumbling isotropically on a well separated time scale (relative to ESR time scales).23–25 It was also
seen from the ESR work that the coupling of the ESR probe tosuch slow processes is dependent on the size and anisotropyof the probing molecule–bigger the size of the ESR probe, thedramatic were the slow motional effects on the spectral features. In the present case the fluorine bearing liquid crystal
molecule itself is a (nuclear) probe, thus the coupling to such
SRLS mechanism is expected to be at its maximum. We thus
are led to explore this dynamic process as a possible sourceof the observed strong coupling of the fluorines to the lattice
via the SR interaction.
There is a quantitative difference, however, in adapting this model to the present case. The relaxation of the
ESR probe (typically a free radical of a chosen moleculargeometry) is sensitive to the reorientational dynamics ofthe spin bearing molecule, and thus reports on its timecorrelations. Within the SRLS model, interpretation of the
ESR observations then relates to evaluating the effects ofthe corresponding stochastic processes, accounted for at thelevel of Smoluchowski equation, coarse graining the angularmomentum variables. And, the assumed isotropic diffusionof the local structures represents a simplified representationof these modes under Brownian approximation. In the present
case, involving a nuclear spin sensitive to the (random)time variations of local magnetic fields generated due to the244507-5

Slow molecular processes in liquid crystals

J. Chem. Phys. 135, 244507 (2011)

(a)

(b)

(c)

(d)

FIG. 6. Proton spin-lattice relaxation rate dispersions fitted to Eq. (1) at different temperatures: (a) TNI = 0.5 ◦ C, (b) TNI = 1 ◦ C, (c) TNI = 4 ◦ C, and(d) TNI = 11 ◦ C.The solid line represents the fit of Eq. (1) to the experimental data. The color dotted lines indicate the contributions of different relaxation
mechanisms. The values of the fitted parameters are given in Table I.

corresponding reorientational dynamics of the spin-bearing
probe molecule, it is the time correlations of these fields that
influence the nuclear spin-lattice relaxation. Consequently
the angular momentum correlation times that appear in theexpression for the nuclear spin relaxation rate18 are in effectconnected, in the present system, in a more subtle mannerto these underlying collective motions. While a quantitative
analysis of such effects requiring Fokker-Planck-Kramer
level description does not seem feasible at this time, it is still
useful to consider the applicability of that model in qualitative
terms to interpret the present work. We consider thus thecorrelation time (say, τC ) derivable from the low frequency
dispersion of fluorine relaxation rates with a Lorentzian fit,
as indicative of the time scale of slow modulations of thespin-rotation interaction, and propose that the underlying
dynamics is associated with the SRLS mechanism. We thus
analyze the corresponding data based on this simplified
model, by representing its contribution as
R1FC = AFCτC.
1 + ω2 τC2(3)Here AFC is a (temperature dependent) measure of thecoupling of the fluorine to these slow modes.
We interpret the experimental data on fluorine by expressing the observed relaxation rate R1F as
R1F = R1F0 + R1FR + R1FCF + R1FC .

(4)

The first term comprises of contributions from all fast
processes, including contribution from the SR mechanism un-

der fast angular momentum relaxations (Eq. (2), with ωτSR
 1).

IV. ANALYSIS AND DISCUSSION

The proton dispersion data are fitted to Eq. (1) at the four
temperatures, as shown in Fig. 6. Each dispersion is decomposed into a frequency-independent contribution (from fast
processes, like molecular reorientations around the long-axes
and alkyl chains). We also find from our analysis that theinter-molecular dipolar interaction modulated by the translational diffusion is also relatively fast on the experimental timescale, and hence leads to a constant contribution along withthe other such processes. This leaves the possible dispersion
mechanisms as: reorientations around the short-axes andcritical fluctuations of the order parameter of the transition.
The model parameters are determined based on a nonlinear
least squares procedure26 and are summarized in Table I. The
experimental data as well as the analysis (Table I) clearlyshow that correlation times of critical fluctuations of thenematic order (τCF ) are significantly high near the transition(Fig. 6(a), TNI = 0.5 ◦ C, τCF ≈ 363 ns), and persist evenat elevated temperatures in the isotropic phase (Fig. 6(d),TNI = 11 ◦ C, τCF ≈ 35 ns)—somewhat unusual in typicalliquid crystals. For comparison, the correlation times obtained from this technique for other liquid crystals near theircorresponding transition temperatures are much lower: e.g.,in 4 -butoxy-2-fluoro-4-isothiocyantotolane (4OTOLFm,
Fig. 1) and 4 -octyloxy-4-cyanobiphenyl (8OCB) these are
68 ns and 180 ns, respectively. Further, their dispersion

244507-6

Rajeswari et al.

J. Chem. Phys. 135, 244507 (2011)

TABLE I. Optimized values of the parameters extracted by fitting the 1 H NMR spin-lattice relaxation rate
dispersions in the isotropic phase at different temperatures to Eq. (1).
−3

TNI (◦ C)

R1H0 (s−1 )A1HR (109 s−2 )τR (ns)AHCF (103 s 2 )τCF (ns)

0.5
2
4
11

0.61 ± 0.01
0.68 ± 0.01
0.7 ± 0.008
0.9 ± 0.006

0.79 ± 0.024
0.87 ± 0.03
0.79 ± 0.01
0.54 ± 0.16

3.96 ± 0.07
3.51 ± 0.07
3.49 ± 0.07
2.38 ± 0.25

18.8 ± 4.4
18.1 ± 6.1
17 ± 2.5
12.9 ± 3.0

362.9 ± 10.7
203.0 ± 7.4
110.7 ± 1.8
35.1 ± 6.3

effects on the relaxation rate are not seen to persist beyond
5–6 ◦ C above the transition.27, 28 Dielectric studies on
4OFTOL (Ref. 6) suggest that the transverse dipole momentof this molecule is not completely averaged out due tohindering of the wing rotation about the para axes of themolecules, arising from the location of fluorine atom on thecore (Fig. 1). In 4OTOLFm on the other hand fluorine islocated in the meta position on the aromatic core and theaveraging was found to be more effective. We note that thisobserved partial averaging of the molecular transverse dipole
moment in 4OFTOL reported from the dielectric studies, andits relatively longer life times of the nematic clusters inferredfrom the present work are perhaps correlated. The remnant
transverse dipoles on the molecules of 4OFTOL could be
providing additional mechanism in promoting the formationof the nematic clusters during the pretransitional behavior,reflected in their enhanced life times. It may further be notedthat this difference between the structures of 4OFTOL and
4OTOLFm also leads to a difference in their order parametervalues.6 The value of τ R is found to vary from about 4.0 ns to
2.4 ns as the sample is heated above the transition temperature(Table. I). In comparison its variation over a corresponding
range of temperature in a related system, 4OTOLFm, isfrom about 1.7–1.4 ns,27 suggesting a higher activationenergy for the rotational dynamics in the present system.
The temperature dependence of the relaxation rate at low
frequencies (Fig. 4) is consistent with the above observations.
While analyzing the more pronounced fluorine dispersion based on Eq. (4), we use the correlation times (τR andτCF ) as inputs obtained from the proton results. An attempt
to fit the data based on the processes included in Eq. (4),with relevant correlation times consistent with the proton
dispersions, immediately suggested that a distinction need
to be made between the contribution from mechanisms that
operate via the dynamic processes included in Eq. (4) andan extra path to the lattice created at very low frequencies(below ∼ 400 kHz) by the onset of overlap of the resonancelines of the two nuclear species. In terms of the nuclear spin
Hamiltonian, this corresponds to gradual change from weakcoupling (WC) regime (i.e., when the dipolar interaction
strength between the fluorine and proton spins is muchsmaller than the difference in their resonance frequencies)
to strong coupling (SC) regime.10 In the latter limit, theenergy-conserving mutual spin flip-flop terms of the twonuclear spins in the dipolar Hamiltonian become significant,providing an additional path for the equilibration of the fluorine system. We therefore proceed with the analysis of thisdata in two steps. We first attempt to fit the fluorine data in
the WC regime (with Eq. (4)) utilizing of course the dynamic

information from the proton results. The relaxation rates
computed based on these best fit parameters, extrapolated
to the SC regime (very low frequency region) then correspond to predicted values based on these proposed dynamic
processes. The observed differences between these values
and the experimental data (below ∼ 400 kHz) are due to theadditional mechanism of the cross-relaxation arising fromthe overlap of the two resonance lines.11 The second step ofanalysis thus involves interpretation of these deviations basedon considerations of overlap of the resonance lines.
Analyzing the fluorine data in the WC regime, we findthat R and CF mechanisms are not adequate, particularly evident considering the strong dispersion even at TNI = 11 ◦ C
where the CF mechanism is at its weakest in effecting a dispersion as per the proton data. Observed strong dispersion
extending monotonically into the low frequency regime thus
requires invoking a slow enough process modulating a qualitatively different spin-lattice coupling, different from that ofthe protons. It is known that the spin-rotation coupling constant of fluorine is much stronger (typically 5 times) than thatof proton,20 and the relaxation rate varies proportional to its
square. This mechanism is qualitatively different in its coupling to the dynamic processes, in that it connected to correlations of molecular angular momentum components, rather
than of the reorientations. The experimental evidence, in conjunction with the analysis of the proton data, thus indicatesthat the observed strong fluorine-lattice coupling is perhaps
due to a very effective spin-rotation interaction (relative toprotons), and the observed dispersion extending to the low
frequencies should thus be due to the presence of very slow
time correlations of the angular momentum correlations. It isin this context that the ESR results seem to provide a clue tothe origin of slow torques needed to effect such slow angularmomentum dynamics. We interpret the results now in termsof Eq. (4), which incorporates additional slow mechanism
which is expected to reflect the effect of the SRLS dynamics(isotropic Brownian rotational diffusion of local structures).
We present the best fit strengths of the fluorine spin systemwith the lattice from different mechanisms, i.e., AFR (shortaxes reorientations), AFCF (order parameter critical fluctuations), AFC (effect of slowly relaxing local structures), alongwith τC (time scales of slow torques arising from the underlying slow modes) in Table II. We observe that τC is in therange of 260–300 ns, and is practically unaffected by temperature within the errors of estimation. We also find that the corresponding coupling strength (AFC in the Table II) increases
as the system is heated into the isotropic phase. The contributions AFR and AFCF represent the coupling of the fluorine
system to the lattice via the hetero-nuclear intra-molecular

244507-7

Slow molecular processes in liquid crystals

J. Chem. Phys. 135, 244507 (2011)

TABLE II. Optimized values of the parameters extracted by fitting the 19 F NMR spin-lattice relaxation rate
dispersions in the isotropic phase at different temperatures to Eq. (4).
−3

TNI (◦ C)

A1FR (109 s−2 )τR (ns)AFCF (103 s 2 )τCF (ns)

AFC (108 s−2 )τC (ns)

0.5
2
4
11

0.41 ± 0.11
0.45 ± 0.01
0.62 ± 0.01
0.41 ± 0.20

3.96
3.51
3.49
2.38

14.2 ± 1.9
14.4 ± 0.13
15.0 ± 0.14
16.4 ± 0.36

362.8
203.0
110.7
35.1

0.086 ± 0.009
0.113 ± 0.006
0.091 ± 0.006
0.22 ± 0.01

292.2 ± 61.9
261.8 ± 29.1
260.1 ± 33.5
273.0 ± 26.5

dipolar interaction with protons, arising due to the different
time modulations. We assume that the inter-molecular homoor hetero-nuclear interactions involving fluorine (modulatedby translational diffusion) are negligible, and if any would be
essentially frequency independent in keeping with the observations based on the proton data. As may be seen from Figs. 6
and 7, the relaxation rates of proton and fluorine differ bymore than an order of magnitude at low frequencies. The constant background term (R1F0 ) was found to be relatively verysmall, with comparable errors, and hence their values are set
to zero (not quoted in Table II). The case of protons is different, since they have significant homo-nuclear inter-molecularinteractions and hence could lead to discernible (though) constant contribution to their relaxation rate.
The analysis of the data in the SC limit (typicallybelow 400 kHz in the present sample) qualitatively differsfrom above. We find that the extrapolated relaxation rates
based on the fit parameters of the dynamic model used in
the high frequency regime (WC) deviate substantially, andsystematically, from the experimental observations (Fig. 3).

The additional contribution in the SC regime arises from aqualitatively different relaxation path available to the fluorine
system, once the fluorine and proton resonance lines startsignificantly overlapping. In terms of nuclear spin Hamiltonian comprising of the Zeeman term and a perturbing nuclear
dipole-dipole interaction, and with the valid assumption ofconservation of the total energy of the Hamiltonian, the question is connected with finding out the probability of resonant
absorption at a given frequency. This in turn inquires into therate at which the Zeeman and dipolar energies could comeinto mutual equilibrium.11 A higher order perturbation calculation is in principle necessary, with repeated operation of theinteraction Hamiltonian on the system. This involves operator
products of alternating terms comprising of either of A and B(secular and pseudo-secular terms in the dipolar interaction10 )on one hand, and one of the transition inducing terms (C,D, E, and F) on the other. This could be simplified,11 witha combination of perturbation calculation for the transition
inducing terms, and method of moments for A and B terms.
The resulting transition probability for the Zeeman energy of

(a)

(b)

(c)

(d)

FIG. 7. Fluorine spin-lattice relaxation rate dispersions fitted to Eq. (4) at different temperatures: (a) TNI = 0.5 ◦ C, (b) TNI = 1 ◦ C, (c) TNI = 4 ◦ C, and(d) TNI = 11 ◦ C. The solid line represents the fit of Eq. (4) and cross-relaxation contribution to the experimental data. The color dotted lines indicate thecontributions of different relaxation mechanisms. The values of the fitted parameters are given in Table II.

244507-8

Rajeswari et al.

a spin to be converted into dipolar energy leads to a characteristic time (T21 ), which is intermediate in nature between the
spin-spin relaxation time and the spin-lattice relaxation time.
This process will become rapidly ineffective if the Zeeman
splitting of the spins becomes larger than the spin-spin (dipolar) coupling among them. This interesting manifestation iseffective in the SC regime, wherein the differences in the Zeeman energy splittings of the two distinct spin species becomecomparable to the line width. In other words if the applied
Zeeman field is low enough for the two spins systems to haveoverlapping resonance lines, then a fresh mechanism comes
into play in relaxing the Zeeman energy to equilibrium. The−1, assuming specific line shapes
simplified expression for T21
for the two NMR lines (of proton and fluorine) is given by11 
−1
T21 = AOL
(5)gH (ν  )gF (ν  )δ(ν  − ν  )dν  dν 
withdAOL = |EH , EF |HHF
|EH ± hν, EF ± hν |2 .

Here AOL is related to the square of the matrix element ofHamiltonian involving simultaneous energy-compensating
flip-flops of the two spin species mediated by the B termin the dipolar interaction. To proceed further, we assume
Lorentzian shapes for both the resonance lines with widths
H and F , for proton and fluorine respectively. The
resultant contribution to the fluorine relaxation rate, effective−1,
only at low enough frequencies, is denoted by R1OL = T21
and is directly dependent on the extent of overlap of the twolines. With this additional mechanism, we express the total
computed contribution as
T
R1F= R1F + R1OL .

We interpret the differential contribution in the SC regime,(R1EXPT − R1F ), as due to R1OL , and fit the corresponding
data to Eq. (5) to obtain the three variable parameters (AOL , H , and F ). In Fig. 7, we include R1OL ,along with contributions from dynamic models (R1F from

J. Chem. Phys. 135, 244507 (2011)

Eq. (4)) to compare with the experimental data. Figure 8
depicts exclusively the effect of the contribution from thismechanism very near the transition. In view of the limited
number of data points pertaining to this region of overlap(due to instrument limitations) the data fit to R1OL in termsof Eq. (5) can at best be viewed as a demonstration of thisinteresting mechanism that could be detected in the FCNMR
experiment, and the values of the fit parameters are thus
providing reasonable order-of-magnitude estimates. We
observe that the width of the Lorentzian line of the protons
remains essentially constant with temperature (about 4 kHz)
while that of the fluorine increases from about 20 kHz at thehighest temperature to nearly 24 kHz very near the transition
point. For comparison let the deviations of the experimental
data at a given temperature from the prediction of dynamic
models (R1F in Eq. (4)) take place at some threshold valueof the frequency (νT , say) at the onset of discernible overlapof the lines. We find that νT increases progressively withdecrease of temperature and is maximum near the transition
temperature. This is indicative of expected line broadening
due to pretransitional effects, and the observed temperature
variation of F seems to support this. It may be further
pointed out that the line widths obtained here are intrinsic
and are free from the instrument artifacts (i.e., different fromapparent line width values due to field inhomogenieties),
since the observations pertain to dipolar interactions between
nuclei residing on the same molecule. The experimentallymeasured values of these widths obtained from the respective
free induction decays, which include contributions from fieldand sample inhomogenieties as well, are found to be about
45 kHz (for 19 F) and 16 kHz (for 1 H). These values are
consistent with the observed (intrinsic) line width data, if we
introduce identical correction factors to make up for the artifacts of the experiment. Finally, while the fluorine relaxationis seen to be profoundly influenced in the SC regime, such is
not the case with protons—this mechanism does not lead todiscernible additional contribution in this limit. This reflects
the fact that there are several protons residing on the molecule
having homo-nuclear dipolar interaction all through the dispersion regime (and thus having appreciable overlap integrals
always), and hence addition of one more such coupling withfluorine is not making an observable difference.
V. CONCLUSIONS

FIG. 8. Fluorine spin-lattice relaxation rate from dynamical models andcross-relaxation (overlap integral) at TNI = 0.5 ◦ C.

The present study attempts to interpret qualitatively different spin-lattice relaxation dispersions of two very similar
nuclei with close gyromagnetic ratios ( γ H ≈ γ F , within6%) in the isotropic phase of a liquid crystal. While there
are a large number protons (1 H) residing on the liquid crystal
molecule, a lone fluorine (19 F) is attached to the aromatic coreof the molecule. Despite the limited access of the fluorine
to couple to the lattice via the less ineffective hetero-nuclear
dipolar interaction with protons, its relaxation rate shows
strong dispersion and much stronger coupling to the lattice(by over an order of magnitude). This is interpreted as dueto the effective coupling of fluorine to the lattice throughspin-rotation interaction, and to the presence of slow enoughmodes in the system to produce appropriate correlations in the244507-9

Slow molecular processes in liquid crystals

angular momentum components of the molecules. Thus thechosen system is in a fortuitous circumstance of sampling two
types of correlations of the dynamic processes of the system:
molecular reorientational correlations through proton dataand its angular momentum correlations through fluorine data.
Interpretation of the experimental data is carried out recognizing this distinction. The FCNMR experiment also allows aneffective contact of the fluorine system with protons (despite
their differing γ  s) at low enough frequencies. In this regime,
called the strong-coupling limit, the hetero-nuclear coupling
acquires an effective homo-nuclear component due to theoverlap of the resonance lines. This leads to cross-relaxation
between the two spin systems, and its effect is especiallydramatic on the spin-lattice relaxation of the lone fluorine in
the molecule. The differing temperature dependences of therelaxation rates of the two nuclear species clearly indicate
the presence of two distinct underlying molecular processes,
but reported differently by the two participating probes.
In that sense it seems profitable to probe the molecular
dynamics through field-cycling NMR relaxometry withdifferent nuclear spins with differing relaxation pathways.
The interpretation of fluorine relaxation data based on SR
interaction necessitated a scenario to include slow torques.
Based on the conclusions of the earlier ESR experiments,23–25
it is proposed that SRLS mechanism could be applicable tothe present case leading to observed slow modes, even thoughquantitative analysis of the composite stochastic process
at this stage is not feasible. Finally, the present FCNMR
experiments27 provide perhaps the first evidence of such slow
modes with a direct mapping of their representative spectral
density.
ACKNOWLEDGMENTS

We would like to thank Professor Anil Kumar, IISc,
Bangalore for valuable discussions, and M.R. would like tothank the University Grants Commission (UGC), New Delhi
for financial support under RFSMS scheme.
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