Dynamics of a Smectic Liquid Crystal: Director Fluctuations and

J. Phys. Chem. 1996, 100, 15663-15669

15663

Dynamics of a Smectic Liquid Crystal: Director Fluctuations and Reorientation of Biaxial Molecules Ronald Y. Dong Department of Physics and Astronomy, Brandon UniVersity, Brandon, Manitoba Canada R7A 6A9 ReceiVed: May 7, 1996; In Final Form: June 26, 1996X

We report on the measurements of quadrupolar splittings and spectral densities in the mesophases of a partially deuterated smectogen 4-n-butoxybenzylidene-4′-n-octylaniline-2,3,5,6-d4 at two different Larmor frequencies. A broad-band multiple-pulse sequence was used to simultaneously measure the Zeeman and quadrupolar spin-lattice relaxation times. Using a model that combines molecular rotational diffusion of an asymmetric rotor and order director fluctuations, the measured spectral densities for the ring and CR deuterons in the nematic and smectic A phases were accounted quantitatively. The contributions from director fluctuations were taken to second order for both J1(ω) and J2(2ω). A global target approach was used to derive the motional parameters and a molecular prefactor for director fluctuations. It was found that the activation energy for spinning motions is lower than that for tumbling motions of the liquid crystal molecule.

I. Introduction known1,2

as an important Director fluctuations are widely source of nuclear spin relaxation in liquid crystals. These fluctuations can be described as hydrodynamic phenomena involving collective motions of a large number of molecules. Studies of director fluctuations can provide information on molecular properties such as elastic constants and rotational viscosities of the liquid crystalline medium. The first variable frequency proton T1 study3 indicated that the usual Lorentzian frequency dependence expected from the BPP theory4 was not obeyed. This led Pincus5 to derive a ω-1/2 frequency relation for the nuclear spin-lattice relaxation rate. Following Pincus, numerous authors contributed to improve and extend nuclear spin relaxation theories of liquid crystals. Lubensky6 noted the square of the nematic order parameter in the spin-lattice relaxation rate. Brochard7 and Blinc et al.8 used a complete set of elastic constants and viscosity coefficients, while Doane et al.9 considered the effects of high-frequency cutoff in the mode spectrum of director fluctuations. Besides director fluctuations, molecular rotations also occur as in normal liquids. Couplings between director fluctuations and molecular reorientations were shown10,11 to give a small cross-term between these two dynamical processes. Director fluctuations can cause spin relaxation1 of both solvent and solute spins in liquid crystals. All the above models use a small-angle (θ) approximation, where θ is the angle between the instantaneous director and its equilibrium orientation. It is well-known that in this approximation director fluctuations contribute a frequency term in the spectral density J1(ω) and have zero contributions to J2(2ω) and J0(ω). In recent years many NMR studies2 of liquid crystals have used deuterons as spin probes to measure these spectral densities of motion for solvent and/or solute molecules. To explain the observed frequency dependence in J2 for strongly ordered solutes in liquid crystals, higher-order director fluctuations (∝θ2) have been considered by Vold et al.12 and van der Zwan et al.13 The frequency dependence in J2 was calculated to be generally small. On the basis of extensive data of solutes in liquid crystals, Joghems et al.14 recently argued that this could be due to an overestimation of the effect of director fluctuations on J1(ω) X

Abstract published in AdVance ACS Abstracts, September 1, 1996.

S0022-3654(96)01304-4 CCC: $12.00

and have carried out calculations for J1 also to second order. The influence of director fluctuations for solvent spins in liquid crystals can best be studied using the field cycling technique.15 This is due to the fact that director fluctuations contain many long-wavelength modes which are effective for spin relaxation in the kilohertz regime. The rotational diffusion model16,17 has been successfully used to describe molecules reorienting in a pseudopotential of mean torque set by their neighbors. Nordio and co-workers considered reorientation of cylindrical, rigid molecules in uniaxial phases. Each molecule is characterized by a rotational diffusion tensor D C , normally defined in a frame fixed on the molecule. A number of models of increasing complexity have been proposed18-22 for rigid molecules reorienting in uniaxial and biaxial liquid crystalline phases. In particular, reorientation of asymmetric molecules in a uniaxial ordered medium has been examined by several authors, including a rigorous treatment proposed by Tarroni and Zannoni20 (the TZ model). However, applications of the TZ model remain scarce. There is one report23 in which deuterium relaxation of a biaxial solute in phase V was modeled using the TZ model. Recently, we have applied24 the TZ model to study the solvent dynamics of a liquid crystal. In the present study we apply the same model to another smectic liquid crystal 4-n-butoxybenzylidene-4′-n-octylaniline (4O.8). Although director fluctuations normally give small contributions in the megahertz region, there are at least two liquid crystals 5O.724 and 4O.825 in which director fluctuations have been invoked in part for relaxing the deuteron spins. Our choice of 4O.8 is partly motivated by recent interests14,26 on higher-order contributions from director fluctuations. Furthermore, there exists deuterium NMR data of a chain-deuterated 4O.8 sample1,25 and of the deuterated solute 1,4-diethynylbenzene (DEB)27 in 4O.8. This data indicates that both J1(ω) and J2(2ω) depend noticeably on frequency. Although the solute and the solvent data exhibit similar features, only the former has been quantitatively analyzed based on a model which combines solute reorientations and director fluctuations as relaxation mechanisms. The data analysis for the solvent has not been carried out, partly due to additional internal degrees of freedom in the end chains. The apparent minima in J2(2ω) for the chain deuterons at the nematic-smectic A phase transition, more pronounced at the CR site, was also observed © 1996 American Chemical Society

15664 J. Phys. Chem., Vol. 100, No. 39, 1996

Dong

for the acetylenic deuterons of DEB. The decrease in J2(2ω) with decreasing temperature in the nematic phase was unusual and attributed1 to effects of increasing molecular order. It was also stated that the observed frequency dependences25 in J2’s of the chain deuterons were too large to be accountable by second-order director fluctuations. In this study, we use the 4O.8-d4 sample studied in a previous study.28 The sample was specifically deuterated at the aniline ring but was also partially labeled at the first carbon (CR) in the octyl chain. This was confirmed based on the observed quadrupolar splittings.25 The ability to observe signals from both the ring and CR sites at 15.2 and 46 MHz proves to be crucial in the present study. These deuterons reflect the dynamics of the molecular core, thereby avoiding any complications of treating internal carbon-carbon bond rotations.2 The existence of the Raman data29 of nematic order parameter 〈P2〉 is useful for checking the 〈P2〉 values calculated from the quadrupolar splittings of the CR deuterons. In section II, the basic theory for rotational diffusion of asymmetric molecules and director fluctuations is outlined. Section III contains a brief description of the experimental method. The last section presents results and a global target analysis31,32 of our deuterium relaxation data at temperatures within the nematic and smectic A phases. The data in the smectic B phase is not treated here, for it may require special attention.30 II. Theory The evolution of a spin system is governed by a spin Hamiltonian. There are fluctuating terms in the Hamiltonian because of rotational and/or collective motions of liquid-crystal molecules. From the standard spin relaxation theory33 for deuterons (I ) 1), the Zeeman and quadupolar spin-lattice relaxation rates are given in terms of spectral densities JmL(mLω) by -1

TIZ

TIQ

) 3J1(ω0)

(1)

where ω0/2π is the Larmor frequency. The spectral density is simply the Fourier transform of the time autocorrelation function GmL(t)

JmL(mLω) )

2

∞ 3π (q )2 ∫0 GmL(t) cos(mLωt) dt 2 CD

(2)

where qCD ) e2qQ/h is the quadrupolar coupling constant (η ) 0), and GmL(t) may be written in terms of the Wigner rotation 2 (Ω) in the fluctuating Hamiltonian matrix Dmn

GmL(t) ) 〈Dm2 L0(ΩLQ(0)) Dm2L*0(ΩLQ(t))〉

a superimposed rotations model,34 the orientational correlation functions are 2 2 Gm(R)(t) ) ∑∑∑[d2p0(βR,Q)]2d2np(βM,R) dn′p (βM,R) gmnn′ (t) g(R) p n

n′

p

(t) (4) where m and n represent the projection indices of a rank two tensor in the L and M frames, respectively. βR,Q is the angle between the C-D bond and the ring para axis, while βM,R is the angle between the para axis and the molecular zM axis. 2 gmnn′ (t) and g(R) p (t) are the reduced correlation functions describing molecular reorientation and internal ring rotation, respectively. For the CR deuterons, there is no internal degree of freedom given our choice of the M frame, and 2 2 (βM,Q) gmnn′ (t) Gm(R)(t) ) ∑∑d2n0(βM,Q) dn′0 n

) J1(ω0) + 4J2(2ω0) -1

Figure 1. (a) Molecular structure of a 4O.8 molecule and the locations of various coordinate frames. (b) A typical deuterium NMR spectrum of our 4O.8-d4 sample.

(3)

where the Euler angles ΩLQ specify the orientation of principal axes of the electric-field-gradient (EFG) tensor with respect to the external magnetic field. The z axis of a laboratory (L) frame is defined by the external magnetic field. To evaluate GmL(t), one needs to transform the EFG tensor through successive coordinates to account for possible internal rotations and reorientation of the molecule. The case in hand is the deuterons on the aniline ring which rotates freely about its para axis. Two intermediate coordinate frames are introduced, viz. a local (R) frame whose z axis is along the ring para axis and a molecular (M) frame that is fixed onto a rigid segment (Caromatic-CR-D) of the biaxial molecular core (Figure 1). The equilibrium director of 4O.8 is along the external magnetic field. Assuming

(5)

n′

where βM,Q is the angle between the CR-D bond and the zM axis. Now gmnn′(t), the reduced correlation functions for a “rigid” asymmetric molecule in a uniaxial phase, is given by20 2 2 2 (t) ) ∫∫dΩ0 dΩ Dmn (Ω0) Dmn′ *(Ω) P1/2(Ω0) P1/2(Ω) × gmnn′

Pˆ (Ω0|Ωt) (6)

where Pˆ (Ω0|Ωt) is a symmetrized conditional probability which gives the probability of finding a molecule with an orientation Ω (≡R,β,γ) at time t if orientation of the molecule at t ) 0 was Ω0, and P(Ω), the equilibrium probability of finding a molecule at an orientation Ω, is connected to the potential of mean torque U(Ω) through the Boltzmann distribution:

P(Ω) )

exp[-U(Ω)/kT]

∫dΩ exp[-U(Ω)/kT]

(7)

It is generally accepted2 that a second rank biaxial potential can be used. For uniaxial media such as nematics, the orienting potential is independent of the Euler angle R and

U(Ω) 3 1 ) a20 cos2 β - + a22 kT 2 2

(

)

x32 sin β cos2γ 2

(8)

where a2,-2 ) a22 is used. The biaxial parameter ξ ) a22/a20 is a measure of deviation from cylindrical symmetry of the molecule and is zero for uniaxial molecules. In principle, the second rank coefficients a20 and a22 can be determined from a knowledge of order parameters Szz(〈P2〉) and Sxx - Syy. Since

Dynamics of a Smectic Liquid Crystal

J. Phys. Chem., Vol. 100, No. 39, 1996 15665

the order parameter tensor for the 4O.8 molecule is not fully known in the present study, we assume that ξ is independent of temperature and choose to input ξ to give best agreements between calculated and experimental relaxation and splitting data. The quadrupolar splitting ∆νR of the CR deuterons is given by

∆νR ) 1 3 (R) 2 d00(θbz)〈P2〉 + (Sxx - Syy)(cos2 θbx - cos2 θby) (9) - qCD 2 2

[

]

where θbx and θby are angles between the C-D bond (the principal b axis of the EFG tensor) and the molecular xM and yM axes, respectively, and θbz ) βM,Q defined above. We used (R) (R) ) 165 kHz and qCD ) 185 kHz. We have chosen yM axis qCD to lie in the plane of Caromatic-CR-D (xM axis is normal to this plane) and θbz ) ∠CCD - βM,R where ∠CCD ) 107.5°. The choice of the molecular frame and βM,R ) 2° gave 〈P2〉 values that were similar to those determined by the Raman study.29 Given the orienting potential (a20 and ξ), the rank L order parameters can be calculated from L (Ω) 〈DL0n〉 ) δm0∫dΩ P(Ω) Dmn

(10)

and 〈P2〉 ) 〈D200〉 and Sxx - Syy ) x6 Re〈D202〉. For an input ξ, 〈P2〉 was derived from ∆νR using the method of bisection.35 This of course specified the a20 and a22 values in the orienting potential at each temperature. To evaluate the correlation functions for reorientation of asymmetric molecules given by eq 6, one needs to find the conditional probability by solving the rotational diffusion equation

(11)

where the symmetrized rotational diffusion operator Γˆ is given by eq 14 of ref 20. The rotational diffusion tensor in the molecular frame is given in the notation of Tarroni and Zannoni20

[

] [

]

(12)

∑K

2 2 (βmnn′ )K(Rmnn′ )K 2 (Rmnn′ )K2 + m2ω2

gp(R)(t) ) exp[-(1 - δp0)DRt]

2 )K/F, the decay constants, are the eigenvalues and where (Rmnn′ 2 (βmnn′)K, the relative weights of the exponentials, are the corresponding eigenvectors from diagonalizing the Γˆ matrix whose elements are formed using a Wigner basis set. The solutions of the rotational diffusion equation20 could take ranks (L) up to 40 in the basis set. By combining eqs 2, 5, and 13, the spectral densities of the CR deuterons due to molecular rotations are

(15)

Again using eqs 2, 4, 13, and 15, we obtain the spectral densities for the ring deuterons

Jm(R)(mω) )

3π2 (R) 2 (qCD) ∑∑∑[d2p0(βR,Q)]2d2np(βM,R) × 2 n n′ p

∑K

2 (βM,R) dn′p

2 2 (βmnn′ )K[(Rmnn′ )K + (1 - δp0)DR]

2 [(Rmnn′ )K + (1 - δp0)DR]2 + m2ω2

(16)

In addition to molecular reorientation and ring rotation, fluctuations in the orientation of director2,12-14 may also contribute to the spectral densities of motion. Standard theories of spin relaxation in nematics by director fluctuations are based on the notion that the mean-square amplitude 〈θ2〉 of the director’s displacement is small such that terms of this and higher orders can be neglected. Recently, second-order contributions from director fluctuations are predicted in J0(ω), J1(ω), and J2(2ω):12-14 (i) (i) (ω) ) J0,DF (ω)/3 J2,DF

(17)

where

L(ω) ) ∫0 dq ∫0 qc

q2 + q′2 dq′ (q + q′2)2 + (η/K)2ω2

qc

2

(18)

with the cutoff wavevector qc ) (η/K)1/2ωc1/2, ωc is a upper cutoff frequency, K is the average elastic constant, and η is the average viscosity. S0 is a nematic order of the molecule relative to the local director and is related to 〈P2〉 according to13

(19)

where the parameter R ) kTqc/2π2K is a measure of the magnitude of director fluctuations. The standard prefactor in director fluctuations is2

3kT (η/K3)1/2 4x2π ) 3πR/x8ωc

A)

(13)

K

(14)

The reduced correlation functions for internal ring rotations about a single axis with a diffusion constant DR may be written as36

S0 ) 〈P2〉/(1 - 3R)

where F ) (Dx + Dy)/2,  ) (Dx - Dy)/(Dx + Dy), η ) Dz/F, and Dx ≡ Dxx, Dy ≡ Dyy, and Dz ≡ Dzz are the principal diffusion elements. The asymmetry parameter for diffusion  is zero for uniaxial molecules. In general, the orientational correlation functions can be written17,20 as a sum of decaying exponentials: 2 2 2 (t) ) ∑(βmnn′ )K exp[(Rmnn′ )Kt] gmnn′

3π2 (R) 2 2 (βM,Q) × (qCD) ∑∑d2n0(βM,Q) dn′0 2 n n′

(i) 2 2 2 2 (i) ) 4(qCD )]2L(ω) ) A S0 [d00(βM,Q

1 ∂Pˆ (Ω0|Ωt) ) Γˆ Pˆ (Ω0|Ωt) F ∂t

Dx 0 0 1+ 0 0 1- 0 D C ) 0 Dy 0 ) F 0 0 0 η 0 0 Dz

Jm(R)(mω) )

(20)

By integrating over a circle in {q, q′} space rather than a square, Vold et al.12 obtained from eq 18

L(ω) ) (π/8) ln[1 + (ωc/ω)2]

(21)

and for the ith deuterons (i) J2DF (2ω) )

3π2 (i) 2 2 2 2 (i) 2 1 (q ) A S0 [d00(βM,Q)] ln[1 + (ωc/2ω)2] 2 CD 3π (22)

15666 J. Phys. Chem., Vol. 100, No. 39, 1996

Dong

When second-order contributions are included in J1DF(ω), Joghems et al.14 found that it has a correction factor (1 - 4R) which reduces to one when R is very small. Thus (i) J1DF (ω) )

3π2 (i) 2 2 2 (i) 2 (q ) AS0 [d00(βM,Q)] (1 - 4R)U(ωc/ω)/xω 2 CD (23)

where the cutoff function11 U(ωc/ω) is to limit coherent modes in the director fluctuation spectrum by having a low cutoff wavelength λc of the order of the molecular length,

U(x) )

[

]

x - x2x + 1 1 + ln 2π x + x2x + 1 1 [tan-1(x2x - 1) + tan-1(x2x + 1)] (24) π

In our calculations of director fluctuations for the ring deuterons, the small βM,R angle could safely be neglected. We note that eqs 22-24 are derived for the nematic phase. In a smectic A phase, propagation of director fluctuations is restricted to the plane of the layer; the square-root law given in eq 23 should change to a linear frequency dependence and the prefactor A has to be replaced by2,38

A′ ) 3kT/2K11ζ

(25)

where ζ is a coherence length for the layers along the planar normal and K11) K22 . K33. Furthermore, if director fluctuations are slow in comparison with molecular rotations, couplings between these two motions produce10,11 a small cross-term contribution to J1(ω). Following Freed,11 this negative cross-term can be shown as (i) (ω) ) J1CR

-

3π 2

2 (i) 2 (i) )]2 ) AS02[d200(βM,Q (qCD

x8ωc π

The sample clearing point is 78.8 °C, while the transition temperatures for smectic B (SB)-smectic A (SA) and smectic A-nematic (N) are 48.8 and 63.8 °C, respectively. A homebuilt superheterodyne coherent pulse spectrometer was operated for deuterons at 15.2 MHz using a Varian 15 in electromagnet and at 46.05 MHz using a 7.1 T Oxford magnet. The sample was placed in a NMR probe whose temperature was regulated either by an air flow with a Bruker BST-1000 temperature controller or by an external oil bath circulator. The temperature gradient across the sample was estimated to be better than 0.3 °C. A typical π/2 pulse length was 4 µs. Pulse control and signal collection were done using a General Electric 1280 minicomputer. Following procedures similar to those described elsewhere,37 data reductions, including fast Fourier transform, to obtain partially relaxed spectra were carried out on a PC using Spectral Cal (Galactics Industries), while linear regressions to find the Zeeman (T1Z) and quadrupolar (T1Q) spin-lattice relaxation times were performed using Origin (Micro Cal). A broad-band Jeener-Broekaert sequence40 with the appropriate phase cycling of radio-frequency and receiver phases was used to simultaneously measure T1Q and T1Z for both labeled sites. The pulse sequence was modified using an additional monitoring 45° pulse to minimize any long-term instability of the spectrometer. This pulse was phase-cycled to have the net effect of subtracting an M∞ signal from the Jeener-Broekaert signal. Signal collection was started about 5 µs after each monitoring 45° pulse and averaged up to 2048 scans at 15.2 MHz and 400 scans at 46 MHz. The experimental uncertainty in the spin-lattice relaxation times was estimated to be within (5%. The quadrupolar splittings were measured from deuterium NMR spectra obtained by FFT the free-induction-decay signal after a π/2 pulse. These splittings were accurate to better than 1%. IV. Results and Discussion

∑K bK

(R2100)K (R2100)K2 + ω2 (26)

where the subscript CR is to denote the cross-term and bK ) (β2100)K/(β2100)1 are relative weights of exponentials which describe molecular reorientation. Here cross-term from secondorder director fluctuations are neglected. Finally, the calculated spectral densities for the ith deuterons are obtained from eqs 14, 16, and 22-26, i.e. (i) (1) (ω) ) J(i) J(i)calc 1 1 (ω) + J1DF(ω) + J1CR(ω) (i) J(i)calc (2ω) ) J(i) 2 2 (2ω) + J2DF(2ω)

When the time scales separation is inappropriate, one may need to use14,39 high-ordering models. A recent solute study14 seems to indicate that these high-ordering models are slightly inferior in explaining J2(2ω). The spin-relaxation theory outlined in this section contains some refinements including asymmetric rotational diffusion and high-order director fluctuations and is used below in an effort to understand relaxation behaviors of the ring and CR deuterons in the nematic and smectic A phases of 4O.8. III. Experimental Method The 4O.8-d4 sample was the one used in a previous study.28 It was found that the R position of the octyl chain was slightly deuterated, giving rise to a small signal as shown in Figure 1.

The spectral densities Jm(R)(mω) and Jm(R)(mω) derived from T1Q and T1Z using eq 1 are plotted versus the reciprocal temperature in the N and SA phases (SB data not shown) of 4O.8 in Figures 2 and 3, respectively. As seen from these figures, both J1(ω) and J2(2ω) are frequency dependent for the CR deuterons in all mesophases. In the case of ring deuterons both J1(ω) and J2(2ω) have a small frequency dependence in the N and SA phases. A comparison of Figures 2 and 3 with those reported in the literature25 shows that there is a general agreement on the temperature behaviors of J1(ω) and J2(2ω). However, direct matching of spectral densities show some disagreements, even allowing for different spectrometer frequencies. We believe that the differences are mainly due to a difference in the sample purity. We analyze simultaneously the CR splitting and the spectral densities of the ring and CR deuterons. For a given biaxiality ξ in the orienting potential, 〈P2〉 (or a20) is determined from ∆νR using eqs 9-10. The derived biaxial potential is then used to solve the rotational diffusion equation using the computer program described by Tarroni and Zannoni.20 In the present study, we take a global target approach,32 that is, to treat the spectral density data at different temperatures in the N and SA phases in the same fitting procedure. Smoothed data were used to obtain the J(i)exp (2ω) values at these tempera(ω) and J(i)exp 1 2 tures. The global target analysis takes advantage of the fact that target parameters of the model vary smoothly with temperature. In addition, this was found31 particularly useful when the parameters in the model were highly correlated and/ or affected by large statistical errors. An optimization routine41 (AMOEBA) was used to minimize the sum-squared error F,

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J. Phys. Chem., Vol. 100, No. 39, 1996 15667

Figure 2. Plots of experimental and calculated spectral densities for the CR deuterons in the N and SA phases. Solid curves denote calculated values at 15.2 MHz, while dashed curves at 46 MHz. O and 0 denote J1(ω0) and J2(2ω0) at 15.2 MHz, while 3 and 4 denote the corresponding J’s at 46 MHz.

Figure 4. Plots of 〈P2〉 versus the temperature (a), where the solid and dashed lines represent NMR and Raman values, respectively, and Sxx - Syy versus 〈P2〉 (b) in the N and SA phases of 4O.8.

NMR 〈P2〉 values are lower, and its value approaches the Raman 〈P2〉 just above the SB-SA transition, more so with a smaller ξ. In the global approach, the rotational diffusion constants can be assumed to obey simple Arrhenius-type relations, giving

Dx ) D0x exp[-E⊥a /RT]

(29)

Dy ) D0y exp[-E⊥a /RT]

(30)

Dz ) Dz0 exp[-E|a/RT]

(31)

DR ) DR0 exp[-ERa /RT]

(32)

where a common activation energy E⊥a is assumed24 for Dx and Dy in a first approximation. In eqs 29-32, the global parameters are preexponentials D0x , D0y , Dz0, and D0R and the activation energies for the diffusion constants E⊥a , E|a, and ERa . When such a relation does not exist for a target parameter like the prefactor A or A′ for director fluctuations, it is still possible to introduce an interpolating relation24 linking its values at different temperatures, i.e. Figure 3. Plots of experimental and calculated spectral densities for the ring deuterons in the N and SA phases. Solid curves denote calculated values at 15.2 MHz, while dashed curves at 46 MHz. Symbols are same as Figure 2.

F ) ∑∑∑∑ k

ω

i

[Jm(i)calc(mω)

-

Jm(i)exp(mω)]k2

(27)

m

where the sum over k is for nine temperatures, ω for two frequencies and m ) 1 or 2. The fitting quality factor is given by the percent mean-squared deviation,

Q ) 100F/∑∑∑∑[Jm(i)exp(mω)]k2 k

ω

i

(28)

m

To give a minimum F, ξ was found to be 0.53. The derived 〈P2〉 values based on the CR splittings in the N and SA phases are compared to those from the Raman study in Figure 4a, while Sxx - Syy versus 〈P2〉 is shown in Figure 4b. It is noted that the

A ) a0 + a1(T - Tmin) + a2(T - Tmin)2

(33)

where the temperature Tmin is taken at the SA-SB transition, and a0, a1, and a2 are the global parameters for the prefactor A. In the present study, a2 was found to be quite small and could be neglected if necessary. In order to limit the number of global parameters, we have made a (admittedly) poor approximation of using eqs 22-24 to describe order fluctuations in the SA phase. We note, however, that to distinguish between the square-root and linear frequency dependences over a narrow frequency range (15-46 MHz) is difficult. Furthermore, the high-frequency cutoff was assumed to vary linearly with temperature from 300 MHz at 350 K to 80 MHz at 322.5 K. In minimizing F, the range of cutoff frequencies appeared not to be too sensitive. These cutoff frequencies at different temperatures are consistent with those found in the solute study27 of 4O.8. We used 72 spectral densities (four at each of the nine

15668 J. Phys. Chem., Vol. 100, No. 39, 1996

Dong TABLE 2: Calculated Spectral Densities for the Cr Deuterons Due to Reorientations, Director Fluctuations, and the Cross-Term in the Global Minimizationa T (K)

J1(ω)

J1DF(ω)

J1CR(ω)

J2(2ω)

J2DF(2ω)

350

9.01 (8.97) 9.86 (9.81) 10.55 (10.49) 10.95 (10.89) 11.31 (11.25) 12.04 (11.96)

8.62 (4.03) 7.68 (3.52) 7.33 (3.26) 7.06 (3.02) 6.76 (2.73) 6.32 (2.30)

-1.04 (-1.03) -1.04 (-1.04) -1.03 (-1.02) -0.97 (-0.96) -0.88 (-0.88) -0.81 (-0.81)

4.68 (4.66) 4.86 (4.85) 5.26 (5.25) 5.78 (5.77) 6.35 (6.34) 6.91 (6.90)

1.01 (0.54) 0.31 (0.16) 0.18 (0.09) 0.12 (0.05) 0.09 (0.03) 0.06 (0.02)

345 340 335 330 325

a The numbers [in s-1] within parentheses are for 46 MHz, while those without parentheses are for 15.2 MHz.

TABLE 3: Calculated Spectral Densities for the Ring Deuterons Due to Reorientations, Director Fluctuations, and the Cross-Term in the Global Minimizationa Figure 5. Plots of derived rotational diffusion constants versus the reciprocal temperature in the N and SA phases of 4O.8.

TABLE 1: Effective Potential Coefficients of the Biaxial Potential, the Prefactor A, and r at Several Temperatures for 4O.8 Used in Our Calculations T (K)

a20

a22

A (s1/2)

350 345 340 335 330 325

-1.609 -2.283 -2.843 -3.534 -4.219 -4.658

-0.853 -1.210 -1.507 -1.873 -2.236 -2.469

1.18 × 10 1.09 × 10-4 9.24 × 10-5 7.80 × 10-5 6.95 × 10-5 6.73 × 10-5 -4

T (K)

J1(ω)

J1DF(ω)

J1CR(ω)

J2(2ω)

J2DF(2ω)

350

7.20 (7.19) 8.85 (8.84) 10.68 (10.67) 12.93 (12.91) 15.42 (15.39) 17.91 (17.87)

0.98 (0.46) 0.87 (0.40) 0.83 (0.37) 0.80 (0.34) 0.77 (0.31) 0.72 (0.26)

-0.12 (-0.12) -0.12 (-0.12) -0.12 (-0.12) -0.11 (-0.11) -0.10 (-0.10) -0.09 (-0.09)

5.61 (5.59) 5.95 (5.94) 6.35 (6.34) 6.69 (6.67) 7.05 (7.04) 7.59 (7.57)

0.11 (0.06) 0.04 (0.02) 0.02 (0.01) 0.014 (0.006) 0.010 (0.004) 0.007 (0.002)

345 340

R

335

0.127 0.064 0.043 0.030 0.023 0.018

330

chosen temperatures and of the two Larmor freuencies) to derive nine global parameters for a given ξ. When choosing the nine global parameters, two schemes were iteratively used to achieve the best F. Scheme A involved varying D0x , Dz0, D0R, E⊥a , E|a, ERa , a0, a1, and a2 with a fixed input D0y /D0x , while scheme B involved varying D0x , D0y , Dz0, D0R, E⊥a , E|a, ERa , a0, and a1 with a2 set to zero. We found that D0y /D0x ratio could be set at three (i.e.,  ) -0.5), and scheme A was finally used to achieve a Q of 0.6%. From the global minimization, the calculated spectral densities at 15.2 MHz are shown as solid curves, while those at 46 MHz as dashed curves for the CR deuterons (Figure 2) and the ring deuterons (Figure 3). These fits between the calculated and experimental spectral density data are in general acceptable except for J(R) 2 (2ω) in the SA phase. For the CR deuterons, the model predicts a shallow J2 minimum which seems to occur more toward the isotropic phase than the experimental J2 minimum. Given the assumptions used in our model, the agreement between the theory and experiment is quite satisfactory. The peculiar temperature behaviours of J1 and J2 for the CR deuterons could be the results of rotational diffusion of asymmetrical molecules in uniaxial phases. Director fluctuations have a larger effect on the CR deuterons than the ring deuterons because of their slightly favorable C-D orientations. Their second-order contributions to J(R) 2 (2ω) are indeed small, but give a small and observable frequency dependence in the N phase. We summarize the model parameters by plotting the rotational diffusion constants in Figure 5 and listing the prefactor A and R for director fluctuations in Table 1. Both A and R were found to decrease with temperature. In particular, the temperature behavior of R/A reflects our assumption that the high-frequency cutoff decreases linearly with temperature. Using η ) 1.5 × 10-2 Pa‚s at 340 K,1 the derived

325

a The numbers [in s-1] within parentheses are for 46 MHz, while those without parentheses are for 15.2 MHz.

A value at this temperature yields a K ) 0.9 ×10-12 N. This average K value appears to be less than the typical value of 5 × 10-12 N. The activation energies for Dx, Dz, and DR were found to be E⊥a ) 22.7 ( 0.9 kJ/mol, E|a ) 11.6 ( 0.4 kJ/mol, and DRa ) 45.2 ( 1.2 kJ/mol, respectively. The error limit for a particular global parameter was estimated by varying the one under consideration while keeping all other parameters identical to those for the minimum F, to give an approximate doubling in the F value. It is interesting to note that E⊥a is larger than E|a, indicating the spinning motion of the 4O.8 molecule is less hindered than the tumbling motion. This is in contradiction with the findings for the DEB solute in 4O.8. The preexponentials D0x , D0y , Dz0, and D0R were found to be 3.37 × 1012, 1.0 × 1013, 3.1 × 1011, and 2.8 × 1016 s-1, respectively. The error limits for D0x ranged from 2.6 × 1012 to 4.8 × 1012 s-1, Dz0 from 2.7 × 1011 to 3.6 × 1011 s-1, and D0R from 1.7 × 1016 to 4.2 × 1016 s-1. It would seem that both the preexponentials and activation energies were fairly well-determined for 4O.8 in the present approach. The high ERa value seems to suggest that ring rotations may be highly hindered in the SB phase.30 To compare various contributions to the spectral densities, we have listed their theoretical values for the CR deuterons in Table 2 and for the ring deuterons in Table 3 at several temperatures. As seen from these tables, molecular reorientation produces little frequency dependences in J1(ω) and J2(2ω) and is therefore in the fast motion limit. Because of the orientation of the C-D bond in the molecular frame, director fluctuations make a much bigger contribution to the spectral densities of the CR deuterons. Both J1CR(ω) and J2DF(2ω) for the ring deuterons were small enough and could be neglected. Finally, to see the importance of asymmetric diffusion, we have

Dynamics of a Smectic Liquid Crystal calculated the Q value by setting the biaxial parameter ξ ) 0 and D0x ) D0y ) 0.67 × 1013 s-1 (the average of D0x and D0y ) and leaving all other target parameters identical to those for the minimum F. Making ξ ) 0 gave larger P2 values, which range from 0.83 at 322.5 K to 0.48 at 350 K and a Q value equal to 1.95%. No attempt was made to find a minimum Q for the case of Dx ) Dy ) D⊥. In summary, our data seem to support the idea that the 4O.8 molecule diffuses like an asymmetric rotor. Even by taking director fluctuations to second order for both J1(ω) and J2(2ω), some systematic deviations between the experimental and calculated spectral densities still exist. We found that Dz > Dy > Dx and  ) -0.5 in the nematic and smectic A phases of 4O.8. The value of  is likely due to the sizable biaxiality (ξ ) 0.53) in the potential of mean torque. The global target approach is proven to be useful in determining reliable activation energies for molecular motions. When a chain-deuterated 4O.8 sample is available to us, the biaxiality ξ could be determined from modeling the chain quadrupolar splittings. This avoids inputting the unknown ξ and allows us to check whether ξ is indeed independent of temperature. Acknowledgment. The financial support of the Natural Sciences and Engineering Council of Canada is gratefully acknowledged. We are grateful to Prof. C. Zannoni for providing us his program for calculation of spectral densities for rotational diffusion of asymmetric molecules and thank N. Finlay for his technical assistance. References and Notes (1) Vold, R. L.; Vold, R. R. In The Molecular Dynamics of Liquid Crystals; Luckhurst, G. R., Veracini, C. A., Eds.; Kumer Academic: Dordrecht, 1994. (2) Dong, R. Y. In Nuclear Magnetic Resonance of Liquid Crystals; Springer-Verlag: New York, 1994. Wade, C. G. Annu. ReV. Phys. Chem. 1977, 28, 47. (3) Weger, M.; Cabane, B. J. Phys., Colloq. 1969, 30, C4-72. (4) Bloembergen, N.; Purcell, E. M.; Pound, R. V. Phys. ReV. 1948, 73, 679. (5) Pincus, P. Solid State Commun. 1969, 7, 415. (6) Lubensky, T. Phys. ReV. A 1970, 2, 2497. (7) Brochard, F. J. Phys. (Paris) 1973, 34, 411. (8) Blinc, R.; Vilfan, M.; Luzar, M.; Seliger, J.; Zagar, V. J. Chem. Phys. 1978, 68, 303. (9) Doane, J. W.; Tarr, C. E.; Nickerson, M. A. Phys. ReV. Lett. 1974, 33, 620.

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