Muon Spin Spectroscopy of the Nematic Liquid Crystal 4-n-Pentyl-4

Jul 6, 2009 - ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, U.K. OX11 0QX, Institut für Physikalis...
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J. Phys. Chem. B 2009, 113, 10135–10142

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Muon Spin Spectroscopy of the Nematic Liquid Crystal 4-n-Pentyl-4′-cyanobiphenyl (5CB) Iain McKenzie,*,† Herbert Dilger,‡ Alexey Stoykov,§ and Robert Scheuermann§ ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, U.K. OX11 0QX, Institut fu¨r Physikalische Chemie, UniVersita¨t Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany, and Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen, Switzerland ReceiVed: March 21, 2009; ReVised Manuscript ReceiVed: June 3, 2009

Avoided level crossing muon spin resonance (ALC-µSR) spectroscopy has been used to study the four cyclohexadienyl-type radicals produced by the addition of muonium (Mu) to the rodlike liquid crystal 4-npentyl-4′-cyanobiphenyl (5CB). ALC-µSR spectra have been obtained over a wide temperature range in the isotropic, nematic, and crystalline phases. Four ∆0 resonances were observed in the ALC-µSR spectra, from which the methylene proton hyperfine coupling constants (hfcs) of the Mu adducts of 5CB were determined as a function of temperature. The methylene proton hfcs of two of the radicals have unusual temperature dependence in the nematic phase and have smaller values than would be predicted from extrapolating the data in the isotropic phase. We have used the Maier-Saupe theory for rodlike liquid crystals to explain the temperature dependence of the methylene proton hfcs, which results from the ordering of the 5CB molecules, the alignment of the molecules with the external magnetic field, and fluctuations that average the anisotropic hyperfine coupling constants. There are no ∆1 resonances in the ALC-µSR spectra of the nematic phase due to the radicals rotating rapidly around the long molecular axis and fluctuations about the local director. The ∆0 resonances broaden substantially as the temperature is lowered due to the slowing down of the fluctuations, which have an average activation energy of approximately 15.9 kJ mol-1. Cooling the sample below 275 K stopped the rotation around the long molecular axis and led to the appearance of ∆1 resonances. Introduction Liquid crystals (LCs) are materials that exhibit phases in which there is a degree of molecular order between the complete disorder of the isotropic (I) liquid phase and the long-range, three-dimensional positional and orientational order of the crystalline (Cr) phase.1-3 Thermotropic LCs form mesophases over a specific temperature range where the LC molecules show a significant, long-range orientational order due to anisotropic intermolecular forces acting on every molecule. LCs are categorized in terms of the shape of the molecules (or mesogens) with calamitic LC being composed of rodlike molecules and discotic LC having disk-shaped molecules. Calamitic LC can form a large number of mesophases of which the simplest, and most disordered, is the nematic phase (N) where the molecules are spatially disordered but have long-range orientational order about a particular direction, called the director (n). The order in a LC is quantified using the order parameter, S, which is defined by

S)

∫ 21 (3 cos2 θ - 1)f(θ)dΩ

(1)

where θ is the angle between the symmetry axis of a rodlike molecule and the director, and f(θ)dΩ is the fraction of molecules in a solid angle dΩ that are oriented at an angle θ.1 There are many experimental techniques, such as polarizing optical microscopy, that are invaluable for studying the bulk * Corresponding author. E-mail: [email protected]. † STFC Rutherford Appleton Laboratory. ‡ Universita¨t Stuttgart. § Paul Scherrer Institute.

properties of liquid crystals but do not provide information about the motion or ordering of calamitic LC on the molecular scale. Magnetic resonance techniques have been shown to provide information about the dynamics and ordering of the LC molecules. Nuclear magnetic resonance (NMR) has been used extensively to study ordering in calamitic LC using rigid solutes like 1,3,5-trichlorobenzene and flexible solutes like propane as probes as well as the LC molecules themselves.4-8 Deuterium NMR is frequently used for studying the ordering of LC but often requires site-specific isotopic labeling, which can be both difficult and expensive. Electron paramagnetic resonance (EPR) has been used to study stable spin probes dissolved in LC materials, and this provides useful information about ordering and dynamics of LC phases.9-18 The experimental work has been complemented with theoretical studies modeling the behavior of the spin probes at the molecular level.19-21 A drawback to the EPR and some of the NMR experiments is that the structures of the spin probes are very different from the LC in which they are dissolved, so it can be argued that the measurements reflect the perturbed LC and that the location of the spin probe is unknown. A series of experimental techniques collectively known as muon spin rotation/resonance/relaxation or µSR are powerful tools for probing molecular dynamics of free radicals in soft matter, solids, liquids, and gases.22,23 These techniques involve injecting spin-polarized positive muons into a sample and detecting the positron produced by the decay of each muon. The muons can pick up a radiolytic electron and form muonium (Mu), which is a light isotope of hydrogen.24 Mu reacts with unsaturated molecules to produce a muoniated radical, where the muon is a 100% polarized spin label. The parity-violating decay of the muon provides a convenient way to monitor the

10.1021/jp9025656 CCC: $40.75  2009 American Chemical Society Published on Web 07/06/2009

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evolution of the muon spin, and its lifetime (2.2 µs) is comparable to many molecular processes. µSR has many features in common with EPR but is comparatively advantageous because the spin labeling occurs in situ, the muon results in a much smaller perturbation than the stable free radicals currently used in EPR, the location of the spin probe can be accurately determined, and the high polarization of the muon makes µSR a more sensitive technique. Avoided level crossing muon spin resonance (ALC-µSR) has been used extensively to study soft matter systems, particularly the partitioning of cosurfactants in lamellar phase dispersions.25-27 The ALC-µSR technique involves measuring the asymmetry of the muon decay as a function of a magnetic field applied parallel to the initial direction of the muon spin. The asymmetry parameter is defined as (nB - nF)/(nB + nF), where nF is the total number of positrons detected in the forward counters and nB is the total number of positrons detected in the backward counters and is proportional to the muon polarization, Pz. In high magnetic fields, the eigenstates of the radical can be approximated by pure Zeeman states, so there is no evolution of the muon’s spin with time, and the asymmetry is independent of the magnetic field. At specific values of the applied magnetic field, nearly degenerate pairs of spin states can be mixed through the isotropic and anisotropic components of the hyperfine interaction. The muon polarization oscillates between the two mixing states, and this leads to a loss of time-integrated asymmetry. There are three types of resonances, which are characterized by the selection rule ∆M ) 0, ( 1, and ( 2, where M is the sum of the mz quantum numbers of the muon, electron, and proton spins. The resonances are referred to as ∆0, ∆1, and ∆2 resonances, respectively. The ∆1 resonance field is given by28

∆1 Bres )

Aµ Aµ 2γµ 2γe

[

Aµ + Ap 1 Aµ - Ap 2 γµ - γp γe

]

(3)

where γp is the proton gyromagnetic ratio. The ∆2 resonance is extremely weak and is rarely observed. The muon and methylene proton hfcs of muoniated cyclohexadienyl-type radicals can be calculated solely from the ∆0 resonance field since it has been shown for many radicals of this type that Aµ and Ap are approximately proportional to each other.29,30

Aµ ) K · Ap

The main contribution to K is the γµ/γp ratio, which has a value of 3.183, and the rest results from the mass dependence of the molecule’s vibrational modes. K was found to be 4.123 for the cyclohexadienyl radical in benzene at 298 K, and we will use this value throughout this work. The relationship between the methylene proton hfc and the ∆0 resonance field is given by

(2)

where Aµ is the muon hfc, γµ is the muon gyromagnetic ratio, and γe is the electron ratio. The ∆1 resonance arises from mixing between spin states with the same electron and proton spins but different muon spin directions. These spin states are mixed only in the presence of anisotropy, so the ∆1 resonance can be considered to be diagnostic of a frozen state or anisotropic motion. The ∆0 resonance is due to mixing between spin states that have the same electron spin but opposite muon and proton spins and is observed for muoniated radicals in the solid, liquid, or gas phases. The ∆0 resonance field depends on both the muon hfc and the proton hfc, Ap, and is given by28

∆0 Bres )

Figure 1. Structures of the four substituted muoniated cyclohexadienyl radicals formed by the addition of muonium to 5CB.

(4)

Ap )

∆0 Bres Θ

(5)

where Θ is a constant term given by

Θ)

[

1 (K - 1) (K + 1) 2 (γµ - γp) γe

]

(6)

and has a value of 1.67 × 10-2 T MHz-1 around room temperature. The only LC to have been studied by the various forms of muon spin spectroscopy is 4-n-pentyl-4′-cyanobiphenyl (5CB).31,32 This was one of the first LCs to be used in display devices due to its low melting point, relatively low viscosity, and chemical stability. 5CB exhibits a N phase between 297 and 308 K, although the melting point can be lowered depending on the rate at which the sample is cooled. Lovett et al. investigated the structure and dynamics of the Mu adducts of 5CB using transverse field muon spin rotation (TF-µSR), ALC-µSR, and longitudinal field muon spin relaxation (LF-µSR). Four types of substituted cyclohexadienyl radical were formed by Mu addition to the biphenyl moiety of 5CB, and their muon hfcs were measured using TF-µSR. The structures of these radicals are shown in Figure 1. Four ∆0 resonances due to the methylene protons of the four radical isomers were observed in the ALCµSR spectrum of 5CB, but no corresponding ∆1 resonances were observed by Lovett et al. at any of the temperatures studied (the expected ∆1 position can be calculated from the Aµ values measured by TF-µSR). The absence of ∆1 resonances in the N

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phase indicates that the dipolar interaction between the muon and the unpaired electron is small for all four radicals. The small dipolar hfcs arise from averaging of the anisotropy by the motion of the radicals, such as the rapid rotation of the radicals about the long molecular axes, small-amplitude oscillations of the long axes around their average orientation, and possibly intramolecular motion. The measurements made by Lovett et al. represent an important first step in applying µSR to studying LC materials, but there is room for improvement. Lovett et al. used a step size of 10 mT, which is not a sufficient resolution to obtain information about dynamics from the line shape of resonances with a fwhm of 20-50 mT. ALC-µSR spectra were only obtained at four temperatures, so it was not possible to fully understand the ordering and dynamics of 5CB as a function of temperature. We have made further ALC-µSR measurements on 5CB, both improving the resolution of the spectra and obtaining spectra at a large number of temperatures in the Cr, N, and I phases. Experimental Section 5CB was purchased from TCI Europe and was used without further purification. Oxygen was removed by melting the sample in an oxygen-free atmosphere and bubbling with nitrogen gas for approximately two hours. It is necessary to remove O2 from the sample because paramagnetic species can broaden the resonances in the ALC-µSR spectra by Heisenberg spinexchange.33 The degassed samples were sealed in brass cells with an internal volume of 17 mL and 50 µm titanium foil windows. The ALC-µSR experiments were performed using the ALC spectrometer at the πE3 beamline of the Paul Scherrer Institute in Villigen, Switzerland. The experimental setup and technique have been described in detail in previous publications.23,34 The samples were initially melted, and the ALC-µSR spectra were obtained at several temperatures as the sample was slowly cooled. The magnetic field was always greater than 1.4 T to produce an oriented monodomain with the director aligned along the magnetic field.35,36 The samples were left for 15-30 min to equilibrate at a given temperature before measurements were made, and several scans were performed at each temperature (each taking at least 45 min) to obtain a sufficient signal-tonoise ratio. No changes were observed in the ALC-µSR spectra as a function of time at a given temperature, so we conclude that the sample was at thermal equilibrium when the measurements were made. The ALC-µSR spectra of 5CB in the I and N phases were obtained between 1.65 and 2.15 T with a step size of 1 mT, while the ALC-µSR spectra in the Cr phase were obtained between 1.2 and 2.2 T with a step size of 2 mT. ALC-µSR spectra have a large field-dependent background that is very sensitive to the stopping position of the muons. It was not possible to remove the background by subtracting the spectrum of a substance that does not have resonances in the ALC-µSR spectrum (such as water) as the background is very sensitive to the density of the sample (which affects the stopping distribution of the muons). Each resonance was fit with a single Lorentzian function, and a fifth-order polynomial was used to model the background. The fitting was performed with the MINUIT function minimization library in the ROOT package from CERN.37 All of the ALC-µSR spectra in the I and N phases were fit between 1.72 and 2.08 T. Ab initio calculations were performed using the Gaussian 03 package of programs.38 The geometries of the radicals were optimized with the unrestricted B3LYP density functional and

Figure 2. ALC-µSR spectra of the Mu adducts of 5CB in the isotropic phase (330 K) and the nematic phase (306 and 280 K). The solid lines represent the best fits to the data.

the 6-311G(d,p) basis set, and the hfcs were calculated using the unrestricted PBE0 functional and the EPR-II basis set. These methods have been demonstrated to give hfcs close to the experimental values.39 A correction factor of 0.94 was applied to the methylene proton hfcs to account for the effect of the light mass of the muon on the structure of the radicals. This procedure for including isotope effects has been used successfully in the calculation of the hfcs of several muoniated cyclohexadienyl-type radicals.40,41 The C5H11 group was replaced by a methyl group for the calculations to reduce the computation time. The torsional potentials for the four radicals were calculated by partially optimizing the structures as a function of dihedral angle using the UB3LYP functional and the 6-31G(d,p) basis set. Results and Discussion Representative ALC-µSR spectra of 5CB in the N and I phases are shown in Figure 2. In all of the spectra in the I and N phases, we observed four resonances between 1.8 and 2.0 T, which agree with the results of Lovett et al. The TF-µSR measurements of Lovett et al. indicate that these resonances are the methylene proton ∆0 resonances of the four Mu adducts of 5CB. We did not observe any ∆1 resonances at any temperature in the N phase. The ∆0 resonance fields and methylene proton hfcs of 5CB are listed in Table 1. Lovett et al. assigned the resonances by comparing the experimental hfcs with values calculated using the empirical substituent effect method described by Roduner.22 We have performed DFT calculations to confirm their assignment and agree with the assignment of Mu-2-5CB and Mu-3-5CB. It has not been possible to differentiate between Mu-1-5CB and Mu-4-5CB based solely on the DFT calculations as the calculated hfcs are so similar (∆Ap ∼ 0.6 MHz), so we will continue to use the assignment of Lovett et al. There was no change in the ALC-µSR spectra near the literature value of the N-Cr phase transition (297 K), but we

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TABLE 1: Assignment of Resonances in the ALC-µSR Spectra of 5CB in the Isotropic Phase calculated Ap/MHz

experimental results

radical

Rodunera

DFTb

Bres330 K/mT

Ap330 K/MHzc

Mu-3-5CB Mu-2-5CB Mu-1-5CB Mu-4-5CB

113.0 115.0 119.7 120.7

120.2 126.8 136.5 135.9

1809.42 ( 0.06 1865.83 ( 0.09 1965.43 ( 0.09 1989.97 ( 0.09

108.327 ( 0.004 111.704 ( 0.005 117.667 ( 0.006 119.136 ( 0.006

a Calculated using the empirical substituent rules of Roduner (ref 22). b Calculated by Gaussian03 using the UB3LYP/6-311G(d,p)// UPBE0/EPR-II method and a factor of 0.94 to account for the secondary isotope effect. c Calculated using eq 5.

Figure 4. Temperature dependence of the methylene proton hyperfine coupling constants of the Mu adducts of 5CB: Mu-1-5CB (∆), Mu2-5CB (b), Mu-3-5CB (0), and Mu-4-5CB (1). The dotted vertical line shows the isotropic to nematic phase transition. The solid lines are the fits in the isotropic phase, extrapolated to the nematic phase. Figure 3. ALC-µSR spectrum of the Mu adducts of 5CB in the crystalline phase at 270 K. The solid line represents the best fit to the data.

observed a large change in the spectra when the sample was cooled below 275 K (Figure 3) with several broad and overlapping resonances between 1.4 and 2.0 T. The ∆1 resonances of the four types of muoniated radical are predicted to lie between 1.64 and 1.80 T and can therefore account for most of the new resonances. The resonance at ∼1.45 T is at too low a field to be due to any of the four muoniated radicals discussed previously. A ∆1 resonance at this field would correspond to Aµ ∼ 400 MHz, but we are unable to suggest a suitable radical with a muon hfc of this magnitude. The appearance of ∆1 resonances indicates that the dipolar hfc is no longer small and that the molecules are no longer rotating rapidly about their long axis. It is possible that the slowing down of the molecular motion coincides with the phase transition from the supercooled N phase to the Cr phase, but we have been unable to independently determine the temperature at which the sample crystallized. The temperature at which the sample crystallizes (TCrN) depends on the rate at which the sample is cooled, although there appears to be a discrepancy on the effect of cooling rate on TCrN in the literature. Oweimreen and Morsy determined TCrN to be 261.7 K for a cooling rate of 10 K min-1 and 264.6 K for a cooling rate of 5 K min-1, while Mansare´ et al. have reported that 5CB crystallizes at the unusually low temperature of 252 K with a cooling rate of 0.2 K min-1.42,43 A phase transition between 270 and 275 K could be consistent with our slow cooling rate (∼0.2 K min-1) and the length of time the sample is maintained at a fixed temperature. This raises the possibility that the ALC-µSR spectrum of 5CB at 284 K obtained by Lovett et al. was on a sample that was still in the N phase rather than in the Cr phase, which would explain why no ∆1 resonances were observed.

The four ∆0 resonances broadened and shifted to higher fields as the temperature was lowered in the I and N phases, but the line shape appeared to be Lorentzian at all temperatures. The methylene proton hfcs of the four Mu adducts of 5CB are shown as a function of temperature in Figure 4, and these increased linearly with decreasing temperature in the I phase, although |dAp/dT| in Mu-2-5CB (-(1.06 ( 0.05) × 10-2 MHz K-1) and Mu-3-5CB (-(6.4 ( 0.5) × 10-3 MHz K-1) is less than that of Mu-1-5CB (-(2.17 ( 0.06) × 10-2 MHz K-1) and Mu4-5CB (-(1.45 ( 0.04) × 10-2 MHz K-1). The muon and methylene proton hfcs of cyclohexadienyl-type radicals are known to increase linearly with decreasing temperature due to the vibrational motion of the CHMu group, with the slope depending on the solvent; dAp/dT of the C6H6Mu radical is -1.25 × 10-2 MHz K-1 in the gas phase,44 -2.19 × 10-2 MHz K-1 in benzene, and -1.83 × 10-2 MHz K-1 in aqueous solution.30 The dAp/dT values of Mu-1-5CB and Mu-4-5CB are within the range of values measured for the C6H6Mu radical. The methylene proton hfcs of Mu-1-5CB and Mu-4-5CB also increased approximately linearly with decreasing temperature in the N phase, while the methylene proton hfcs of Mu-2-5CB and Mu-3-5CB deviate from linearity below the isotropic-nematic phase transition temperature (TNI). The proton hfcs of Mu-2-5CB and Mu-3-5CB are lower in the N phase than one would predict based on an extrapolation of the values measured in the I phase. We define the shift of the hfc, ∆Ap, to be the difference between the measured hfc and the value extrapolated from the isotropic phase measurements. The temperature dependence of ∆Ap can be fit using a critical exponent model of the form

∆Ap ) C(TNI - T)R

(7)

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Figure 5. Temperature dependence of the hyperfine coupling constant shift, ∆Ap, for two of the Mu adducts of 5CB in the nematic phase: Mu-2-5CB (b) and Mu-3-5CB (0). The dotted vertical line shows the isotropic to nematic phase transition. The solid lines are the best fits to the data using eq 7.

with critical exponents of 0.34 ( 0.05 for Mu-2-5CB and 0.58 ( 0.10 for Mu-3-5CB (Figure 5). The hfcs of Mu-2-5CB and Mu-3-5CB shift from their isotropic values because the molecules align with the magnetic field and the hfcs are anisotropic. The hfcs are the sum of an isotropic component due to the Fermi contact interaction (AXiso) and an anisotropic component due to dipolar coupling between the electron and nuclear spins (denoted DX if the hyperfine tensor has axial symmetry). The dipolar hfcs make it possible to determine the orientation of the molecules with respect to the applied magnetic field. The Mu adducts of 5CB have axially symmetric muon and methylene proton hyperfine tensors due to fast rotation about the long axis of the molecule; one component (DX|) is parallel to the axis of rotation, and two components are perpendicular, DX⊥. The hyperfine tensor is by definition traceless so

DX|| ) -2DX⊥

(8)

Figure 6. Calculated torsional potentials (UB3LYP/6-31G(d,p)) for the Mu adducts of 5CB: Mu-1-5CB (∆), Mu-2-5CB (b), Mu-3-5CB (0), and Mu-4-5CB (1).

Wertz, and Bolton and assuming that the rotation axis is coincident with the long axis of the mesogens.45 The Dp| values for all four isomers are between 0.3 and 0.6 MHz. The fact that no shift in the value of the hfc was observed in Mu-1-5CB and Mu-4-5CB suggests that there is an additional type of motion in these two radicals that further averages the dipolar coupling and leads to Dp| ∼ 0. We have calculated the torsional potentials for all four isomers, and these are shown in Figure 6. The torsional barriers in Mu-1-5CB and Mu-4-5CB are similar to the unsubstituted biphenyl (equilibrium dihedral angle of 44.4 ( 1.2° and barriers at 0° and 90° of 6.0 ( 2.1 and 6.5 ( 2.0 kJ mol-1, respectively46-48) and are much lower than in Mu-2-5CB and Mu-3-5CB. There is a larger torsional motion of the biphenyl moieties in Mu-1-5CB and Mu-4-5CB than in Mu-2-5CB and Mu-3-5CB, and we suggest that this motion further averages the dipolar interaction and results in Dp| ∼ 0. The differences in the torsional barriers and the equilibrium dihedral angle also account for the difference in dAp/dT values. The ∆0 resonance position of the Mu adducts of 5CB will also depend on the angle that the radicals make with the magnetic field

The hfcs of the muon and the methylene proton depend on the angle between the long axis of the muoniated radicals and the magnetic field, θ, such that

1 || ∆0 2 Bres (θ) ) Θ Aiso p + Dp (3 cos θ - 1) 2

1 Aµ(θ) ) Aµiso + Dµ||(3 cos2 θ - 1) 2

The field dependence of the muon polarization due to a ∆0 resonance in the I phase is given by28

(9)

[

]

and

1 || 2 Ap(θ) ) Aiso p + Dp (3 cos θ - 1) 2

j z(B) ) 1 P

(10)

We have assumed that the angular dependence of the muon and methylene proton hfc tensors is the same. The decrease of the hfcs in the N phase of Mu-2-5CB and Mu-3-5CB indicates that these radicals have nonzero, although small, dipolar hfcs, and the absence of a shift in Mu-1-5CB and Mu-4-5CB indicates that the dipolar hfc in these radicals is close to zero. We have calculated Dp| values for the four isomers from the DFT calculated hyperfine tensors of the minimum energy structures using the method reported by Weil,

(11)

(2fR /N)ω2r ∆0 2 λ2 + ω2r + [2π(γµ - γp)(B - Bres )]

(12) where fR is the fraction of muons that have formed the radical; N is the dimension of the spin matrix; λ is the muon spin relaxation rate (1/T1µ); and ωr is the ALC transition frequency, which is given by

ωr )

πAµ Ap ∆0 Bres γe

(13)

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The ∆0 resonance has a Lorentzian line shape in the I phase, but in a powder, the shape of the resonance is obtained by integrating from θ ) 0 to π and weighting according to the probability of the molecule being at the angle θ with respect to the magnetic field, f(θ)dΩ.

j z(B) ) 1 P

(4πf /N)ω2r f(θ)sin θdθ

R ∫0π λ2 + ω2 + [2π(γ r

µ

∆0 - γp)(B - Bres (θ))]2 (14)

We have made the approximation that the ∆0 resonances of the Mu adducts of 5CB are approximately Lorentzian due to the small anisotropy, the director being aligned with the applied magnetic field, and the transition frequency being independent of field. The maximum amplitude of the resonance occurs when [f(θ) sin θ] is at a maximum. The angular distribution of 5CB molecules in the N phase can be obtained using the Maier-Saupe (MS) theory, which is a very successful model for explaining the ordering of molecules in uniaxial calamitic LC.49-52 The normalized distribution of molecules as a function of angle, f(θ), according to the MS theory is given by1 2

f(θ) )

e(m cos θ) 4πZ

(15)

where the normalization constant, Z, is given by

Z)

∫01 emx dx 2

(16)

The angular distribution depends on the order parameter, the interaction energy between molecules (U), and the temperature1

m)

3U · S 2kBT

(17)

This approach is different from that of Lovett et al., who assumed that there was an equal probability of the director lying between the angles of 0 and θM and that the director fluctuated about its mean position with a mean-squared amplitude that is proportional to temperature. The MS theory is a better approach because it has been demonstrated many times to describe the ordering in LC. Lovett et al. assumed that Dp| had a fixed value of 2 MHz and that Dµ| changed with temperature, which we consider an inappropriate approximation due to the structure of the radicals. The muon and methylene proton dipolar hfcs of cyclohexadienyl radicals should be proportional to each other, so it is not realistic to have the fitted Dµ| values smaller than the fixed Dp| value. The angle corresponding to the maximum amplitude of the ∆0 resonance, θmax, is given by

θmax ) sin-1

( ) 1

√2m

(18)

The proton hfc shift in the N phase is given by

1 ∆Ap ) Dp||(3 cos2 θmax - 1) 2

(19)

Combining eqs 17, 18, and 19, we produce an expression for the temperature dependence of ∆Ap that is valid below TNI

[

∆Ap ) Dp|| 1 -

kBT (2U · S)

]

(20)

The proton hfcs of Mu-2-5CB and Mu-3-5CB in the N phase are lower than expected based on the isotropic values because kBT/2US > 1 in the N phase. This can be demonstrated just below TNI, where the order parameter is 0.44 and kBTNI/U(TNI) ) 4.55.1 At very low temperatures, ∆Ap becomes positive and approaches Dp|, which is the expected value when all the molecules are aligned with the magnetic field (S ) 1). There is no sudden change in the magnitude of the proton hfc at TNI, which indicates that Dp| is temperature dependent and equals zero at the phase transition. The temperature dependence of Dp| is due to fluctuations of the radicals about the local director and fluctuations of the director. The slowing down of this motion is responsible for the broadening of the ∆0 resonances with decreasing temperature, which will be discussed later in this paper. We have no independent information on the temperature dependence of Dp| or U, so it is not possible to calculate S(T) solely from ALC-µSR spectra. The order parameter of 5CB has been measured by a variety of experimental techniques: birefringence,53 magnetic susceptibility,54 NMR,55,56 and IR adsorption.57 The value of S(T) depends on the experimental technique that was used, although it has been demonstrated that the different experimental methods can give fairly close temperature dependence over the nematic range after normalization to a chosen value of S well below TNI. The temperature dependence of S(T) for several classes of LC can be approximated to within one percent with the following expression, with ε approximately 1 for cyanobiphenyl LC.53,58

[

( )]

S(T) ) 0.1 + 0.9 1 - 0.99

T TNI

ε 0.25

(21)

We have assumed that U is independent of temperature (Maier and Saupe’s approximation) to calculate the temperature dependence of Dp|. The dipolar hfcs increase approximately linearly with decreasing temperature, and the slopes are proportional to the calculated dipolar hfcs for rotation solely about the long molecular axis (DFT calculations: Dp| ) 0.51 MHz for Mu-2-5CB and 0.38 MHz for Mu-3-5CB) (Figure 7). The temperature dependence of spin relaxation rates of the muoniated radicals can provide information about molecular reorientation and fluctuations of the director and the local order parameter. We have analyzed the temperature dependence of the fwhm line width of the ∆0 resonances (∆B1/2) of Mu-2-5CB and Mu-3-5CB to explore the motion of 5CB in the N phase. We have not performed a similar analysis for Mu-1-5CB and Mu-4-5CB because the resonances overlap considerably, making it much more difficult to accurately determine the line width. ∆B1/2 increases significantly with temperature in the N phase due to a decrease in motional averaging (Figure 8). The line width of the ∆0 resonances in the N phase depends on both the spin relaxation rate and the dipolar hfc, although since the latter is assumed to be very small we can obtain λ from ∆B1/2 using the isotropic formula

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Wij ) Mij2

Figure 7. Temperature dependence of the dipolar hyperfine coupling constants (Dp|) of two of the Mu adducts of 5CB in the nematic phase: Mu-2-5CB (b) and Mu-3-5CB (0).

τ 1 + ωij2 τ2

where Mij is the matrix element responsible for the transition; ωij is the transition frequency; and τ is the motional correlation time. Lovett et al. suggested, and we concur, that the muon spin relaxation is due to the modulation of the dipolar hfc by molecular reorientation, in which case Mij2 ∼ (γµBD)2, where BD is the effective fluctuating field at the muon and equals (3/2)1/2Dµ/γµ (Dµ is the averaged dipolar hfc for rapid rotation around the long molecular axis only). At most longitudinal fields, the analysis of spin relaxation is complicated as there are several transitions that are important, but at level crossings the analysis is considerably simpler because there is one dominant transition between the mixing spin states (1 S 2), so λ ) W12. Further simplifications are possible because the absence of ∆1 resonances leads us to conclude that τ < ∼50 ns and that the radicals are in the fast motional regime (ωr2τ2 , 1). The fluctuation rate under these circumstances is inversely proportional to the correlation time, so we can rewrite eq 23 to give

Mij2 1 ) τ λ

Figure 8. Temperature dependence of the fwhm line width (∆B1/2) of the ∆0 resonances for two of the Mu adducts of 5CB: Mu-2-5CB (b) and Mu-3-5CB (0). The dotted vertical line shows the isotropic to nematic phase transition. The inset graph is an Arrhenius plot of ln(fluctuation rate) versus inverse temperature for these radicals in the nematic phase.

∆B1/2 )

√ω2r + λ2

π(γµ - γp)

(22)

Freed has developed complex, general expressions for spin relaxation in liquid crystals that take director and order fluctuations into account, but these are somewhat unwieldy to use.59 We have based our analysis on the model for spin relaxation in muoniated radicals developed by Cox and Sivia.60,61 In their model they consider only the four spin states of the muon-electron system and use classical rate equations to compute the evolution of spin state populations. We assume that one type of motion dominates and that the radicals can each be characterized by a single correlation time. The transition rate between the states i and j, Wij, is given by

(23)

(24)

Lovett et al. found that the temperature dependence of τ-1 can be fitted with the Arrhenius equation and an energy barrier of 13.3 ( 0.6 kJ mol-1. We are also able to fit the temperature dependence of the fluctuation rate in the N phase with the Arrhenius equation and energy barriers of 16.2 ( 1.3 kJ mol-1 for Mu-2-5CB and 15.5 ( 1.1 kJ mol-1 for Mu-3-5CB. The fluctuation rates are very sensitive to the dipolar hfc, but this does not affect the activation energies obtained from the Arrhenius equation. The activation barriers measured with ALCµSR are slightly larger than those determined with LF-µSR, but the advantage of ALC-µSR is that we can determine the barrier for each radical individually. The barriers for Mu-2-5CB and Mu-3-5CB are the same within the experimental error. The spin relaxation rate in the I phase just above TNI is proportional to (T - T*)-1/2, where T* is a temperature slightly below TNI, due to short-range ordering and semislow motion of the correlated region.1,62 Conclusions ALC-µSR spectra of the Mu adducts of 5CB were obtained over a wide temperature range. No ∆1 resonances were observed in the N phase due to the rapid rotation of the radicals about the long axis of the molecule and fluctuations of the molecular alignment, which greatly reduces the dipolar hfcs. The ∆0 resonances have approximately Lorentzian line shape, and two of the resonances shift to lower fields in the nematic phase compared with their expected values obtained from extrapolating the temperature dependence of the hfcs in the isotropic phase. This shift is due to the ordering of the 5CB molecules with the director aligned along the magnetic field and increased dipolar coupling due to decreased molecular motion. The dipolar hfcs increase linearly with decreasing temperature in the nematic phase. The ∆0 resonances narrow substantially at higher temperatures in the N phase due to fluctuations of the LC material with an average activation barrier of ∼15.9 kJ mol-1. Acknowledgment. Support from the staff at the Laboratory for Muon Spin Spectroscopy at the Paul Scherrer Institute is

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