Molecular Hydrogens Dissolved in Liquid Crystals
E. Elliott Burnell1 & Cornelis A. de Lange2
1 University of British Columbia, Vancouver, British Columbia, Canada
2 Vrije Universiteit, Amsterdam, The Netherlands
The NMR spectra of molecular hydrogen isotopologs dissolved and orientationally ordered in nematic liquid-crystal solvents provide a wealth of
information. Because they are quantum rotors, the various isotopologs H2 , D2 , T2 , HD, HT, and DT have different order parameters, with larger
internuclear distance having smaller order parameter. The signs and magnitudes of the order parameters are liquid-crystal dependent, this
being a direct result of the dominant anisotropic intermolecular interaction being that between the solute quadrupole moment and the mean
liquid-crystal electric-field gradient (EFG) felt by the solute. Thus, the mixing of different liquid crystals can produce a solvent where molecular
hydrogen feels a zero EFG, called a magic mixture. The remaining, dominant anisotropic interaction (for solutes larger than hydrogen) in these
magic mixtures involves short-range anisotropic forces that depend on solute size and shape. Studies involving the very symmetrical methane
H, D, and T isotopologs point out the important effect on dipolar couplings of molecular vibrations and interactions between molecular vibration
and reorientation, effects which make it impossible to utilize fully the high accuracy with which dipolar couplings in general can be measured.
Keywords: liquid crystal, ordered liquids, anisotropic, hydrogen, orientation, vibrations, methane, NMR, order parameter, electric
field gradient
How to cite this article:
eMagRes, 2016, Vol 5: 901–912. DOI 10.1002/9780470034590.emrstm1477
Introduction
Since the discovery of Saupe and Englert in 1963 that benzene
dissolved and orientationally ordered in nematic liquid crystals
showed 1 H NMR spectra dominated by anisotropic interactions
that in isotropic solvents average to zero,1,2 the method has
been applied to a multitude of solutes. These anisotropic
interactions involve the direct dipolar couplings between each
pair of magnetic nuclei in the solute molecule, anisotropic
contributions to the chemical shielding and indirect spin–spin
coupling tensors, and for nuclei with I ≥ 1 the quadrupolar
interaction.3 – 5 In the early years, many solutes were studied
and a large amount of information was gathered about solute
geometries in orientationally ordered environments and about
their various anisotropic tensor properties. The results were
commonly interpreted in terms of rigid solute structures,6 and
for a long time an understanding of what precise solute–solvent
interaction mechanisms caused solute orientation and to what
extent was almost completely lacking. This situation persisted
for quite some time.
Gradually, it was realized that in order to make progress
another approach was needed. Instead of studying solutes with
the aim of obtaining novel information about them in an
anisotropic liquid environment, research turned to the use of
small, well-characterized solutes instead. In these studies, the
purpose was no longer to learn about solute properties but to
use these small solutes as probes in the orientationally ordered
phases. It was hoped that through systematic studies of this
kind a better understanding could be achieved of the detailed
Volume 5, 2016
mechanisms that cause orientational order. In addition, the
role of internal molecular motions such as small- and largeamplitude solute vibrations became a topic of interest.
In order to address these questions, experiments were initiated on molecular hydrogen7 and its isotopologs8 in various
nematic liquid-crystal solvents. These and related experiments on other simple solutes have led to a breakthrough
in understanding many fundamental questions on NMR of
orientationally ordered solutes. Molecular hydrogen as an isolated molecule is very well known and all its degrees of freedom
(electronic, vibrational, and rotational) are extremely well
characterized. Despite its low solubility, this molecule is very
attractive and in fact ideal for study as a solute in a liquid crystal.
The interaction between solute and solvent can be modeled and
taken into account using perturbation theory. This leads to the
insight that the rotational levels of molecular hydrogen are perturbed by the environment but that vibrational and electronic
degrees of freedom remain mostly unaffected.8,9 Subsequent
studies on deuterated and tritiated solutes confirm and extend
the original findings.10
Our research involving the deuterated isotopologs focused
on two NMR observables, the direct dipolar and the quadrupolar couplings. The intramolecular contribution to both these
couplings is proportional to the molecular order parameter.
Hence, the ratio of these couplings should be independent of
orientation and a property of the solute alone. However, the
value of this ratio in the solvent differs from its value for molecular hydrogen isolated in the gas phase. Initially, this was ascribed
to environmental effects owing to the solvent. Later is was
© 2016 John Wiley & Sons, Ltd.
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EE Burnell & CA de Lange
realized that this observation provided the key to elucidating
the prevailing ordering mechanism for the hydrogens.11 Moreover, a significant isotope effect on the degree of orientational
order is observed due to quantum-mechanical effects involving
the liquid-crystal field and the solute rotational and vibrational
wave functions.8
In subsequent studies, it was discovered that the degree of
orientational order of molecular hydrogen and its isotopologs
could be manipulated by employing mixtures of liquid-crystal
solvents. It was found that for certain mixture compositions
the direct dipolar couplings that normally dominate the NMR
spectra could be reduced to essentially zero, suggesting zero
orientational order for a solute in an anisotropic solvent.
Somehow one or more mechanisms that cause orientational
order could be switched on and off at will.12,13
Built on the success of these experiments subsequent research
focused on detailed studies in which D2 played a key role in
the investigation of other small, well-characterized solutes.
From this work, it became clear that solute orientational order
is essentially induced by two mechanisms: (i) an interaction
between the solute molecular quadrupole moment with an
average nonzero electric-field gradient (EFG), which is a property of the liquid-crystal solvent; and (ii) a solute–solvent
interaction that depends solely on the size and shape of
the solute. For special liquid-crystal mixtures termed magic
mixtures, the average EFG reduces to zero and only the sizeand-shape mechanism remains. For certain solutes with small
electronic moments (such as the alkanes), the first mechanism
is also unimportant. Hence, such solutes are termed magic
solutes.14
A special extension of our research on molecular hydrogen
was found in studies on methane and its isotopologs. As it
had been observed experimentally that solutes with tetrahedral
symmetry (that should by symmetry show zero average orientational order in a liquid crystal) showed unexpected spectral
splittings, our studies revealed that a fundamental interaction
between solute vibrational and reorientational motions was at
the root of these observations.15 This vibration–reorientational
interaction was found to play an important role in larger
solutes as well, thus limiting the accuracy with which geometrical information for solutes dissolved in liquid crystals can be
obtained.16
In summary, studies of the extremely well-characterized
molecular hydrogen and its isotopologs dissolved in a variety
of liquid-crystal solvents have played a key role in obtaining a
fundamental understanding of the physics behind orientational
order of solutes in anisotropic phases. The purpose of this article
is to tell this exciting story.
Liquid-crystal Solvents
In our studies, different liquid-crystal solvents were
employed, mainly EBBA, 1132, and 5CB. EBBA stands for
N-(p-ethoxybenzylidene)-p -n-butylaniline. The commercial
mixture Merck ZLI-1132 consists of p-butylcyclohexyl-p cyanophenyl (38.4%), p-pentylcyclohexyl-p -cyanophenyl
(34.2%), p-heptylcyclohexyl-p -cyanophenyl (18.1%), and
p-pentylcyclohexyl-p -cyanobiphenyl (9.3%). 5CB stands
902
for p-n-pentyl-p -cyanobiphenyl. In our initial experiments,
H2 , HD, and D2 were dissolved under pressure in these
liquid-crystal solvents. Later, for reasons to be discussed,
mixtures of these nematic solvents were also employed.
Theoretical Considerations
In the gas phase, H2 occurs in two modifications, one with total
nuclear spin I = 1 that only combines with odd-numbered
rotational states J (ortho-H2 ) and the other with total spin state
I = 0 in combination with even-numbered rotational states
(para-H2 ). For D2 , a similar distinction exists between orthoD2 (total spin I = 0 or I = 2, with even J states) and para-D2
(total spin I = 1 with odd J states). The rate of conversion
between ortho and para modifications is usually slow. On the
time scale of our experiments, the spectra of H2 arise only
from the ortho modification, while separate spectra for both
ortho- and para-D2 are observed. Spin-lattice relaxation rates
have been measured for D2 in several nematic phases and
values found for para-D2 are about twice that for ortho-D2 in
agreement with theory.17
For HD, a single NMR spectrum arises as there is no
restriction on the spin and rotational states accessible to the
molecule.
The high-resolution Hamiltonian for an orientationally
ordered spin system with the director of the nematic phase
parallel to the external magnetic field Z is (in Hertz)3 – 5 :
Ĥ = −
νi (1 − σi,ZZ )Îi,Z
i
+
Jij Îi · Îj
i<j
+
Dij (3Îi,Z Îj,Z − Îi · Îj )
i<j
+
B
i
i
3
2
− Î2i )
(3Îi,Z
(1)
We shall be concerned with direct dipolar couplings Dij and
quadrupolar couplings Bi and neglect anisotropies in the
indirect couplings Jij . As the relevant indirect couplings predominantly arise from the spherically symmetric Fermi-contact
mechanism, they are positive in sign and their anisotropies can
be safely neglected.
For a cylindrically symmetric diatomic with all nuclear spins
I ≤ 1, the direct dipolar and quadrupolar couplings are given
by Burnell et al.8
hγi γj 1 3cos2 θ − 1
Dij = −
(2)
4π 2 2
rij3
∂2V 3 e ∂z2 i Qi 3cos2 θ − 1
Bi =
4
h
2
and
(3)
with θ the angle between internuclear axis z and magneticfield
2 direction Z, V the intramolecular potential in the solute,
∂ V
(often designated as eq in the literature) the negative of
∂z2
i
© 2016 John Wiley & Sons, Ltd.
Volume 5, 2016
Molecular Hydrogens Dissolved in Liquid Crystals
the EFG at deuteron nucleus i, and eQi the deuteron nuclear
quadrupole moment. The angular brackets denote averaging
over all intramolecular and intermolecular motions.
If the intramolecular and intermolecular averages in
equations (2) and (3) can be performed separately, an
assumption which is by no means valid in general, we obtain8
hγi γj 1
Dij = −
S
(4)
4π 2 rij3
and
Bi =
∂2V 3 e ∂z2 Qi
i
4
h
S
(5)
with orientation parameter S defined as
3cos2 θ − 1
S=
(6)
2
The value of r13 averaged over all internal motions can be
ij
evaluated from experimental gas-phase data about equilibrium internuclear distance, vibrational anharmonicities, and
centrifugal distortion.18 The average should be taken over all
vibrational and rotational levels, but as excited vibrational
levels are hardly occupied, the averaging can be restricted to
the populated rotational levels8 :
J(J + 1)(2J + 1)e−BJ(J+1)/kT
J(J + 1) = J (7)
−BJ(J+1)/kT
J (2J + 1)e
The primes on the summation symbols indicate that summations have to be taken over even J (ortho-D2 ), odd J (ortho-H2 ,
para-D2 ), or all J (HD).
2 for the temperature
The intramolecular average ∂∂zV2
r=re
average of the EFG at the site of the deuteron nucleus in D2 and
HD, consisting of a contribution due to the other nucleus and
another one due to the electrons, can also be calculated from
available data.19,20
Averaging over all intermolecular motions requires assumptions about the possible interactions between solute and
surrounding solvent. As a priori nothing is known about such
mechanisms, we assume that the effect of the solvent on the
rotational motion can be viewed as a perturbation of the form8 :
3 2
1
cos θ −
(8)
Ĥ = Δ
2
2
The quantity Δ signifies the solute–solvent interaction
between a single liquid-crystal ‘field’ and a single solute property
in its simplest bilinear second-rank tensorial form. Although
in principle the interaction could be much more complicated,
we shall investigate the problem with equation (8) as a perturbation to the well-known gas-phase quantum-mechanical
rotational problem. In this approach, the degeneracy of the mJ
levels is lifted, and the orientation parameter S can be predicted
with Δ as the only adjustable parameter. It is important to
note that for different isotopologs the values of S come out
Volume 5, 2016
differently. Hence, we predict a quantum-mechanical effect on
the degree of orientational order even at room temperature.
In that sense, the hydrogens are exceptional because of their
relatively large rotational constants B. For molecules with small
rotational constants, summations are replaced by integrals and
isotope-independent S values are predicted8 :
2
1
3 cos2 θ e−3Δcos θ/2kT dΩ
−
S=
(9)
2 θ/2kT
−3Δcos
2
2
e
dΩ
Expanding the exponentials to first order in Δ, equation (9)
reduces to the classical limit for S8 :
1 Δ
(10)
S=−
5 kT
First Experimental Results
In order to give an idea of how the quality of our NMR spectra
improved over time, in Figure 1 we present the 1 H spectra of
H2 (a, recorded in 1968) and a mixture of H2 and HD (b,
recorded in 1982), and in Figure 2, an 2 H NMR spectrum of D2
(recorded in 1987).21 The spectra can easily be analyzed and
dipolar and quadrupolar couplings can be extracted. The 1 H
spectrum of H2 is a doublet with splitting 3DHH and that of
HD is a triplet (because the nuclear spin of D is 1) with splitting
2DHD + JHD . The labels in Figure 2 indicate how BD , DDD , and
JDD can be obtained from the spectrum of D2 . The ratios of the
dipolar couplings depend predominantly on the product of the
magnetogyric ratios of the H and D involved.
The NMR spectra of H2 , HD, and D2 dissolved and orientationally ordered in several nematic liquid crystals (EBBA, 1132,
and a few others) have yielded surprising results: (i) the signs
of the order parameters are liquid-crystal dependent; (ii) the
molecules with larger internuclear distance (rH2 > rHD > rD2 )
have smaller order parameters; (iii) the average order parameters in ortho- and para-D2 are equal; (iv) the same interaction
parameter Δ serves to interpret all the results for the three solute
molecules; and (v) the ratio B : D of dipolar to quadrupolar
couplings in HD and D2 is typically about 6% smaller than
expected from gas-phase data. These observations need further
explanation.
First, we focus attention on the spectrum of D2 (Figure 2),
where through a happy coincidence the sign of the order
parameter S can be determined without difficulty. The reason
is that the spectrum contains two lines of unequal intensity
separated by 3JDD , which is known to be positive. The sign of S
can be directly related to the sign of the indirect coupling and
the order in which the two lines appear. From Figure 3, it can
be seen immediately that the order parameters in the liquid
crystals 1132 and EBBA must be of different sign.
The three order parameters of the isotopologs H2 , HD, and
D2 are not equal to each other and are in an order (SHH <
SHD < SDD ), opposite to what the conventional wisdom of the
day required. Our observations nicely demonstrate that the
orientational order of these solutes is a quantum-mechanical
effect (associated with the relatively large rotational constants
B) that persists even at room temperature. The small quantummechanical differences predicted between Sortho-D2 and Spara-D2
are not resolved experimentally.
© 2016 John Wiley & Sons, Ltd.
903
EE Burnell & CA de Lange
Isotropic
100 Hz
H2
Proton NMR
in 1132
HD
H2
TMS
1500 Hz
Nematic
−5000
(a)
H2
TMS
H2
(b)
0
5000
10000
ν (Hz)
Figure 1. (a) Continuous-wave 100 MHz (Varian HA-100) proton NMR spectra of molecular hydrogen dissolved in the isotropic phase (140 ◦ C,
top) and the nematic phase (80 ◦ C, bottom) of the nematic liquid-crystal mixture 60% p -ethoxy-p-hexanoyloxyazobenzene and 40% p -ethoxy-pvaleryloxyazobenzene. The extra peaks in the isotropic-phase spectrum are from the liquid-crystal solvent. Sufficient hydrogen to generate an overpressure
of about 60 atm was condensed (using liquid helium) into a thick-walled 5 mm outer diameter NMR tube containing the liquid crystal, and the tube was
flame sealed. (Reproduced from Ref. 22.) (b) Proton NMR spectrum of a mixture of H2 , HD, and D2 in 1132 (298 K). The remaining lines are from an
impurity. (Reprinted figure with permission from E.E. Burnell, C.A. de Lange and J.G. Snijders, Phys. Rev., A25, 2339, 1982. Copyright © 1982, American
Physical Society)
6D
2H
NMR of D2 in 1132
3J
2B
0
1000
2000
3000
H2 the molecules behave as quantum-mechanical rotors with J
a good quantum number. Our results show that although the
perturbation proportional to Δ mixes different J values, the
degree of mixing is limited and that J remains a reasonable
quantum number.
From equations (4) and (5), it is apparent that the ratio B : D
should be independent of the degree of orientational order and
therefore a property of the HD and D2 molecules alone. The
experimental B/D values observed for our isotopologs in a
liquid-crystal environment differ from the gas-phase value.
The observed change could arise because a liquid-crystal environment might perturb the electronic orbitals of the solute
species to a small extent. However, this explanation is not very
satisfactory in view of the fact that rotational and vibrational
levels are only slightly perturbed.
Hz
Figure 2. 61.4 MHz deuteron NMR spectrum of D2 in 1132. (Reprinted
with permission from E. E. Burnell and C. A. de Lange, Chem. Rev.
(Washington, D.C.), 1998, 98, 2359, and references therein. Copyright ©
1998, American Chemical Society)
The fact that the same value of Δ explains the experimental results for the three different isotopologs may come as a
surprise. The mathematical form of the solute–solvent interaction assumed is as simple as possible, and there is no a priori
evidence that this simple picture will hold. The fact that it does
has important implications that we shall expand upon later.
The hydrogens are unusual molecules in the sense that their
rotational levels are so widely spaced that even in liquid or solid
904
Killing Several Birds with One Stone
In the above section, we encountered a number of issues that
deserve further consideration: (i) the sign of S varies from liquid
crystal to liquid crystal; (ii) the deviation of the B : D ratio from
the gas-phase value is not properly understood; and (iii) a single
solute–solvent interaction mechanism appears to be adequate
to explain the results for the various isotopologs in a particular
liquid-crystal solvent. It would be very useful if these loose ends
could be tied together with a single explanation.
Many different mechanisms that describe the orientational
order of solutes in nematic phases have been suggested in
the literature. Dispersion forces,23 polarizability anisotropy,24
© 2016 John Wiley & Sons, Ltd.
Volume 5, 2016
Molecular Hydrogens Dissolved in Liquid Crystals
Deuterium NMR of HD and D2
EBBA
1132
0
1000
2000
3000
4000
ν (Hz)
Figure 3. 2 H NMR spectra of a mixture of H2 , HD, and D2 in EBBA and 1132 (298 K). The outermost lines of each set of multiplets are from HD, and
the inside lines are from D2 . (Reprinted figure with permission from E.E. Burnell, C.A. de Lange and J.G. Snijders, Phys. Rev., A25, 2339, 1982. Copyright
© 1982, American Physical Society)
size-and-shape effects,25 moments of inertia26,27 of the solute as
the dominating cause, and electrostatic forces derived from the
multipole expansion have been advanced as possible sources.
There is no a priori reason why only a single mechanism
should describe orientational order. However, our work on
the hydrogens suggests that a single second-order tensorial
mechanism that is a product of a liquid-crystal ‘field’ and some
solute property suffices.
In a paper by Patey et al.,11 new light was shed on the problem
by assuming that a single interaction between an average liquidcrystal EFG F felt by the solute, and the molecular quadrupole
moment Q of the solute could explain the observations. For a
nematic phase with axial symmetry and a cylindrical solute, the
interaction Hamiltonian is
1
ĤQ = − FZZ Qzz P2 (cos θ )
2
(11)
where θ is the angle between the solute symmetry axis z and the
nematic director direction Z. In the classical limit, this would
lead to a degree of solute order of
S≈
FZZ Qzz
10kT
(12)
In the presence of an average liquid-crystal EFG, a deuteron
nucleus μ in the solute would experience not only an
intramolecular EFG but also an external EFG:
3
B(observed) = B(intramolecular) − eQμ FZZ /h
4
with
B(intramolecular) =
3 2
(e qQμ /h)S
4
(13)
(14)
The value of the quadrupole coupling constant (3/4)(e2 qQμ /h)
at the D nucleus of rigid D2 (at equilibrium geometry) obtained
from virtually exact molecular hydrogen wave functions28
and molecular beam magnetic resonance experiments 29 is
Volume 5, 2016
226 kHz.15 When the nonrigidity of D2 is accounted for, slight
changes occur.8,9
Now it comes as no surprise that the B : D ratio for the
solute in the nematic phase deviates from that in the gas
phase. In fact, the deviation can be used to obtain an estimate
of B(extramolecular) and hence of FZZ . The sign of FZZ is
different for different nematic solvents and the degree of
orientational order S calculated in the presence of the external
EFG agrees convincingly with our previous observations. The
interaction between the average liquid-crystal EFG and the
molecular quadrupole moments appears to account for most
of the orientational order of the hydrogens in nematic phases.
Different signs of FZZ lead to different signs of S [equation (12)].
As we shall discuss later, for molecules other than the
hydrogens, the degree of orientational order can be explained
adequately by two very different mechanisms. In addition
to the interaction between average solvent EFG and solute
quadrupole moment, another mechanism that is determined
by solute size and shape often prevails. As the hydrogens are
essentially spherical, a significant size-and-shape contribution
is not expected, and the interaction discussed in this section
dominates. The results obtained in Ref. 11 were later supported
convincingly by those of ab initio calculations.9 Moreover,
these calculations confirm the notion that the hydrogens
dissolved in a nematic liquid crystal are only slightly perturbed
by the anisotropic environment and that J remains a reasonably
good quantum number.
Experiments with Magic Results
The fact that the sign of the orientation parameter S is opposite
in nematic liquid-crystal solvents 1132 and EBBA suggests an
interesting possibility. It appears at this point that in a nematic
phase the orientational order of hydrogen and its isotopologs
depends mainly on a single orientation mechanism, namely
© 2016 John Wiley & Sons, Ltd.
905
EE Burnell & CA de Lange
906
Deuterium NMR of D2
49.3 Wt% 1132
61.3 Wt% 1132
76.0 Wt% 1132
200
0
−200
−400 Hz
1000
40
0
0
−1000
−40
0
(b)
D (Hz)
(a) 400
B (Hz)
the interaction between the liquid-crystal average EFG and the
solute molecular quadrupole moment. As the EFGs in 1132 and
EBBA appear to be of opposite sign, it seems feasible that mixing
these nematic phases in an appropriate proportion would lead
to a new nematic phase, but now with a much reduced EFG,
and hence a much reduced degree of orientational order of the
H2 , HD, and D2 solutes. Indeed, mixing 1132 and EBBA does
lead to a new nematic phase. In order to investigate whether
these mixed phases possess a reduced solvent EFG, experiments
with D2 as a solute are required.30
The results of these experiments are shown in Figures 4 and
5. By mixing 1132 and EBBA in varying proportions and by
dissolving D2 in these mixtures, it is clear that we can create
entirely different situations. In a mixture with 49.3 wt% 1132,
the orientational order S of D2 is negative, corresponding to an
average negative EFG due to an excess of EBBA in the mixture.
In a mixture with 76.0 wt% 1132, the orientational order S of
D2 is positive, corresponding to an average positive EFG due
to an excess of 1132 in the mixture. In a mixture with 61.3 wt%
1132 where the amounts of 1132 and EBBA are balanced, the
2 H NMR spectrum of D reduces to essentially a single line
2
with zero splitting. As shown in Figure 4(a), a solute dissolved
in a nematic phase with essentially zero degree of order was
observed for the first time.30
From Figure 4(b), it can be concluded that the quadrupolar spitting B and the dipolar splitting D go through zero at
approximately the same composition of the liquid-crystal mixture. By taking an appropriate mixture of the two solvents, an
overall splitting of several kilohertz is reduced to almost zero.
As demonstrated in Figure 5 for nematic phases of 1132, EBBA
and their mixtures (and as observed later for other nematic
liquid crystals as a function of temperature31,32 and 1132 as a
function of pressure33 ), the order parameters calculated from
the EFG agree quite closely with those measured from the dipolar couplings using equation (4). The slight difference indicates
the presence of a small, extra interaction that consistently gives
rise to a minute negative contribution to the D2 order parameter. A direct result of this contribution is that B and D actually
go through zero at slightly different mixture compositions.
A model has been proposed to rationalize this small effect.34
However, the presence of this small effect does not take away
from the fact that the interaction between the average solvent EFG and the solute molecular quadrupole moment is the
dominant mechanism for the ordering of D2 in these mixtures.
The observation that through mixing component liquid
crystals an important mechanism for causing orientational
order of simple solutes (namely the interaction between solvent EFG and solute molecular quadrupole moment) can be
affected at will constitutes a veritable breakthrough. This novel
insight provides the experimentalist with an invaluable tool
to distinguish among various possible physical mechanisms
contributing to the orientational order of solute molecules and
to obtain information about the internal liquid-crystal field. It
is therefore common usage to speak of magic mixtures when
zero-EFG nematic solvents are discussed. As we shall see later,
these magic mixtures have found extensive use in liquid-crystal
NMR spectroscopy.
20
40
60
Wt% 1132
80
100
Figure 4. (a) 61.3 MHz deuteron NMR spectra of D2 dissolved in nematic
mixtures of EBBA and 1132 at 310 K. (Reprinted from Chemical Physics
Letters, 107, P.B. Barker, A.J. van der Est, E.E. Burnell, G.N. Patey,
C.A. de Lange and J.G. Snijders, NMR of deuterium in liquid crystal
mixtures, 426–430., Copyright 1984, with permission from Elsevier) (b)
Experimental dipolar coupling constant D (right axis, open circles) and
experimental quadrupolar coupling constant B (left axis, stars) of D2
dissolved in nematic mixtures of EBBA and 1132 at 310 K. (Reprinted from
Chemical Physics Letters, 107, P.B. Barker, A.J. van der Est, E.E. Burnell,
G.N. Patey, C.A. de Lange and J.G. Snijders, NMR of deuterium in liquid
crystal mixtures, 426-430., Copyright 1984, with permission from Elsevier)
After the first experiments with magic mixtures described
in Ref. 30, the concept was extended. It was found that also
5CB (with positive EFG) and EBBA (with negative EFG) when
mixed in appropriate amounts lead to zero-EFG mixtures.35
In addition, 3 H NMR of the tritiated isotopologs of hydrogen
(T2 , HT, and DT) dissolved in either component liquid crystals
or in zero-EFG mixtures10 (Figure 6) essentially confirms the
findings of Ref. 30. The 3 H spectrum of T2 is a doublet
with splitting 3 DTT , that of HT is a doublet with splitting 2
DHT + JHT , and that of DT a triplet (because D has nuclear
spin 1) with splitting 2 DDT + JDT ). The splittings depend
© 2016 John Wiley & Sons, Ltd.
Volume 5, 2016
Molecular Hydrogens Dissolved in Liquid Crystals
10
TT
HT
5
DT
S × 103
Theoretical SEFG
0
Experimental S
−5
5000
−10
−5
5
0
Fzz × 10
−11
−5000
Figure 6. 640.12 MHz tritium NMR spectrum of a mixture of T2 , H2 ,
and D2 in EBBA at 300 K after irradiation of the sample tube in order to
achieve isotope scrambling. (Reprinted figure with permission from E.E.
Burnell, C.A. de Lange, A.L. Segre, D. Capitani, G. Angelini, G. Lilla and
J.B.S. Barnhoorn, Phys. Rev. E, 55, 496, 1997. Copyright © 1982, American
Physical Society)
esu
Figure 5. Experimental order parameters S [from DDD and equation (4)
filled circles] and order parameters SEFG as a function of FZZ [equation
(12)]. FZZ values are obtained from experiments with D2 in nematic
mixtures of 1132 and EBBA at 310 K (see text). (Reprinted from Chemical
Physics Letters, 107, P.B. Barker, A.J. van der Est, E.E. Burnell, G.N. Patey,
C.A. de Lange and J.G. Snijders, NMR of deuterium in liquid crystal
mixtures, 426–430., Copyright 1984, with permission from Elsevier)
predominantly on the product of the magnetogyric ratios of T
times that of T, H, or D.
Vibration–Reorientation Interaction
We now return to a point that was already mentioned with
respect to equations (2) and (3), viz. the assumption that
averages over intramolecular and reorientational motions can
be performed separately. Of course, the question arises how
good this assumption is and whether it can be tested. The
hydrogens are not the best testing ground here because the
observables are products of intramolecular and reorientational
averages. A small change in one average must be counteracted
by a change in the other average to lead to the same observable. Moreover, such changes would come on top of fairly
large anisotropic splittings and would therefore be difficult to
extract. A better testing ground would be provided by solutes
that should show zero dipolar and quadrupolar splittings when
averages over intramolecular and reorientational motions can
be separated, but that experimentally show small dipolar
and quadrupolar splittings that can be shown to arise from
vibration–reorientation interaction. An excellent example is
provided by the methanes (solutes with tetrahedral symmetry) that show small a priori unexpected anisotropic splittings.
These splittings can be analyzed using methods very similar
Volume 5, 2016
0
Frequency (Hz)
to those developed to understand the behavior of hydrogens
in nematic phases and that are described in previous sections.
This important spin off for larger molecules will be discussed.
We start by generalizing equation (8). The Hamiltonian
that describes the interaction between a liquid-crystal ‘field’
E in an axial nematic phase and a solute is again written
in its simplest form, namely as a single bilinear second-rank
tensorial interaction. In equation (8), this interaction is given
for a cylindrically symmetric rigid solute. Here we consider a
more general case15 :
3
1
1 (15)
cos θi cos θj − δij
βij (Qm )
Ĥ = − G
3
2
2
ij
with
G = E|| − E⊥
(16)
The quantity β signifies the solute property that interacts
with the field and is taken to depend on internal motions
through the solute normal modes Qm and θi is the angle between
the solute i axis and the field direction. As before, a single
interaction mechanism is assumed although it is not specified
which. The interaction of equation (15) is now considered as
a perturbation of the zero-order vibrational–rotational wave
functions of the solute.
In order to proceed, we develop the β(Qm ) tensor into a
Taylor series around the equilibrium (Qm = 0):
∂β kl
Qm + · · ·
βkl (Qm ) = βkl (Qm = 0) +
∂Q
m Qm =0
m
(17)
The observables that we are concerned with and that could be
obtained from splittings in the NMR spectra such as dipolar
© 2016 John Wiley & Sons, Ltd.
907
EE Burnell & CA de Lange
and quadrupolar couplings have the following general form:
3
1
akl (Qm )
A(Qm , Ω) =
cos θk cos θl − δkl
(18)
2
2
kl
In order to calculate the quantity A(Qm , Ω) in the presence of
a nonzero interaction, we apply standard perturbation theory
to second order. After some algebra, we predict anisotropic
splittings even for the fully tetrahedral species CH4 and CD4 :
⎛
1 −1 ⎝ ∂aij ∂βij
nrig
A (T) = G
Fkl
10
∂Sk ∂Sl
ij
kl
⎞
1 ∂aii ∂βjj ⎠
(19)
−
3
∂Sk
∂Sl
i
j
where Sk are symmetry modes and Fkl is the isotopically
invariant harmonic force field matrix related to the vibrational
frequencies ωm by:
2
Lkm Fkl Llm = ωm
(20)
kl
where L is the matrix that transforms normal modes to symmetry modes. The derivatives are all taken at equilibrium
geometry.
For the nonfully tetrahedral solutes CH3 D, CH2 D2 , and
CHD3 , an additional contribution to the anisotropic splittings
is obtained:
⎞
⎛
G ⎝ 1 eq eq eq eq ⎠
rig
A (T) =
aii
βkk +
aij βij
(21)
−
10kT
3
i
k
ij
As in the case of the hydrogens, methane and its isotopologs
are very well-characterized molecules and accurate rotational
and vibrational data are available. Therefore, we can compare
our theoretical predictions to a large body of experimental
data that we have obtained for the entire set of protonated
and deuterated methanes.36 Methane possesses four normal
modes (symmetric stretch A1 , bend E, asymmetric stretch F2 ,
and asymmetric bend F2 ). The symmetric stretch does not contribute to any anisotropic splittings, so we are left with three
unknown derivatives of the solute β tensor with respect to normal modes Qm or, preferably, the isotope-invariant symmetry
modes Sm . Hence, all the anisotropic splittings observed in
the methanes are described by three parameters Gβ1 , Gβ2 , and
Gβ3 that can be obtained by fitting the theoretical expressions
to the observed splittings. Excellent fits are obtained for each
liquid-crystal solvent.
From the theoretical expressions for the quadrupolar
splittings, it follows that with knowledge of the parameters
eq
Gβ1 , Gβ2 , and Gβ3 the solute properties V|| and the derivatives
∂V|| /∂Sk can be obtained. From ab initio calculations, it is clear
that the derivatives with respect to the bends with symmetry
eq
E and F2 are negligible.37 In addition to V|| , this leaves
only the derivative with respect to the stretch mode F2 to be
extracted. Both quantities are solute properties and therefore
liquid-crystal independent. However, the values that we obtain
vary significantly with liquid crystal.24
908
There is one more factor to be considered. For the hydrogens,
we found that the deviations observed for the B : D ratio can be
rationalized by the presence of an average external liquid-crystal
EFG that interacts with the molecular quadrupole moment of
the hydrogen isotopologs. For the methanes, a similar isotopeindependent contribution to the quadrupolar splittings can
be expected. It is tempting to transfer the same EFG values
estimated for the various liquid-crystal solvents in which the
hydrogen solutes were studied to the case of the methanes.
The underlying assumption here is that the average EFG felt
by hydrogens and methanes in nematic solvents is essentially
identical. If with this assumption we now include the average
liquid-crystal EFGs in our fitting procedure for the methanes, it
eq
is pleasing to see that the quantities V|| and the derivative with
respect to the F2 stretch mode are much more liquid-crystal
independent.
Comparing the results of our studies on hydrogen and
methane and their deuterated isotopologs, a consistent picture
emerges and the following conclusions can be drawn: (i) a simple single interaction of second-rank tensorial form accounted
for by standard perturbation theory is in excellent agreement
with our observations and suffices in all cases; (ii) assuming
an average external solvent EFG that interacts with the solute
quadrupole moments leads to consistency, thus supporting
this notion; and (iii) vibration–reorientation interaction is
the cause of anisotropic splittings in the methanes in nematic
phases.
Later work on a large variety of solutes has shown that (in
addition to the orientation mechanism that arises from the
interaction between the average liquid-crystal EFG and the
solute quadrupole moment) a second mechanism that involves
solute size and shape is important. Experiments on methanes
in magic mixture have confirmed that for the methanes the
size-and-shape mechanism is unimportant, as expected from
their tetrahedral or near-tetrahedral shape. Experiments on the
tritiated methanes38 are in complete agreement with the results
found for the protonated and deuterated isotopologs.
Once the existence and importance of the vibration–
reorientation interaction were established, it was realized that
this mechanism also plays a role in larger molecules. A case
in point is acetylene studied in various nematic solvents.39
From a general point of view, it must be concluded that,
despite the high accuracy with which anisotropic couplings
can be measured with NMR, it is difficult if not impossible to translate this experimental accuracy to molecular
properties such as molecular structure. In that sense, it is
finally realized that vibration–reorientation interaction is a
real hindrance in molecular structure determination using
liquid-crystal solvents.16
More on Magic Mixtures
Experiments on D2 dissolved in an appropriate mixture of the
liquid crystals 1132 or 5CB and EBBA have led to the concept
of magic mixtures. In order to describe most of the orientational order of D2 , a single mechanism is found to suffice. This
mechanism involves the interaction between an average external liquid-crystal EFG and the solute molecular quadrupole
© 2016 John Wiley & Sons, Ltd.
Volume 5, 2016
Molecular Hydrogens Dissolved in Liquid Crystals
moment. As this EFG is temperature dependent, a certain composition of the mixture possesses a zero-EFG at only one specific
temperature. The use of these liquid-crystal mixtures allows
the experimentalist to ‘tune’ the EFG-molecular-quadrupolemoment mechanism at will. The concept of magic mixtures
also has important implications for molecules larger than D2 ,
as will be discussed in this section.
Since the initial experiments on D2 , the use of magic mixtures
has become widespread and a variety of larger molecules has
been studied in such mixtures. In general, the behavior of
larger molecules dissolved in magic mixtures differs from that
of D2 . In D2 , the degree of orientational order can be ‘tuned’
through virtually zero, whereas for larger molecules anisotropic
splittings usually remain. This could be due to a number of
reasons. First, the same EFG that the small solute D2 experiences
in a liquid-crystal solvent might not be felt by larger solutes.
On average, the sampling of the liquid-crystal environment
by solutes could depend on solute size and shape. It could
be that D2 samples regions inside the solvent that would not
be accessible to larger solutes. Second, an explanation for the
observations could be that the anisotropic shape that is usually
associated with larger molecules is at the root of a significant
size-and-shape-dependent orientation mechanism that does
not occur in D2 , which is close to being spherically symmetric.
As we shall see, the second of these two possibilities essentially
applies.
When a solute is dissolved in a nematic phase, the
solute–solvent interaction can be described in terms of a
multipole expansion.40,41 Such an expansion breaks down at
short intermolecular distances where the electron clouds of
solute and solvent molecules start to overlap. This type of
overlap is subject to restrictions based on the Pauli principle
and leads to a strong repulsive interaction in which anisotropic
molecular shape plays a key role. In addition to all the possible
bilinear interactions that could be imagined on the basis of the
multipole expansion, a hard-core anisotropic shape-dependent
repulsion potential is therefore to be expected. Experiments
on D2 in magic mixtures have demonstrated conclusively that
the electrostatic interaction between average solvent EFG and
molecular quadrupole moment dominates over all other terms
in the multipole expansion. For larger solutes, a large body of
experimental evidence shows that, in addition to this single
electrostatic interaction, a single size-and-shape mechanism
plays a key role.
Since the advent of magic mixtures, a large number of solutes
have been studied in component liquid crystals and their zeroEFG mixtures. All these studies provide ample experimental
evidence that a single orientation mechanism is sufficient to
account for the additional size-and-shape-dependent mechanism present for molecules other than D2 . All these studies
have been summarized extensively before in a book13 and an
extensive review article.12 For further details, we refer the reader
to these publications.
The size-and-shape contribution to solute orientational
order arises from the anisotropic Pauli repulsion potential
between solute and solvent molecules. As this amounts to a
breakdown of the multipole expansion, this hard-core interaction has no quantum-mechanical Hamiltonian associated
Volume 5, 2016
with it and therefore cannot be treated with perturbation theory. The only pragmatic way to account for this interaction
is through the use of phenomenological models. Trying to
develop such models has a long history. Early models were
based on solute moments-of-inertia considerations,26 describing solutes as rectangular parallelepipeds (the Straley model),42
and the use of Stuart’s ‘Wirkungsradien’.25 Later, most of the
research groups active in the field of solutes in nematic solvents
have developed models based on solute molecular structure.
Although details vary, all these models are in the same spirit
and show similar results.12,43 In the literature, research groups
often use their own favorite brand. We mention the chord and
modified-chord models,44 – 46 the C and CI models,12 and a
model that describes the interaction between the solute surface normal and the liquid-crystal mean field.47 For a more
complete description of these phenomenological models, we
again refer to the literature.12,13 With these models, we can
routinely predict solute orientation parameters to an accuracy
of about 10%.
As an aside, we mention a special class of solutes that are
termed magic solutes. These solutes happen to possess small
multipole moments, in particular small molecular quadrupole
moments. Even in component liquid crystals with nonzero
EFG, the orientation mechanism that dominates for D2 is
relatively unimportant for these magic solutes. Representative
examples are, e.g., methane and higher alkanes. The orientational order of such solutes in component liquid crystals and in
magic mixtures is completely dominated by the size-and-shape
mechanism in all cases.
Flexible Molecules as Solutes in Nematic
Phases
We shall conclude with a separate class of solutes, viz. molecules
that do not exist in a single conformation but that undergo rapid
conformational change, either between symmetry-related or
symmetry-unrelated conformations.12,48 – 50 The present interest in this class of solutes merits a separate section.
We discuss conformational change in terms of an interconversion between separate ‘rigid’ conformations. This interconversion is assumed to be fast on the NMR timescale. The notion
of an ‘average’ solute molecule whose orientational order can
be described with a limited number of at most five orientation
parameters is found to be fallacious.51 Unfortunately, if the
different conformations are not related by symmetry, in the
general and most unfavorable case we require five independent
orientation parameters for each conformation. The anisotropic
observables take the form:
(i) (i)
Pi
Skl Ckl
(22)
O=
i
kl
where O is the dipolar or quadrupolar coupling, Pi the conformer probability, S(i)
kl the kl element of the order tensor,
and Ckl(i) a constant appropriate for conformer i. As Pi and
S(i)
kl occur as a product in equation (22), they cannot be separated in a single experiment. This is a serious limitation, but
there appears to be a way around this dilemma. The solution
lies in a temperature-dependent study of the solute. However,
© 2016 John Wiley & Sons, Ltd.
909
EE Burnell & CA de Lange
flexible solutes usually contain many proton spins, and a very
complicated 1 H NMR spectrum results.
Recently, the 1 H NMR spectra of the flexible magic solutes
n-butane,49,52 n-pentane,50,53 and n-hexane54 were studied in
nematic phases and solved with the use of novel evolutionary
strategies.55 The fact that anisotropic spectra of these alkanes
(that in the past were much too complex) can now be solved
routinely using these new spectral fitting techniques allows
temperature-dependent studies. The spectral analysis is only
possible in practice because we can use the phenomenological
models discussed earlier to predict orientation parameters for
each conformation to a degree of accuracy that is sufficient
to provide a good starting point in the fitting procedure. By
changing the temperature, the population Pi of the different
conformations can be varied, and a separation of Pi and S(i)
kl
can be achieved from a series of spectra. This separation also
relies on results from the solute ethane in the same nematic
phase56 and the observation that the liquid-crystal field in the
longer solutes is proportional to that in ethane. For the first
time, information on conformational probabilities of flexible
solutes dissolved in nematic liquid crystals can now be obtained
experimentally.49,50
Conclusions
Experimental and theoretical work on hydrogen and its isotopologs has been the key to understanding the interactions
that take place between solutes and their nematic environment.
Experiments on D2 have been particularly enlightening because
from its 2 H NMR spectrum the sign of its average orientation
parameter can be obtained. Surprisingly, depending on the
nematic solvent, this sign can be either positive or negative. In
addition, it has been found that by mixing nematic phases a new
nematic phase is obtained in which the degree of orientational
order of D2 can be manipulated and even be made zero. Moreover, the single orientation mechanism that describes most of
the orientational order of D2 and the other hydrogens is the
interaction between the average electric-field gradient present
in the solvent with the solute molecular quadrupole moment.
Liquid-crystal mixtures that at a certain temperature show
zero-EFG are termed magic mixtures and have found extensive
use in liquid-crystal NMR.
When methane and its isotopologs are studied in nematic
phases and appropriate mixtures thereof, convincing evidence
has been found for a coupling between reorientational and
vibrational motion. This coupling causes anisotropic splittings
even in tetrahedral solutes. The interaction between solute
reorientation and solute internal motions is always present and
severely limits the accuracy with which structural information
can be extracted for solutes.
Dissolving larger molecules in mixtures of nematic phases
has demonstrated that a second interaction between solute and
liquid-crystal solvent plays a dominant role, viz. the size-andshape mechanism caused by the anisotropic solute–solvent
Pauli repulsion that dominates at short intermolecular range.
This interaction cannot be treated quantum-mechanically and
is therefore dealt with using phenomenological models. Such
910
models are capable of predicting orientation parameters at
approximately the 10% level.
Solutes that possess small molecular moments and whose
orientational order is therefore dominated by the size-andshape mechanism are termed magic solutes. A special class is
formed by the longer alkanes that moreover undergo interconversion among many conformations. The very complex 1 H
NMR spectra of these solutes can nowadays be solved when
employing novel, sophisticated evolutionary strategies. This
development shows particular promise for obtaining a better understanding of the behavior of flexible molecules in an
anisotropic environment.
In summary, experiments that started with hydrogen and
its isotopologs dissolved in nematic phases have given an
enormous impetus to understanding the fundamentals of orientational order caused by the solute–solvent interactions.
This and related developments have moved liquid-crystal NMR
spectroscopy into a new and promising area.
Acknowledgments
The authors are grateful to David Buckingham (University of
Bristol, later University of Cambridge) for suggesting experiments on the hydrogens long before remotely feasible. We
thank the late Jaap Snijders (University of Amsterdam) and
Grenfell Patey (University of British Columbia) for their invaluable contributions over many years. Finally, we wish to thank
several generations of Ph.D. students and postdocs who did
most of the hard work.
Biographical Sketches
E. Elliott Burnell, b. 1943. BSc, 1965 and MSc, 1968, Memorial
U. of Newfoundland, PhD, 1970, University of Bristol. Following
Postdocs in Physics (UBC) and Physics (Basel), joined UBC Chemistry
in 1972 (currently Professor). Approximately 150 papers on various
aspects of the application of NMR techniques to the investigation
of orientationally ordered liquids, especially the understanding of
anisotropic intermolecular forces.
Cornelis A. de Lange, b. 1943. BSc, 1963 and MSc, 1966 (both cum
laude), University of Amsterdam, PhD 1969, University of Bristol.
Employment: Shell Research Amsterdam, Vrije Universiteit Amsterdam, and University of Amsterdam. Emeritus Professor of Physics and
former Member of the Dutch Senate. Approximately 200 papers on the
application of NMR techniques to the investigation of orientationally
ordered liquids, and laser spectroscopy, in particular photoelectron
spectroscopy.
Related Articles
Emsley, James W.: Having Fun with Liquid Crystals; Khetrapal,
C. L.: Development of NMR of Oriented Systems; Analysis
of Spectra: Automatic Methods; Liquid Crystalline Samples:
Spectral Analysis; Liquid Crystalline Samples: Structure of
Nonrigid Molecules; Liquid Crystals: General Considerations;
Multiple Quantum Spectroscopy in Liquid Crystalline
Solvents; Structure of Rigid Molecules Dissolved in Liquid
Crystalline Solvents; Two-Dimensional NMR of Molecules
Oriented in Liquid Crystalline Phases; Analysis of Complex
High-Resolution NMR Spectra by Sophisticated Evolutionary
© 2016 John Wiley & Sons, Ltd.
Volume 5, 2016
Molecular Hydrogens Dissolved in Liquid Crystals
Strategies; Liquid Crystalline Samples: Orientational Order of
Small Rigid Solutes in Nematic Mesophases
30. P. B. Barker, A. J. van der Est, E. E. Burnell, G. N. Patey, C. A. de Lange, and
J. G. Snijders, Chem. Phys. Lett., 1984, 107, 426.
31. A. Weaver, A. J. van der Est, J. C. T. Rendell, G. S. Bates, G. L. Hoatson, and
E. E. Burnell, Liq. Cryst., 1987, 2, 633.
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Volume 5, 2016
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© 2016 John Wiley & Sons, Ltd.
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Molecular Hydrogens Dissolved in Liquid Crystals E. Elliott Burnell 1

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