Anisotropic self-diffusion in thermotropic liquid crystals

Anisotropic self-diffusion in thermotropic liquid crystals studied by 1H and 2H pulse-field-gradient spin-echo NMR S. V. Dvinskikh,1,* I. Furo´,2 H. Zi...

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PHYSICAL REVIEW E, VOLUME 65, 061701

Anisotropic self-diffusion in thermotropic liquid crystals studied by 1 H and 2 H pulse-field-gradient spin-echo NMR S. V. Dvinskikh,1,* I. Furo´,2 H. Zimmermann,3 and A. Maliniak1 1

Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University, SE-10691 Stockholm, Sweden 2 Division of Physical Chemistry, Department of Chemistry, Royal Institute of Technology, SE-10044 Stockholm, Sweden 3 Department of Biophysics, Max-Planck-Institut fu¨r Medizinische Forschung, D-69120 Heidelberg, Germany 共Received 25 January 2002; published 10 June 2002兲 The molecular self-diffusion coefficients in nematic and smectic-A thermotropic liquid crystals are measured using stimulated-echo-type 2 H and 1 H pulse-field-gradient spin-echo nuclear magnetic resonance 共PGSE NMR兲 combined with multiple-pulse dipolar decoupling and slice selection. The temperature dependence of the principal components of the diffusion tensor in the nematic phase follows a simple Arrhenius relationship except in the region of nematic-isotropic phase transition where it reflects, merely, the decrease of the molecular orientational order. The average of the principal diffusion coefficients in the isotropic-nematic phase transition region is close to the diffusion coefficient in the isotropic phase. At the nematic–smectic-A phase transition the diffusion coefficients change continuously. The results in nematic phase are best described in terms of the affine transformation model for diffusion in nematics formed by hard ellipsoids. In the smectic-A phase the data are interpreted using a modified model for diffusion in presence of a periodic potential along the director. DOI: 10.1103/PhysRevE.65.061701

PACS number共s兲: 61.30.⫺v, 66.10.Cb, 82.56.Lz, 76.60.⫺k

I. INTRODUCTION

Anisotropic translational diffusion in thermotropic liquid crystals 共LC兲 has attracted considerable attention in the past. This information is essential for understanding various aspects of the anisotropic molecular interactions and ordering in mesophases 关1– 6兴. Diffusion in LCs has been studied by a variety of experimental techniques including quasielastic neutron scattering 共QENS兲, magnetic resonance 关electron spin resonance 共ESR兲 and nuclear magnetic resonance 共NMR兲兴, forced Rayleigh scattering 共FRS兲, optical microscopy, and radioactive tracer diffusion. To date, a general understanding of the relationship between the diffusion processes and the orientational order in various types of thermotropic liquid crystals has been reached. Further progress in revealing the details of molecular diffusion, however, has been limited due to experimental problems. In particular, there is a lack of data in phase transition regions that are reliable enough to be confronted with theoretical models. Recently, we have reported new pulsed-field-gradient spin-echo 共PGSE兲 NMR experiments that enable accurate measurements of diffusion coefficients in anisotropic systems, such as LCs or soft solids 关7–12兴. Compared to other methods, PGSE NMR is unique because it is noninvasive, molecularly selective, and requires no foreign probes 关1,3,13–18兴. In this technique, no change of molecular properties is required, instead, the uniform labeling of the position of all molecules in the sample is achieved by spatial encoding of the NMR resonance frequency in the presence of the magnetic field gradient. The method in its original form

*Corresponding author; Electronic address: [email protected]; on leave from the Institute of Physics, St. Petersburg State University, 198904 St. Petersburg, Russia. 1063-651X/2002/65共6兲/061701共9兲/$20.00

is, however, less suitable for liquid crystalline materials where anisotropic spin interactions, such as the nuclear dipole-dipole coupling, are not averaged to zero by molecular motions. Hence, spin coherences created by radiofrequency pulses decay quickly, which leaves insufficient time for the encoding/decoding gradient pulses 关1,3兴. Our approaches are based on the early concept of combining the 1 H multiple-pulse dipolar decoupling and the PGSE technique 关19–24兴. Furthermore, a PGSE experiment, which involves deuterium stimulated echo sequence, was recently developed 关12兴 and used to determine the slow molecular diffusion (⬃10⫺14 m2 /s) in a columnar, liquid crystalline phase 关25兴. For LCs with molecular diffusion coefficients ⬎10⫺12 m2 /s these new experiments provide the diffusion coefficients with an accuracy of few percents, as demonstrated on lyotropic 关7–10兴 and thermotropic 关11,12兴 liquid crystals. In the present work, we extend our previous investigations of thermotropic nematic phases. In addition, we report diffusion measurements in a smectic-A phase. We focus here on the analyses of the experimental diffusion coefficients and, in particular, their correspondence to different dynamical models for both nematic and smectic phases. The paper is organized as follows: in Sec. II experimental details are provided, the main features of the diffusion models are presented in Sec. III. Finally, the experimental results and validity of the various models are discussed in Sec. IV. II. EXPERIMENT

The measurements were performed on three thermotropic liquid crystals formed by ethoxy-benzylidene-butyl-aniline 共EBBA兲, 4-pentyl-4 ⬘ -cyanobiphenyl 共5CB兲, and partially deuterated 4-octyl-4 ⬘ -cyanobiphenyl 共8CB兲. The EBBA sample was obtained from NIOPIK, St. Petersburg. The sample of 5CB was a kind gift from Merck, UK. The 8CB

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sample, synthesized according to previously described procedure 关26兴, was deuterated in all positions except the ␤ -CH2 group of the aliphatic chain. All samples were used without further purification. The phase transition temperatures, estimated from NMR spectra and by polarizing microscopy, were in agreement with previously tabulated values 关27兴. Both nematic and smectic phases orient homogeneously with the director parallel to the external magnetic field of the NMR spectrometer. The measurements were performed on a Bruker DMX-200 spectrometer 共4.6 T兲, operating at 200, 50, and 31 MHz for 1 H, 13C, and 2 H nuclei, respectively. A home-built multiple-tuned gradient probe was used 关8,28兴. The probe was equipped with two interchangeable gradient coils that produced magnetic field gradients either along the z 共parallel to B 0 ) or x 共perpendicular to B 0 ) axes 关28兴. The length of the 90° radio-frequency pulses were 2.3, 7.0, and 5.0 ␮ s for 1 H, 13C, and 2 H, respectively. The gradient coils were calibrated by measuring the diffusion coefficients of D2 O in the temperature range of 5 –45 °C by 2 H PGSE NMR and comparing them to literature data 关29兴. The sample temperature was stable and reproducible within ⫾0.15 K. Sample heating by the decoupling sequences was estimated by observing the shift in the isotropic-nematic transition temperature and the recycling delay was adjusted to keep the heating effect to less than 0.4 K. The typical random error of the diffusion coefficient values, estimated from the reproducibility, were about ⫾2% near the phase transition to isotropic phase and ⫾5% at the lowest temperatures. The diffusion in the isotropic phase was measured by conventional Hahn-echo or stimulated-echo-type PGSE NMR 关13,14兴. In the mesophase of protonated samples 5CB and EBBA most of the data were obtained by 1 H PGSE NMR combined with homonuclear multiple-pulse dipolar decoupling 关8兴, while a few points were also measured by the heteronuclear 13C兵 1 H其 analog of this technique 关7兴 and by method based on 1 H magic-echo 关24兴. In the partially deuterated 8CB sample most of the measurements were performed by stimulated-echo-type 2 H PGSE NMR on the signal from the methyl group 关12兴 with supplementary experiments by 1 H homonuclearly decoupled PGSE NMR on the signal from the ␤ -CH2 group.

A. Nematic phase

Chu and Moroi (CM) model. The self-diffusion coefficient in nematic phases has been derived via a parametrized form of the linear momentum time autocorrelation function in the limit of perfectly ordered clusters 关30兴. The theory relates D 储 and D⬜ , the principal components of the diffusion tensor parallel and perpendicular to the phase director, to the uniaxial molecular geometry and the order parameter S by D 储 ⫽ 具 D 典 关 1⫹2S 共 1⫺ ␳ 兲 / 共 2 ␳ ⫹1 兲兴

共1a兲

D⬜ ⫽ 具 D 典 关 1⫺S 共 1⫺ ␳ 兲 / 共 2 ␳ ⫹1 兲兴 ,

共1b兲

and

with the isotropic average defined by

具 D 典 ⫽ 共 2D⬜ ⫹D 储 兲 /3,

共1c兲

where ␳ ⫽ ␲ /(4Q) is a geometrical factor for a rodlike molecule of length L and diameter d with the axial ratio Q ⫽L/d. The theory predicts that the isotropic average is independent of S and at the transition to the isotropic phase 共i.e., in the limit S→0) coincides with D iso , the diffusion coefficient in the isotropic phase. Consequently, 具 D 典 is supposed to follow the temperature dependence extrapolated from the diffusion coefficients in the isotropic phase, with no discontinuity at the isotropic-nematic phase transition. The model has been previously applied with varying success to nematic liquid crystals: p-azoxyanisole 共PAA兲 关30,35兴, 5CB 关11,36兴, and methoxy-benzylidene-butyl-aniline 共MBBA兲 关21,35,37兴. Particularly, some agreement with the radio-tracer results in PAA was found 关30兴. Hess-Frenkel-Allen (HFA) model. The following expressions have been obtained for the diffusion in nematic phase by the affine transformation from the isotropic diffusion of hard spheres to the space of aligned uniaxial ellipsoids 关31兴: D 储 ⫽ 具 D 典 g ␣ 关 Q 4/3⫺2/3Q ⫺2/3共 Q 2 ⫺1 兲共 1⫺S 兲兴

共2a兲

D⬜ ⫽ 具 D 典 g ␣ 关 Q ⫺2/3⫹1/3Q ⫺2/3共 Q 2 ⫺1 兲共 1⫺S 兲兴 ,

共2b兲

and

where III. DIFFUSION MODELS

␣ ⫽ 关 1⫹2/3共 Q ⫺2 ⫺1 兲共 1⫺S 兲兴 ⫺1/3关 1⫹1/3共 Q 2 ⫺1 兲

In this section, a number of theoretical models for molecular diffusion in the nematic and smectic phases will be discussed. For the former, three different approaches are compared: 共i兲 a model based on a properly parametrized form of the velocity correlation function proposed by Chu and Moroi 关30兴, 共ii兲 an affine transformation model suggested by Hess et al. 关31兴, and 共iii兲 a hydrodynamic approach developed by Franklin 关32兴. In the smectic phase the modified model of Volino and co-workers 关33,34兴 for diffusion in the presence of a periodic potential will be considered. While many models have been described in the literature, those selected here have already been used by other authors for interpreting experimental diffusion results in LCs and some have been tested using molecular dynamics simulation.

⫻ 共 1⫺S 兲兴 ⫺2/3

共2c兲

for a molecule with axial ratio Q. Note that according to this model the isotropic average 具 D 典 , as defined in Eq. 共1c兲, is sensitive to the molecular orientational order and geometry. Instead, the geometric average

具 D 典 g ⫽ 共 D⬜2/3D 1/3 储 兲

共2d兲

is predicted to be independent of the molecular shape and order and becomes D iso at S⫽0. This model has been successfully tested using molecular dynamics computer simulation 关31,38兴 and also recently by comparison to NMR results 关11兴.

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Franklin model. The model is based on the hydrodynamic theory for isotropic liquids and the diffusion coefficients are defined by the following expressions 关32兴: D 储 ⫽kT 关 1/␮ f ⫹ 共 2⫹S 兲 / 共 6 ␲ ␮ 2 ⌽ ␣ 储 兲兴

共3a兲

D⬜ ⫽kT 关 1/␮ f ⫹ 共 5⫺S 兲 / 共 12␲ ␮ 2 ⌽ ␣⬜ 兲兴 ,

共3b兲

and

where k is the Boltzmann’s constant, f is the scalar friction constant, ␮ and ⌽ are geometrical parameters related to the molecular structure. Viscosity parameters ␣ 储 and ␣⬜ are given by the linear combination of five Leslie coefficients ␣ i , i⫽1 –5 共for the explicit formulas see the original paper 关32兴 and also the correction by Urbach et al. 关35兴兲. Apparently, the temperature dependence of the anisotropic radio-tracer diffusion in the nematic phase of PAA was correctly predicted using this model 关32兴. The theory involves many parameters 共friction and viscosity constants兲, which are often not readily available with sufficient accuracy. Therefore, the analysis using this model is not always feasible. In the present work we restrict it to 5CB, for which the required material properties have been reported elsewhere. B. Smectic-A phase

Volino and co-authors 关33,34兴 have modeled the diffusion in smectic-A phase with the assumption of the existence of periodic potential along the smectic director. This creates a potential barrier, which influences the molecular diffusion along the director (z coordinate兲, while the in-layer diffusion remains essentially unaffected. Thus, the temperatureindependent symmetric potential becomes V⫽⫺V 1 /2 cos共 2 ␲ z/d 兲

共4兲

where d is the layer spacing. With this potential, the equations for the diffusion in smectic-A phase are given by 共 D 储 兲 sm ⫽ 共 D 储 兲 nem 关 I 0 共 V 1 /2RT 兲兴 ⫺2

共5a兲

共 D⬜ 兲 sm ⫽ 共 D⬜ 兲 nem ,

共5b兲

and

where I 0 is the modified Bessel function of first kind and of zero order, (D 储 ) nem and (D⬜ ) nem are the diffusion coefficients in the absence of smectic positional ordering but in presence of the nematic orientational order. For the nonzero potential amplitude V 1 this model predicts discontinuity of D 储 at the nematic–smectic-A phase transition. This approach has successfully described the diffusion measured by QENS and NMR methods in some smectic LCs with a pronounced discontinuity of D 储 at the nematic-smectic phase transition 关1,39兴. IV. RESULTS AND DISCUSSION

The results of diffusion experiments in isotropic and liquid crystalline phases for 5CB, 8CB, and EBBA samples are collected in Figs. 1 and 2.

In the nematic and smectic-A liquid crystals the secondrank diffusion tensor has, due to the uniaxial symmetry of the liquid crystals, two principal values. Since the director is homogeneously oriented along the magnetic field of the spectrometer, the two principal diffusion coefficients D 储 and D⬜ are measured by magnetic field gradient held, respectively, parallel and perpendicular to the main magnetic field. In the isotropic phase the experimental diffusion coefficients for the two gradient directions coincide. A. Isotropic and nematic phases

The molecular diffusion in isotropic and nematic phases of the three compounds exhibits some common qualitative features that can be observed in Fig. 1, and summarized as follows: 共i兲 temperature dependence of the diffusion coefficient in the isotropic phase exhibits Arrhenius-type behavior with D⫽D 0 exp(⫺Eiso a /RT), 共ii兲 the relation D 储 ⬎D⬜ holds for all liquid crystalline phases; 共iii兲 the diffusion in nematic phase, except in the phase transition region, can be approxi⬜ mated using Arrhenius relationship with E a储 ⬍E iso a ⬍E a , and 共iv兲 the diffusion coefficients in the liquid crystalline and isotropic phases are related by D 储 ⬎D iso ⬇ 具 D 典 ⬇ 具 D 典 g ⬎D⬜ , where D iso is extrapolated from the isotropic phase and 具 D 典 , 具 D 典 g are the average diffusion coefficients as defined in Eqs. 共1c兲 and 共2d兲. This contrasts some previous unexpected observations where D 储 , D⬜ ⬎D iso and D 储 , D⬜ ⬍D iso . Such results obtained by foreign probe molecules 关1,40– 45兴 may be connected to their different mass and geometry, while data obtained on the mesogenic molecules 关2,21,46兴 may rather be attributed to experimental artifacts. The diffusion anisotropy D 储 /D⬜ shown in Fig. 3共a兲 is largest in EBBA; it varies from 1.8 at the phase transition to ⬇3.3 at low temperatures. For 5CB, the corresponding values are 1.6 and 2.7, respectively. In 8CB, the anisotropy reaches a maximum value of ⬇2.1 in the nematic phase and decreases to ⬇1.7 on further cooling in the smectic phase. The activation energies in the isotropic phase for 5CB, 8CB, and EBBA are collected in Table I. They agree, in general, well with other NMR results 关4,5,46 –55兴. Due to the methodical difficulties, the diffusion coefficients obtained by different methods in the nematic phase differ by as much as one order of magnitude. For 5CB, the literature diffusion data in the isotropic and nematic phase have been compiled in Ref. 关11兴. Recent measurement of diffusion 共only D 储 ) in these phases of 5CB by stray field static gradient NMR 关54兴 agree well with our results. Also, deuteron magic-echo PGSE NMR 关12兴 measurements performed recently in chain deuterated 5CB-d 11 关76兴 confirmed our previous proton data 关11兴. For 8CB, the diffusion coefficients obtained by FRS technique on probe molecules 关44兴 are underestimated by a factor of up to 2 and the observed diffusion anisotropy is significantly lower than our present value. Diffusion measurements on impurity molecules, generally, underestimate the diffusion anisotropy of the solvent, even where the guest molecule closely matches the mass and geometry of the host molecule 关41– 44,55,56兴. Moreover, the relation between average diffusion coefficients in nematic phase and D iso is am-

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FIG. 1. The temperature dependence of the diffusion coefficient D iso (䊉) in the isotropic phase and principal diffusion coefficients D⬜ (䊐) and D 储 (䉮) in the nematic liquid crystal. Their averages, 具 D 典 ⫽(2D⬜ ⫹D 储 )/3 (〫) and 具 D 典 g ⫽(D⬜ ) 2/3 (D 储 ) 1/3 共䊊兲, are also included. The dotted lines are the Arrhenius fits to the isotropic diffusion coefficient D iso , also extrapolated into the mesophase region. Dashed and solid lines are the fits to the CM model 关30兴 关Eqs. 共1兲兴 and to the HFA model 关31兴 关Eqs. 共2兲兴, respectively. 061701-4

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FIG. 2. Experimental diffusion data D⬜ (䊐) and D 储 (䉮) in the nematic and smectic-A phases of 8CB together with theoretical fits 共lines兲. D iso (䊉) is the diffusion coefficient in the isotropic phase. In the nematic phase the HFA model 关31兴 is applied with parameters as in Fig. 1共b兲. In the smectic phase and near the phase transition in the nematic phase the modified Volino and co-author’s model 关33,34兴 关Eqs. 共4兲,共5兲兴 with the potential amplitude given in Eq. 共6兲 is applied.

biguous in this case. The results by QENS in nematic chaindeuterated EBBA 关57–59兴, available only for nominally perpendicular direction, disagree with our results by a factor of ⬇2 in magnitude 共after the correction for temperature shift and presumably inhomogeneous sample orientation 关57兴兲. The activation energy, however, agrees well. The temperature dependence of diffusion coefficients within the mesophase can be tentatively decomposed in a conventional thermally activated Arrhenius-type process, analogous to one observed in the isotropic phase, and an anisotropy activated process due to the presence of molecular orientational ordering. The nematic diffusion models of Refs. 关30,31兴 共see Sec. III兲 deal with the latter 共anisotropic兲 contribution. Inspection of the temperature dependences of the average diffusion coefficients reveals that the parameters of thermally activated contribution are similar to that in the isotropic phase. Note that various theoretical models introduce different average diffusion coefficients 共see above兲. However, for small diffusion anisotropy their difference is not significant 共Fig. 1兲 and isotropic and geometric averages, 具 D 典 and 具 D 典 g , are within 5–10 % of the diffusion coefficients extrapolated from the isotropic phase. Also, the apparent activation energies are of similar magnitude. In the models, the theoretical ‘‘diffusion coefficient in the isotropic limit’’ is formally defined as the limiting value of D 储 ,⬜ at S→0. Since S⫽0 implies that the sample is con-

FIG. 3. The temperature dependence of the diffusion anisotropy D 储 /D⬜ in 5CB 共䊊兲, 8CB (䊏), and EBBA (䉱). Solid lines are calculated using the HFA model 共for 5CB and EBBA兲 and the combined HFA and Volino and co-author’s models 共for 8CB兲 with the parameters as in Figs. 1 and 2. 共b兲 The temperature dependence of the order parameter S in 5CB 共䊊兲, 8CB (䊏), and EBBA 共bold line兲 as reported in Refs. 关60– 62兴, respectively. Thin lines are the approximation by Haller function 关63兴, as described in the text.

verted to its isotropic liquid state, this limiting diffusion coefficient is expected to coincide with, or to be close to, the value extrapolated from the isotropic phase. Interpreting the experimental data in terms of the models requires the knowledge of the order parameter profile across the mesophase region. Since it was not determined in the present work, we rely on literature data. Fortunately, for the present samples numerous measurements of S have been reported. Data typically scatter within approximately 20% wide ranges due to experimental scaling problems. The relative profiles, however, are much more accurate. In our simulations, we selected 共somewhat arbitrarily兲 the order parameter profiles of Refs. 关60– 62兴 for 5CB, 8CB, and EBBA, respectively, that fall approximately in the middle of scattering intervals. The data, presented in Fig. 3共b兲, in the nematic phases were approximated by the Haller function S nem ⫽(1 ⫺T/T * ) ␥ 关63兴 with the parameters values T * ⫽308.5 K and ␥ ⫽0.162, T * ⫽312.5 K and ␥ ⫽0.150, T * ⫽351.4 K and ␥ ⫽0.182, for samples 5CB, 8CB, and EBBA, respectively.

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TABLE I. Parameters for the diffusion model simulation in the isotropic and nematic phases. Isotropic

Nematic phase CM model

E iso a

E 具aD 典

共kJ/mol兲 b

共kJ/mol兲 c

5CB

32.8

29.1

8CB

34.2

38.3

EBBA

28.7

28.7

HFA model

Q

a

4.4 共3.6 –5.6兲 3.4 共2.9–3.9兲 6.6 共5.2–9.2兲

具D典 Ea g

共kJ/mol兲 c 32.8 42.4 32.0

Q

a

2.4 共2.2–2.6兲 2.1 共1.9–2.2兲 2.9 共2.5–3.4兲

a

Values in parentheses are the range of variation of the best fit value of Q when the order parameter S is scaled by a factor of 0.9–1.1 corresponding to the scatter in the previously determined experimental order parameter. b (⫾0.5)kJ/mol. c (⫾2.0)kJ/mol.

In the smectic-A phase of 8CB the approximation S sm ⫽0.721⫻(1⫺T/306.07) 0.033 was used. CM model [30]. The two sets of diffusion coefficients, D⬜ and D 储 , were numerically fitted to Eqs. 共1a兲,共1b兲 with the axial ratio Q as the only free parameter. The activation energy E 具aD 典 , required in the model, was determined from the slope of the experimental 具 D 典 temperature dependence 共Fig. 1兲. Note that for 8CB, the points in the nematic phase close to the smectic transition temperature, where formation of pretransitional smectic clusters may influence the diffusion coefficient, were excluded from the fit. The values of Q, derived from the analysis, together with the activation energies E 具aD 典 are collected in Table I and the results are shown in Fig. 1. A number of observations can be pointed out: 共i兲 The isotropic average 具 D 典 ⫽(2D⬜ ⫹D 储 )/3 in the nematic phase can be described by the Arrhenius-type relationship, except for the phase transition region. In 5CB and EBBA samples, a small but significant discontinuity for coefficient 具 D 典 is observed at the nematic-isotropic transition, so that 具 D 典 is approximately 5% larger compared to the isotropic diffusion coefficient D iso near the transition point. No such discontinuity was detected within experimental accuracy for 8CB. 共ii兲 No obvious trend is observed for the activation energies in nematic liquid crystals compared to the isotropic phases. In fact, these are slightly lower, slightly higher, and unaffected for 5CB, 8CB, and EBBA, respectively. This is in agreement with the expectations since the activation energies in the nematic phases reflect the temperature behavior of a hypothetical isotropic diffusion coefficient, 具 D 典 . 共iii兲 The axial ratios Q determined for the three compounds are larger than expected from the molecular geometries (Q⬇3). In particular the value derived for EBBA (Q ⫽6.6) is in contradiction with the molecular picture. HFA model [31]. In analogy with the CM model analysis, the diffusion coefficients D⬜ and D 储 were fitted to Eqs. 共2a兲– 共2c兲 using the axial ratio Q as free parameter and with the 具D典 activation energy E a g determined from the slope of the

experimental 具 D 典 g temperature dependence. Also here, the points in the nematic-smectic transition region 共for 8CB sample兲 were excluded from the fit. The values of Q, derived from the analysis, and the activation energies for 具 D 典 g are included in Table I and results are shown in Fig. 1. The conclusions from the analysis can be summarized as follows: 共i兲 The geometrical average 具 D 典 g calculated from the experimental diffusion coefficients D 储 and D⬜ using Eq. 共2d兲 coincides with the line extrapolated from the isotropic phase 共Fig. 1兲. This agreement is particularly impressive for 5CB. 共ii兲 Once again we note that the activation energies for the average diffusion coefficient are similar to those determined in the isotropic phase. 共iii兲 The axial ratio parameters derived here are essentially in agreement with the values we expect from molecular geometries (Q⬇3). Somewhat low values for 5CB and 8CB may be a consequence of molecular association as was suggested from x-ray data for 5CB 关36,64兴. In a similar type of experiment on EBBA homologs no association phenomenon has been observed 关64兴. The smaller axial ratio for 8CB as compared to 5CB homolog can be explained by the fact that the longer chain inclined to the molecular core results in increasing the effective molecular diameter. Note that, the Q values for nCB are very similar to those recently obtained from the analysis of the viscosity data: Q⫽2.6 and 1.9 for 5CB and 8CB, respectively 关65兴. Franklin model [32]. The analysis using this model requires the five Leslie coefficients ␣ i (i⫽1, . . . ,5) and the scalar friction constants f in the nematic phase, which are scarcely available. The complete and relatively accurate set of Leslie coefficients can only be found for 5CB, as reported in Ref. 关66兴. Hence, analysis of diffusion in this sample only will be considered. However, accurate value of the friction constant f and its temperature dependence is still missing. Therefore the difference of two diffusion coefficients (D 储 ⫺D⬜ ) was evaluated, since in that the term containing f is canceled out 关cf. Eqs. 共3兲兴. The calculated and experimental data are compared in Fig. 4. A clear disagreement in quali-

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FIG. 4. Comparison of the experimental difference of two principal diffusion coefficients (D 储 ⫺D⬜ ) 共䊏兲 in nematic 5CB to the calculation 共䊊兲 by the Franklin model 关32兴 of Eq. 共3兲.

tative behavior is seen. This theory is based on the analogy with polymer solutions in which the number of flexible segments in the molecule is supposed to be large, which is obviously not the case for 5CB molecule. Alternatively, the inconsistence may also be caused by insufficient accuracy of experimental Leslie coefficients, especially in the phase transition region. B. Smectic-A phase

Measurements of molecular diffusion in smectic phases have been reported more frequently in the literature as compared to nematics. The ability of a viscous smectic phase to keep its director orientation at an arbitrary angle to the magnetic field allowed performing conventional PGSE NMR experiment with the director oriented at the ‘‘magic angle’’ 54.7° to the magnetic field. In such a situation, the nuclear dipolar couplings are suppressed 关1,67–70兴. This type of experiment, however, is not feasible for 8CB due to a relatively fast director reorientation characteristic of the smectic phase of this compound 关71兴. In 8CB, the relation D 储 ⬎D⬜ characteristic of nematics, holds also in the smectic-A phase 共Fig. 2兲. This result is in contradiction with theoretical expectations 关1,20,33,34,39,72兴 and also with the majority of experimental results in other compounds. Such unconventional behavior, however, has previously been observed in cyanobenzylidene-octyloxyaniline 共CBOOA兲 关70兴 and terephthal-bis-butylaniline 共TBBA兲 关68兴. Also, recent molecular dynamics simulation of the smectic-A phase of 8CB 共though, performed only on nanosecond time scale兲 predicted D 储 ⬎D⬜ 关73兴. The fact that the smectic layer structure is stable in spite of faster out-of-layer diffusion D 储 compared to in-layer motion D⬜ has been discussed in the literature 关74兴 and can be understood by assuming solid-like jump process diffusion for D 储 and liquidlike small step 共compared to molecular sizes兲 diffusion inside the layer. In contrast to nematics, the apparent activation energy in the smectic phase is larger for diffusion parallel to director 共i.e., along the smectic layers normal兲, E a储 ⬎E⬜a . Hence, if the smectic-phase temperature region would extend to lower temperatures, the two curves D 储 (T) and D⬜ (T) would cross, resulting in a

more ‘‘conventional’’ behavior with faster in-layer diffusion. This has indeed been observed in smectic-A phase of TBBA 关68兴. Another notable experimental result in 8CB is the continuous decrease of both principal diffusion coefficients observed at the nematic-smectic phase transition. On the other hand, the diffusion anisotropy behavior changes from a general increase with lowering temperature in nematic phase to the opposite in smectic phase 关Fig. 3共a兲兴. Note, that the theoretical model 关33,34,39兴 predicts discontinuity of D 储 and its activation energy at the nematic-smectic transition as a result of layered structure formation. By experiments, previous measurements were of insufficient accuracy near the transition point 关1,39,68,69兴. The analysis of the diffusion coefficients using the Volino et al. model 关Eqs. 共4兲,共5兲兴 resulted in large fitting errors and unphysical parameters. Therefore, in order to adapt the model to a situation with continuous change of diffusion coefficients at the nematic-smectic transition we empirically modified it by allowing the parameter V 1 in Eq. 共4兲 to become temperature dependent. Suitably parametrized form of function V 1 (T) is V 1 ⫽V 0 共 1⫺T/T* 兲 ␥ ,

共6兲

which reflects the temperature dependence of McMillan’s smectic order parameter 关75兴. The value of parameter T * in Eq. 共6兲 is close to the smectic-nematic transition temperature and V 0 is the potential amplitude at the low-temperature limit. The modified model converges to the original expressions of Eqs. 共4兲,共5兲 at temperatures within the smectic phase away from the transition point, where the potential of Eq. 共6兲 becomes virtually constant. The experimental results in 8CB in its smectic phase and in pre-transitional region in nematic phase were simulated by Eqs. 共4兲–共6兲, where the coefficients (D 储 ) nem and (D⬜ ) nem are calculated according to the HFA model with the parameters optimized in the nematic phase of 8CB 共see above兲. The best parameters for the potential of Eq. 共6兲 are estimated to V 0 ⫽6.5 kJ/mol, T * ⫽306.13 K, and ␥ ⫽0.150. The potential amplitude can be compared to the values 4.6 and 11.1 kJ/mol, obtained for TBBA 关33,34兴 and ethylacetoxybenzylidene-aminocinnamate 共EABAC兲 关1兴, respectively. The corresponding fit for both the nematic and smectic regions, shown in Fig. 2, demonstrates a good consistency with the experiment. Particularly, it was possible to fit the theory to the data in the nematic–smectic-A phase transition region. V. CONCLUSION

We have presented the measurements of diffusion coefficients for three mesogenic compounds 共5CB, 8CB, and EBBA兲 in their isotropic and liquid crystalline 共nematic and smectic-A) phases. The methods, based on 1 H and 2 H PGSE NMR, allow accurate measurements of diffusion coefficients in the range down to 10⫺12 m2 /s. The measurements provided information on anisotropic diffusion that was previously inaccessible.

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Since the self-diffusion of the molecules constituting the mesophase is measured instead that of probe molecules, the information is direct and the analysis and interpretation is simple. In the nematic phase, three different models were applied to experimental diffusion coefficients: 共i兲 the ChuMoroi model where the self-diffusion coefficient has been derived using a parametrized form of the linear momentum time autocorrelation function 关30兴, 共ii兲 the Hess-FrenkelAllen approach based on the affine transformation from the isotropic diffusion of hard spheres to the space of aligned uniaxial ellipsoids 关31兴, and 共iii兲 the Franklin model that rests on the hydrodynamic theory for isotropic liquids 关32兴. All three models depend on parameters that are related to the orientational order and molecular shape. In the analyses we performed numerical fits of the experimental diffusion coefficients, D⬜ and D 储 . The analysis using the hydrodynamic model 共iii兲 failed completely. Models 共i兲 and 共ii兲 were, on the other hand, able to provide fits with reasonable parameter

values. In fact, the latter produced very realistic molecular axial ratios for all three compounds. For the first time the transformation of the diffusion tensor at the nematic–smectic-A transition is accurately measured. It is found that for the 8CB sample the diffusion coefficients change continuously and ‘‘nematic-like’’ relation D 储 ⬎D⬜ persists in the observed smectic temperature range. The translational dynamics in the smectic-A phase of 8CB can be described by Volino and co-authors’ model 关33,34兴, modified for the case of the second-order nematic-smectic phase transition.

关1兴 G. J. Kru¨ger, Phys. Rep. 82, 229 共1982兲. 关2兴 F. Noack, Mol. Cryst. Liq. Cryst. 113, 247 共1984兲. 关3兴 J. Ka¨rger, H. Pfeifer, and W. Heink, Adv. Magn. Reson. 12, 1 共1988兲. 关4兴 F. Noack, St. Becker, and J. Struppe, Annu. Rep. NMR Spectrosc. 33, 1 共1997兲. 关5兴 F. Noack, in Handbook of Liquid Crystals, edited by D. Demus et al. 共Wiley-VCH, Weinheim, 1998兲, Vol. 1, Chap. 13. 关6兴 S. Miyajima, EMIS Datarev. Ser. 25, 457 共2000兲. 关7兴 S. V. Dvinskikh, R. Sitnikov, and I. Furo´, J. Magn. Reson. 142, 102 共2000兲. 关8兴 S. V. Dvinskikh and I. Furo´, J. Magn. Reson. 144, 142 共2000兲. 关9兴 S. V. Dvinskikh and I. Furo´, J. Magn. Reson. 146, 283 共2000兲. 关10兴 S. V. Dvinskikh and I. Furo´, J. Magn. Reson. 148, 73 共2001兲. 关11兴 S. V. Dvinskikh and I. Furo´, J. Chem. Phys. 15, 1946 共2001兲. 关12兴 S. V. Dvinskikh et al., J. Magn. Reson. 153, 83 共2001兲. 关13兴 E. O. Stejskal and J. E. Tanner, J. Chem. Phys. 42, 288 共1965兲. 关14兴 J. E. Tanner, J. Chem. Phys. 52, 2523 共1970兲. 关15兴 P. Stilbs, Prog. Nucl. Magn. Reson. Spectrosc. 19, 1 共1987兲. 关16兴 P. T. Callaghan, in Principles of Nuclear Magnetic Resonance Microscopy 共Clarendon Press, Oxford, 1991兲. 关17兴 W. S. Price, Concepts Magn. Reson. 9, 299 共1997兲. 关18兴 W. S. Price, Concepts Magn. Reson. 10, 197 共1998兲. 关19兴 R. Blinc, J. Pirsˇ, and I. Zupancˇicˇ, Phys. Rev. Lett. 30, 546 共1973兲. 关20兴 R. Blinc et al., Phys. Rev. Lett. 33, 1192 共1974兲. 关21兴 I. Zupancˇicˇ et al., Solid State Commun. 15, 227 共1974兲. 关22兴 M. S. Crawford, B. C. Gerstein, A.-L. Kuo, and C. G. Wade, J. Am. Chem. Soc. 102, 3728 共1980兲. 关23兴 I. Chang et al., Phys. Rev. Lett. 76, 2523 共1996兲. 关24兴 Wurong Zhang and D. G. Cory, Phys. Rev. Lett. 80, 1324 共1998兲. 关25兴 S. V. Dvinskikh, I. Furo´, H. Zimmermann, and A. Maliniak, Phys. Rev. E 共to be published兲. 关26兴 H. Zimmermann, Liq. Cryst. 4, 591 共1989兲. 关27兴 D. Demus and H. Zaschke, in Flu¨ssige kristalle in Tabellen II 共VEB Deutscher Verlag fu¨r Grundstoffindustrie, Leipzig, 1984兲.

关28兴 I. Furo´ and H. Johannesson, J. Magn. Reson., Ser. A 119, 15 共1996兲. 关29兴 R. Mills, J. Phys. Chem. 77, 685 共1973兲. 关30兴 K.-S. Chu and D. S. Moroi, J. Phys. 共Paris兲, Colloq. 36, C1-99 共1975兲. 关31兴 S. Hess, D. Frenkel, and M. P. Allen, Mol. Phys. 74, 765 共1991兲. 关32兴 W. Franklin, Phys. Rev. A 11, 2156 共1975兲. 关33兴 F. Volino, A. J. Dianoux, and A. Heidemann, J. Phys. 共France兲 Lett. 40, L583 共1979兲. 关34兴 F. Volino and A. J. Dianoux, Mol. Phys. 36, 389 共1978兲. 关35兴 W. Urbach, H. Hervet, and F. Rondelez, J. Chem. Phys. 83, 1877 共1985兲. 关36兴 A. J. Leadbetter, F. P. Temme, A. Heidemann, and W. S. Howells, Chem. Phys. Lett. 34, 363 共1975兲. 关37兴 K. Ohta, M. Terazima, and N. Hirota, Bull. Chem. Soc. Jpn. 68, 2809 共1995兲. 关38兴 M. P. Allen, Phys. Rev. Lett. 65, 2881 共1990兲. 关39兴 R. M. Richardson, A. J. Leadbetter, D. H. Bonsor, and G. J. Kru¨ger, Mol. Phys. 40, 741 共1980兲. 关40兴 G. J. Kru¨ger and H. Spiesecke, Z. Naturforsch. A 28, 964 共1973兲. 关41兴 M. Hara, S. Ichikawa, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys., Part 1 23, 1420 共1984兲. 关42兴 M. Hara et al., Jpn. J. Appl. Phys., Part 2 24, L777 共1985兲. 关43兴 M. Hara, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys., Part 1 25, 1756 共1986兲. 关44兴 H. Takezoe, M. Hara, S. Ichikawa, and A. Fukuda, Mol. Cryst. Liq. Cryst. 122, 169 共1985兲. 关45兴 T. Nishikawa, J. Minabe, H. Takezoe, and A. Fukuda, Mol. Cryst. Liq. Cryst. 231, 153 共1993兲. 关46兴 R. Blinc, B. Marin, J. Pirsˇ, and J. W. Doane, Phys. Rev. Lett. 54, 438 共1985兲. 关47兴 G. Rollmann, K. F. Reinhart, and F. Noack, Z. Naturforsch. A 34, 964 共1979兲. 关48兴 S. K. Ghosh, E. Tettamanti, and A. Ricchiuto, Chem. Phys. Lett. 101, 499 共1983兲.

ACKNOWLEDGMENTS

This work has been supported by the Carl Tryggers Foundation, Magn. Bergvalls Foundation, the Swedish Research Council 共VR兲, and the Deutscher Academischer Austauschdienst together with the Swedish Institute under Project No. 313-S-PPP-7/98.

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关49兴 M. E. Moseley and A. Loewenstein, Mol. Cryst. Liq. Cryst. 90, 117 共1982兲. 关50兴 S. K. Ghosh and E. Tettamanti, Lett. Nuovo Cimento Soc. Ital. Fis. 40, 197 共1984兲. 关51兴 M. Vilfan et al., J. Chem. Phys. 103, 8726 共1995兲. 关52兴 P. Holstein et al., J. Magn. Magn. Mater. 143, 427 共2000兲. 关53兴 N. V. Kashirin, V. D. Skirda, and I. V. Ovchinnikov, Colloid J. 62, 68 共2000兲. 关54兴 M. Vilfan et al., Magn. Reson. Imaging 19, 433 共2001兲. 关55兴 D. R. Spiegel, A. L. Thompson, and W. C. Campbell, J. Chem. Phys. 114, 3842 共2001兲. 关56兴 H. Hervet, W. Urbach, and F. Rondelez, J. Chem. Phys. 68, 2725 共1978兲. 关57兴 M. P. Fontana, M. Ricco, C. J. Carlile, and B. Rosi-Schwartz, Physica B 180-181, 726 共1992兲. 关58兴 M. Ricco et al., Mol. Cryst. Liq. Cryst. 212, 139 共1992兲. 关59兴 M. P. Fontana and G. Gallone, Phys. Scr. T57, 168 共1995兲. 关60兴 J. W. Emsley, G. R. Luckhurst, and C. P. Stockley, Mol. Phys. 44, 565 共1981兲. 关61兴 R. G. Horn, J. Phys. 共Paris兲 39, 105 共1978兲. 关62兴 M. Ylihautala, J. Lounila, and J. Jokisaari, J. Chem. Phys. 110, 6381 共1999兲.

关63兴 I. Haller, Prog. Solid State Chem. 10, 103 共1975兲. 关64兴 A. J. Leadbetter, R. M. Richardson, and C. N. Colling, J. Phys. 共Paris兲, Colloq. 36, C1-37 共1975兲. 关65兴 H. Ehrentraut and S. Hess, Phys. Rev. E 51, 2203 共1995兲. 关66兴 H. Herba, A. Szymanski, and A. Drzymala, Mol. Cryst. Liq. Cryst. 127, 153 共1985兲. 关67兴 G. J. Kru¨ger, H. Spiesecke, and R. Weiss, Phys. Lett. 51A, 295 共1975兲. 关68兴 G. J. Kru¨ger, H. Spiesecke, and R. V. Steenwinkel, J. Phys. 共Paris兲, Colloq. 37, C3-123 共1976兲. 关69兴 G. J. Kru¨ger, H. Spiesecke, R. V. Steenwinkel, and F. Noack, Mol. Cryst. Liq. Cryst. 40, 103 共1977兲. 关70兴 S. Miyajima, A. F. McDowell, and R. M. Cotts, Chem. Phys. Lett. 212, 277 共1993兲. 关71兴 J. W. Emsley, J. E. Long, G. R. Luckhurst, and P. Pedrielli, Phys. Rev. E 60, 1831 共1999兲. 关72兴 O. Parodi, J. Phys. 共France兲 Lett. 37, L-143 共1976兲. 关73兴 Y. Lansac, M. A. Glaser, and N. A. Clark, Phys. Rev. E 64, 051703 共2001兲. 关74兴 G. J. Kru¨ger and R. Weiss, J. Phys. 共Paris兲 38, 353 共1977兲. 关75兴 W. L. McMillan, Phys. Rev. A 4, 1238 共1971兲. 关76兴 S. V. Dvinskikh 共unpublished兲.

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