Iwamoto M., Kwon Y.-S., Lee T. (Eds.) Nanoscale Interface for Organic Electronics

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Time-Averaged Deuterium NMR Studies of the Dynamic Properties 209

sample at 295 K (see Fig. 6) but this difference is not caused by the

change in temperature but rather the increase in the angle between the

two fields from 44.7° to 55.0°. This increase in

causes the limiting

director orientation,

∞

, about which the director fluctuates in the

stationary state to increase (see Eq. (12)). We have predicted the time

dependence of the oscillations in

(t) using the hydrodynamic theory and

in these calculations the parameter

was assigned the same value as for

the higher temperature since

ε χ

∆ ∆

ɶ ɶ

is found to be independent of

temperature.

The magnetic relaxation time has not been determined at

288 K and so was treated as a fitting parameter; the value giving the best

fit to the results in Fig. 7 was 2.60 ms. This result is consistent with

earlier measurements of

for 5CB

as well as with measurements of

both made at higher temperatures. The agreement between the

predicted oscillations and those found experimentally is clearly very

good, again providing strong support for the validity of the analysis, the

choice of parameters and the experiment.

The director oscillations are predicted to increase in magnitude with

decreasing frequency (see Figs. 1 and 2) and to test this aspect of the

theory we have made time-resolved NMR measurements at 10 Hz using

the same 5CB-d

cell and under the original conditions, that is E

equal to

(t)/

t/ms

0 5 10 15 20 25

Fig. 7. The time dependence of the angle,

(

, made by the director with the magnetic

field for 5CB-d

at 288 K subject to a sinusoidal electric field with a frequency of 100

measured over many cycles. The solid line shows the predicted variation in

)

calculated with

= 3.45 and

= 2.60 ms for the angle between the fields equal to 55°.

A. Sugimura and G. R. Luckhurst 210

1.26 MVm

-1

of 44.7° and at 295 K. The quality of the spectra was

found to differ slightly from those shown in Fig. 5 for 100 Hz because

the minor spectral oscillations are reduced in intensity. This occurs

because at the lower frequency for the electric field the director moves

more slowly and so its orientation does not change during the acquisition

of the FID.

The time dependence of the director orientation until just

beyond the first cycle is shown in Fig. 8. The striking feature of these

results is the large angle made by the director at the maximum and the

fact that at the minimum the director returns to be essentially parallel to

the magnetic field. In addition, the form of the oscillations is not

symmetric being significantly broader at the maximum than at the

minimum, this is in qualitative agreement with the theoretical predictions

shown in Fig. 2. It can be understood in the following way. At such low

frequencies the director moves faster than the electric field and so the

director should remain parallel not to the field itself but to the effective

field resulting from the combination of the electric and magnetic fields.

The angle,

∞

, made by this effective field with the magnetic field

direction is given by Eq. (12) but now with

multiplied by a time

dependence originating from that of the electric field (see Eq. (7));

Fig. 8.

The time dependence of the director orientation with respect to the magnetic field,

(t), measured for 5CB-d

at 295 K following the application of a 10

Hz sinusoidal

electric field. The dashed line shows the time dependence of the angle between the

effect

ive field and the magnetic field direction. The solid line shows the predicted

variation with

= 3.45 and

= 1.54

ms obtained from previous measurements and with

equal to 44.7°.

t /ms

(t)/

0 10 20 30 40 50 60 70

Time-Averaged Deuterium NMR Studies of the Dynamic Properties 211

this gives

2 4 2 1/2

1 2 sin (2 )cos2

cos2 .

(1 4 sin (2 )cos2 4sin (2 ) )

ft ft

ρ π α

ρ π α π ρ

∞

+ +

(15)

The time dependence of this angle, calculated for

= 3.45, f = 10 Hz and

= 44.7°, is shown as the dashed line in Fig. 8 and we see immediately

that the director does indeed remain parallel to the effective field except

in the vicinity of the minimum where there is a small phase lag. This

is well-accounted for by the hydrodynamic theory using the same

parameters as in the calculation of

∞

and with

1.54 ms

. The result

is shown as the solid line in Fig. 8 and clearly predicts the small phase

lag. Overall the agreement between theory and experiment is good but

not quite perfect; thus the maximum angle is predicted to be about 2°

smaller than that observed. We have tried to improve the agreement by

modifying the parameters

and

but we could not obtain any

significant improvement over the good agreement that we had already

found. We also note that the calculations were, in fact, relatively

insensitive to the magnetic relaxation time since the director is moving at

almost the same rate as the effective field. The important point to note is

again the quality of the agreement with theory at this significantly lower

frequency for the electric field.

The oscillatory director motion that we have observed seems to have

elements in common with that predicted and observed for a nematic

subject to a static magnetic field and spun about an axis orthogonal to the

field.

39,40

The behaviour of the director, both static and dynamic, in this

experiment can also be obtained from the torque-balance equation

Ω sin 2 ,

d B

θ χ

γ θ

∆

 

− = −

 

 

(16)

again the elastic and inertial contributions to the torque-balance equation

are neglected here. At equilibrium, when d

/dt is zero, the solution to the

torque-balance equation gives the director orientation with respect to the

magnetic field as

Ω

sin2 ,

Ω

(17)

A. Sugimura and G. R. Luckhurst 212

where

Ω

is given by

0 1

Ω

µ γ

∆

(18)

It is of interest that this critical angular velocity,

Ω

, is just one half the

inverse of the magnetic relaxation time,

(see Eq. (2)). The theory

predicts, therefore, that as the director is spun at an angular velocity,

Ω

so the director orientation moves away from being parallel to the

magnetic field until when

Ω

is equal to

Ω

the director is at 45° to the

field. Beyond this point the theory predicts that the director orientation

will now vary with time according to

1/2

*2 1/2 * *

Ω

tan[(

Ω

1) ( )] = tan - ,

Ω

1 4

t t

 +

 

− −

 

 

−

 

 

(19)

obtained by integration of Eq. (16). Here the scaled variables are

Ω Ω

Ω

t t

and

is an arbitrary time origin. To see the

effect of this dynamic process for the director on the spectrum we

show the time dependence of P

(cos

) which is proportional to the

quadrupolar splitting (see Eq. (14)) in Fig. 9 for different scaled angular

velocities. Here the results are plotted as a function of the scaled time

starting at zero where the initial director orientation is at 45° to the

magnetic field corresponding to P

(cos

) of 1/4. For a scaled angular

Fig. 9. The time dependence of P

(cos θ) for the scaled angular velocities,

Ω

*, of 1.1

(solid line) and 2.0 (dot-dashed line) calculated from Eq. (19).

(cos

)

Ω

*=1.1

Ω

*=2.0

0 1 2 3 4 5 6 7

-0.5

0.5

Time-Averaged Deuterium NMR Studies of the Dynamic Properties 213

velocity of 1.1, that is just above the critical velocity, the value of

(cos

) decreases slowly because the magnetic torque restricts the

spinning motion of the director. However, this restraining torque

decreases as

tends to 90° and so P

(cos

) changes more rapidly. After

the director passes 90° when t* is about 2.9 the magnetic torque changes

sign and now assists in the rotation of the director, as is apparent from

the rapid change in P

(cos

) from -1/2 to 1. When P

(cos

) reaches the

value of 1 corresponding to

of 180° the magnetic torque again changes

sign and restricts the rotation of the director. This pattern of behaviour is

repeated and so the director rotation can be envisaged as being slow as it

moves from parallel to orthogonal to the magnetic field to being

extremely rapid as it moves from being orthogonal to parallel to it. The

periodicity in the director rotation is then

and not

/4. The extremes in

the director orientation during the sample rotation are unlike those

predicted for the oscillating electric field experiment which depend on

the angle between the two fields, the frequency of the electric field and

the ratio of the electric and magnetic energies. To reach these extreme

director orientations

would need to be essentially 90°, the electric

energy would have to be greater that the magnetic and the frequency

would need to be low. It seems that these conditions have been

approached but not quite reached.

Just as the dynamic behaviour in the

oscillating field experiment depends on the frequency of the field so that

in the spinning experiment depends on the angular velocity of the

sample. This is clearly apparent from the results, in Fig. 9, obtained with

a larger scaled angular velocity of 2.0. Now the scaled time taken for one

cycle, from 45° to 225°, has decreased from about 6.8 when

Ω

* is 1.1 to

1.8 when

Ω

* is increased to 2.0. In addition, the director rotation from

45° to 90° is still slower than for that from 90° to 180° but the difference

is clearly not as great as for the slower sample rotation. This difference in

the nature of the director rotation induced by sample spinning clearly

results because the larger angular velocity makes the viscous torque

greater in comparison with the magnetic torque. Such a difference is

also reflected by the scaled average angular velocity,

, which theory

predicts to be given by

* *2 1/2

(

Ω

1) ,

= −

(20)

A. Sugimura and G. R. Luckhurst 214

where the director does not rotate as fast as the sample because of an

effective friction resulting from the magnetic torque. In the limit that

Ω

* >> 1 the magnetic friction is small and so the director should rotate

with the angular velocity of the sample. These various predictions have

been studied using ESR spectroscopy in a similar manner to our NMR

experiments.

39,40

One technically important difference between the two

methods is the much smaller magnetic field strength (~0.3 T) used in

ESR relative to NMR spectroscopy (~7.0 T) which means that the

critical spinning speed is about 550 times smaller in ESR than NMR, this

reduction has clear practical advantages. ESR experiments

39,41

have

revealed that immediately the sample is spun above

Ω

the director also

starts to spin but with the slightly smaller average angular velocity

predicted. However, the oscillations in the observed spectral intensity,

caused by the director rotation, were rapidly attenuated and after

approximately 30 cycles the spectrum adopts a two-dimensional powder

pattern. This results from a random distribution of the director in the

plane orthogonal to the spinning axis. It has been postulated

that this

total loss of director coherence during sample spinning could result from

small fluctuations in the spinning speed of the sample. Although there

are clear differences in the two experiments it is certainly of interest to

ask why the director orientation remains coherent in the oscillating

electric field experiment but not in the spinning experiment. If the

postulate

concerning the spinning experiment is applicable then the

coherence might be attributed to the high stability of the electric field,

that is in E

and f. To explore this possibility we have repeated the

oscillating electric field experiment at a frequency of 100 Hz but with

white noise superimposed on this; the relative amount of noise was set at

10%. The time-resolved NMR spectra were then measured and although

there was a slight broadening of the spectral lines no loss of director

coherence was observed at least over eleven cycles. This is apparent

from the time dependence of the director orientation obtained for the

final seven cycles and shown in Fig. 10. It would appear that the stability

of the oscillating electric field employed in our experiments cannot be

responsible for the retention of the director coherence found over many

oscillations in the director orientation. Another explanation is that in

the oscillating electric field experiment the director is subject to a

Time-Averaged Deuterium NMR Studies of the Dynamic Properties 215

homogeneous field which is responsible for its motion. This is,

implicitly, the situation envisaged in the theory for the rotating director

whose motion is taken to be spatially uniform. However, in the ESR

experiment the director is rotated by spinning the sample tube and so

now the torque on the director is not uniform and has to be transmitted

from the surface of the spinning tube through elastic interactions. In

addition, this may lead to the generation of line or wall defects which

together with their migration and annihilation could disrupt the

uniformity of the director.

We turn now to the results obtained for the time-averaged NMR

spectra and these are shown in Fig. 11, for 5CB-d

at a temperature of

288 K, for a range of frequencies of the sinusoidal electric field. It is

apparent that instead of a single quadrupolar doublet expected for a

monodomain nematic there are two doublets with additional spectral

intensity between them indicating that the director adopts a range of

orientations during the application of the electric field with the limiting

values determining the extent of the two spectral halves. This does not

mean that the director orientation is randomised, at any given instant, in

the sample but rather that the individual FIDs being averaged correspond

to different director orientations of the monodomain, unlike the situation

for the spinning experiment. For the lowest frequency of the electric field

Fig. 10.

The time dependence of the director orientation with respect to the magnetic

field, θ(t), measured for 5CB-d

at 295 K after the application of a 100

Hz sinusoidal

electric field with superimposed 10% white noise. The solid line shows the dependence

predicted by the hydrodynamic theory with ρ = 3.45, τ

= 1.54 ms and α = 55.6°.

(t)/

t/ms

25 30 35 40 45 50 55

A. Sugimura and G. R. Luckhurst 216

ν/kHz

10Hz

150Hz

200Hz

250Hz

300Hz

400Hz

500Hz

1kHz

50Hz

75Hz

100Hz

20Hz

30Hz

40Hz

-50 -25 0 25 50

Fig. 11 The time-averaged NMR spectra for 5CB-d

measured for values of the

frequency, between 10 Hz and 1 kHz, of the sinusoidal electric field with E

1.41 MVm

-1

at 288 K.

the spread in the quadrupolar splitting observed in the average spectrum

is significant, corresponding to the large oscillations in the angle made

by the director with the magnetic field. As the frequency of the field is

increased so the oscillations in the director orientation are reduced in

magnitude and consequently the spread in the quadrupolar splittings. At

higher frequencies the difference in the extreme quadrupolar splittings

decreases still further until the two doublets collapse to a single doublet

indicating that the director adopts a unique orientation, in accord with the

analysis given in Sec. 2. The values of the two extreme quadrupolar

splittings can be estimated directly from the spectrum and used to

determine

max

and

min

via Eq. (14). These angles are shown in Fig. 12(a)

as a function of frequency and, as theory suggests, at low frequencies,

when the director is able to follow the oscillations in the effective field

faithfully, the two angles are independent of frequency. However, at

Time-Averaged Deuterium NMR Studies of the Dynamic Properties 217

about 10 Hz the difference in the two angles starts to decrease relatively

rapidly with

min

exhibiting greater changes than

max

, as we had

predicted (see Fig. 3). At frequencies above 300 Hz the two limiting

director orientations are essentially the same.

It is clear from a comparison of the experimental data shown in

Fig. 12(a) and the theoretical predictions given in Fig. 3 that, at a

qualitative level, the agreement between theory and experiment for the

frequency dependence of

max

and

min

is good. However, to make a

quantitative comparison we need to know the values of

and the

angle between the two fields,

. As we have seen already

takes the

value

4.37 and

is measured to be

2.60 ms at this temperature; the angle

between the two fields is 44.7°

. Τ

he frequency dependence of

max

and

min

predicted with these parameters is shown as the solid lines

in Fig. 12(a) and are clearly in good quantitative agreement with

experiment, except for

min

at low frequencies where the experimental

values are slightly larger than those predicted. This could result, at least

in part, from the large errors in the angles extracted from the quadrupolar

splittings when the director is almost parallel to the magnetic field

(see Eq. (14)). To test this possibility we have compared the primary

experimental data, namely the quadrupolar splittings, with theory.

Fig. 12.

The frequency dependence, on a logarithmic scale, of (a) the limiting director

orientations,

max

and

min

for 5CB-d

at 288 K determined from the time-

averaged

spectra and (b) the limiting quadrupolar splittings,

max min

( )

ν θ

∆ ≡

and

min max

( )

ν θ

∆ ≡

measured from the time-averaged spectra and scaled with

∆

. The predicted variation of

both angles and quadrupolar splittings, calculated with the parameters

= 4.37 and

= 2.60 ms determined previously for 5CB and with an angle,

, between the electric

and magnetic fields of 44.7°, are shown as the solid lines.

( f )/

f /Hz

min

max

(a)

(b)

(a)

0.1 1 10 100 1000 10000

∆ν(

f /Hz

∆ν

min

∆ν

max

)/∆ν

(b)

0.1 1 10 100 1000 10000

0.4

0.5

0.6

0.7

0.8

0.9

A. Sugimura and G. R. Luckhurst 218

In Fig. 12(b) we show the frequency dependence of the limiting

splittings,

max

∆

and

min

∆

; in labelling these splittings we have chosen

to use the subscripts max and min employed for the director orientations.

However, because of the relationship between the director orientation

and the splitting this leads to the counter-intuitive situation that

max

∆

smaller than

min

∆

. The frequency dependence of these limiting splittings,

predicted by theory using the same parameter set as for the director

orientation, is shown as the solid line in Fig. 12(b). The prediction of the

splittings is clearly in far better agreement with theory than for the

director orientation which shows that the discrepancy found for the

director orientation originates from the error in determining this from the

quadrupolar splittings.

An additional test of our theoretical model, employed to understand

the oscillatory motion of the director, is to simulate the time-averaged

spectrum from the predicted time dependence of the director orientation.

The theoretical spectrum is given by the time average

( ) ( , ( ( )), ) ,

I L t T dt

ν ν ν θ

−

∫



(21)

where

is the frequency,

( ( ))

ν θ

denotes the resonance frequencies for

the two components of the quadrupolar doublet and these depend on the

director orientation via

0 2

( ( )) (cos ( )) ,

t P t

ν θ ν θ

∆

 

= ±

 

 

(22)

(cf. Eq. (14)). The spectral lineshape is written as

( , ( ( )), )

L t T

ν ν θ

−

where

−

is a measure of the linewidth which we assume to be

independent of the director orientation. However, recent measurements

suggest that allowing for the angular dependence of the linewidth might

be a useful tool when attempting a detailed fit to experiment. The

lineshape is taken to be a Lorentzian, that is

2 2

( , ( ( )), ) .

1 ( ( ( )))

L t T

T t

ν ν θ

−

+ −

(23)

Since the director orientation is calculated for discrete values of time it is

convenient to replace the integral in Eq. (21) with a summation over a