We made two kinds of SSFLC cells: with and without nc-ZnO doped. The ferroelectric LC material used in this experiment was FELIX-017/100 manufactured by Clariant Japan. After rubbing the alignment layer (polyimide RN1182 manufactured by Nissan Chemical Ltd.) coated on the Indium Tin Oxide (ITO) substrates, the ferroelectric LC was infused into the cell made by the above
surface-treated substrates with the 2 μm cell gap. An undoped SSFLC cell was then obtained. On the other hand, ZnO nanocrystals with 3.4 nm average diameter were prepared for being doped into the ferroelectric LC to obtain a nc-ZnO-doped SSFLC cell in the doping level of around 1 wt %.
The experimental setup was nearly the same as that adopted in the measurement of nematic cells in Fig. 2-4-1, but a small change was introduced. In a smectic phase LC, the tilt angle subtended by the LC molecules and the layer normal is sensitive to the ambient temperature [26]. Therefore, to suppress the tilt angle fluctuations caused by the ambient temperature variations, we used an accurate heating stage with
0.1 C
± ° precision. The smectic C phase’s temperature range of the ferroelectric LC
material we used is −28 C to 73 C° ° , so that keeing the measuring temperature at 28 C° is a safe choice. The experimental scheme is given below.
M M
P L D1
DP2 D ˆi 3°
ˆf ˆ1
n
SSFLC cell
Fig. 3-2-1 The scheme is similar to that shown in Fig. 2-4-1. H represents the heating stage. Note that n is the direction in one of the two orientational ˆ1 states, which is shown in Fig. 3-1-4. A function generator was used to switch the SSFLC cell between two bistable states.
As shown in the figure, we still probed the dynamics parallel to both the substrates of the cell and the scattering plane for simplifying our discussion.
3.2.2 Experimental Results and Discussion
We begin the discussion by first examining the dynamical behavior of the LC molecules in the nc-ZnO-doped SSFLC cell.
Having connected the electrodes attached on the ITO of the cell to a function generator, we applied sinusoidal driving voltages (VPP =1 V) to the cell with a series of frequencies, say 10 Hz , 100 Hz , 1 KHz , and 10 KHz to drive the LC molecules in the cell to fluctuate coercively. The corresponding autocorrelation functions calculated by the digital correlator Flex02-01D are shown in the following figure.
Fig. 3-2-2 The autocorrelation functions of the scattered light signals
correspond to externally sinusoidal driving voltages with different driving frequencies. The small hollow circles represent the measured raw data while the red lines represent the fitting curves.
The regularly oscillating behavior of the autocorrelation functions under lower driving frequencies gives us a hint that we are supposed to use a sinusoidal function as our fitting model. The fitting model thus may take the form
( ) ( ) ( )
( 2) cos dt ,
g t = ⋅a ωt ⋅H p ⋅e− +c (3.2.1) where a is the initial fluctuation amplitude, ω is the frequency of the oscillation,
( )
H p is the Heaviside function (unit step function) that stands for taking nothing until t= p, e−dt reflects the decay character of the oscillation, and c is a constant.
The fitting curves are shown as red lines in Fig. 3-2-2 and seem quite matching. The fitting results are tabulated below.
10 Hz 100 Hz 1 KHz
a 0.1394 0.1074 0.07044
ω 62.82 628.2 6282
p 0.01 0.001 0.0001
d 0.4536 5.496 37.16
c 1.006 1.013 1.005
Table 3-2-1 The fitting results of Fig. 3-2-2. Note that the parameters ω and p are correlated.
Driven Parameter Freq.
The fitting results worth noting are the parameters ω and p . The unit of ω is
radian per second. When it is transformed into Hz , we find that the fitting values of ω are completely equal to the driving frequencies of the applied sinusoidal driving
voltages. This observation strongly implies that (1) the autocorrelation function is sensitive to the dynamics under measure, and it provides the equivalent dynamical information existing in the dynamics; (2) the motion of the LC molecules is indeed modulated by the external driving voltage coercively and coherently or else the conclusion (1) will collapse; (3) the LC molecular coherent motion is switched between the two bistable states of the nc-ZnO-doped SSFLC cell with the driving frequency suggested by (2). The fourth case in Fig. 3-2-2 can not be fitted well because the LC molecules can not keep up with such a high frequency voltage modulation any more in such a high viscous environment, and the coherent behavior with the driving frequency mentioned in the conclusion (2) was not be observed any more.
Besides ω, the fitting parameter p also could provide us with something useful. If we take a closer look into Fig. 3-2-2, we could find that the initial positions in which the function profile begins to oscillate are all different under different driving frequencies. This observation suggests that p and the driving frequency may
be correlated with each other, which leads to an assumption that p and ω are correlated with each other since ω and the driving frequency are coherent, which was confirmed by the above conclusion (2). From Table 3-2-1, the fitting results give solid evidence that p and ω are absolutely correlated with each other because of the existence of the relationship ω⋅ =p constant.
Having come this far, we have obtained an informative conclusion: once we find that there exists an oscillation in the measured autocorrelation function, there must be certain kind of periodic fluctuation existing in our probing region, and even further, we can understand how fast this periodic fluctuation is oscillating, which can be simply deduced from the fitting value of either ω or p .
Let us turn our attention to the measurements of the SSFLC cells without applying any external field on them. In these measurements, unlike any before, we adopted a strategy of long time measuring. It is necessary to do so because we have no idea about what kind(s) of fluctuation(s) we shall see and how many fluctuation modes there will be in the ferroelectric LC cell with much more complicated molecular motions than those in the nematics. The knowledge background of subsection 1.3.3 has told us to adopt a measuring time over 10,000 characteristic decay time of the autocorrelation function to attain a statistical uncertainty of 1 % [11].
Compared with the fluctuation mode that corresponds to the longest characteristic
decay time, if the measuring time is not long enough, we may lose the accuracy of the dynamical information about this fluctuation mode to any extent, even lose it completely.
The measuring times we adopted in the measurements of the cases with no external field applied were all kept at 7,200 seconds. The results of the measured autocorrelation functions together with their fitting curves and their corresponding intensity (photocounts per second) traces are shown below.
( ) a ( ) b
-1000 0 1000 2000 3000 4000 5000 6000 7000 8000 90k
-1000 0 1000 2000 3000 4000 5000 6000 7000 8000 15.0k 7,200 seconds for the undoped SSFLC cell while the bottom shows the corresponding autocorrelation function (hollow circle line) and its fitting curve (red line); (b) the case for the nc-ZnO-doped SSFLC cell. Note that in both intensity traces in (a) and (b), they exhibit not only global fluctuation but also local fluctuation.
The difference deserves to be mentioned in Fig. 3-2-3 is the small oscillating ripples existing in the middle region of the autocorrelation function for nc-ZnO-doped SSFLC cell, whereas ripples do not appear for the undoped case. This obliges us to
adopt different fitting models for them, which gives
( )
( 1)1 ( 2)2 ( 3)3( 2) t s t s t s
g t = + ⋅A B e− τ + ⋅C e− τ + ⋅D e− τ (3.2.2)
for undoped SSFLC cell and
( )
( 1)1 ( 2)2 ( 3)3( ) ( )
In measuring the size of Brownian particle in chapter 1 and the anchoring strength of nematic LC cell in chapter 2, we always applied single stretched exponential function g(2)( )t = +A Bexp⎡⎣−(t τ)s⎤⎦ to fit the measured raw data since the dynamical behaviors that were measured were all close to a single fluctuation mode (in Brownian motion case, it is translational motion; in nematic LC molecular fluctuation case, it is twist fluctuation) and a single relaxation time was thus expected.
However, the situation in SSFLC seems much more complicated. In addition to the molecular conic motion that exists in the ferroelectric LC material, the restriction caused by surface constrains the LC molecular motion in switching between the two bistable states, and this may contribute a fluctuation mode. Furthermore, the existence of nc-ZnO affects some physical properties of the ferroelectric LC material [30, 31,
32], which may also contribute a different fluctuation mode. Hence, a model that describes only one single relaxation dynamics is no longer appropriate for describing the dynamics in SSFLC.
The measured raw autocorrelation function profile of the undoped SSFLC cell shown in Fig. 3-2-3 (a) gives us a hint that we can try a triple stretched exponential function of the form shown in Eq. (3.2.2) owing to the three steps appearing in its profile. The fitting results are tabulated below.
nc-ZnO-doped undoped (a) undoped (b)
A 0.9917 0.9917 0.9896
B 0.009248 0.0008995 NA
C 0.02395 0.04127 0.04199
D 0.03119 0.0174 0.01934
E 0.006359 NA NA
s1 1 1 NA
s2 0.5624 0.7454 0.7155
s3 1 0.5181 0.5208
τ1 8.782e-005 1.168e-004 NA
τ2 0.7885 0.2077 0.2058
τ3 206.7 459.9 643.8
ω 73.05 NA NA
p 0.01 NA NA
d 8.699 NA NA
Cell Parameter
Table 3-2-2 The fitting results of the two autocorrelation functions in Fig.
3-2-3. Here,“undoped (a)＂represents the fitting results by using the fitting model Eq. (3.2.2);“undoped (b)＂represents the fitting results by using a double stretched exponential functiong(2)
( )
t = + ⋅A C e−(tτ2)s2 + ⋅D e−(tτ3)s3.Note that adopting triple stretched exponential function means that we believe there may exist three exponentially decaying dynamics inside the probing region, and we mathematically combine these three exponentially decaying dynamics by simply adding up their corresponding stretched exponential functions by virtue of the linearity property of the correlation operation.
Let us try to explain what we really got from these measurements. As we have known that a ferroelectric LC molecule is in a constant conic motion, a spontaneous polarization accompanying it also rotates. However, the spontaneous polarization is induced from the unbalance of the spatial charge distribution inside the molecule. This internal unbalance is unnecessarily correlated to the ensemble behavior of the molecule. For this reason, the uncorrelation in between contributes excess degrees of freedom to the fluctuations in the ferroelectric LC material, just like that shown in Fig.
3-2-4.
ˆx plane. (b) the small fluctuations of the tilt and the polarization [15].
Now that both the tilt part and the polarization part contribute to the ensemble fluctuations as well, it is reasonable for us to treat them as different fluctuation modes.
The parallel fluctuating component (δξ and Pδ ) thus corresponds to the amplitude excitations while the perpendicular fluctuating component ( δξ⊥ and δP⊥ ) corresponds to the phase excitations [15]. It is these two excitations that contribute to the two relaxation times τ2 and τ3 in Table 3-2-2. Moreover, since our experiment was conducted at almost constant ambient temperature (with ± 0.1 C° precision), the fluctuation in tilt angle was supposed to be suppressed, which leads to the difficulty of the existence of the amplitude excitations. Therefore, τ2 was supposed to be relatively small, and this conjecture is strongly confirmed by the fitting results shown in Table 3-2-2. In other words, we reach the point that in the ferroelectric LC the phase excitations are more popular than the amplitude excitations given that the
ambient temperature is kept constant stably. So far, we have explained the physical meaning of τ2 and τ3, and the discussion seems quite reasonable. What about τ1? In our viewpoint, τ1 may possibly come from the correlation of the noises that exist in our measuring system, especially those dependent of frequency, the so-call 1 fα noise. These kinds of noises have the“pink＂character, that is, their power spectral density decays with their frequencies. According to the Wiener-Khinchin theorem, it is this spectrum decay with frequency that produces an equivalent correlation time.
Generally speaking, the correlation time of 1 fα noise is not easy to be observed when it is mixed into a relatively strong signal, but why can we extract this information? This is possibly because (1) the measuring time (7,200 seconds) was long enough (compared with the measuring time in the measurements of nematic cells) for observing it; (2) the mixing signal did not have strong correlation itself (compared with the observed strong correlation in the measurements of nematic cells), as you can see in the comparison between Fig. 3-2-3 and Fig. 1-3-6 (b) .
A final question about the fitting relaxation times may arise: how can we determine which one among τ1, τ2, and τ3 is the minor effect coming from the 1 fα noise that exists in the system? To answer this question, we made a simple test as follows: using a double stretched exponential function as our fitting model to fit the raw data measured from the undoped cell. The fitting result is shown in the third
column of Table 3-2-2. Now, we can easily tell which one is the minor term coming from noise and which ones are the dominant terms coming from the two excitation modes (amplitude and phase).
Last but not least, let us examine the difference between the autocorrelation functions measured from the undoped SSFLC cell and the nc-ZnO-doped SSFLC cell.
With nc-ZnO doped, the improved molecular alignment quality is expectd and was really achieved in our lab. This mainly originates from the dipole-dipole interaction between the nc-ZnO and the C=O groups of the surrounding ferroelectric LC. Each ZnO quantum dot provides a dipolar interaction with the surrounding ferroelectric LC molecules and forces them to align parallel to its electric polarization resulting from the internal dipole moment [29].
Note that in section 3.1, we have understood that the smectic phase has more ordered spatial arrangement of the molecules than nematic phase does, which is revealed by a fact that the smectic phase has somewhat positional order while the nematic does not.
This fact strongly hints that the dynamical behavior in the smectic phase structure is supposed to be closer to that in the solid crystal structure than that in the nematic phase structure. But unlike the underdamped dynamics of the phonons in the soild crystal, molecular fluctuations in the ferroelectric LC are still overdampted, as proved in the autocorrelation function measured from the undoped SSFLC cell shown in Fig.
3-2-3 (a). However, something really interesting happened right after doping nc-ZnO into the pure ferroelectric LC cell. In Fig. 3-2-2, the LC molecules driven by the external electric field switched between the two bistable states regularly, which leads to an periodically oscillating autocorrelation function; in Fig. 3-2-3 (b), the autocorrelation function exhibits the similar oscillating part to that in Fig. 3-2-2 without any externally applied electric field. This observation strongly implies that the fluctuations in the ferroelectric LC experience a dynamical change from completely overdamped to partially overdamped and partially underdamped after doped with nc-ZnO. We attribute this underdamped component of the whole fluctuations to the better spatial molecular arrangement coming from the intrinsically positional order and the doping-induced better orientational order. It is this doping-induced effect that makes this smectic phase LC in the cell be much closer to solid crystal, and makes the molecules in the cell encounter less resistance for oscillating underdampingly. The nc-ZnO helps the molecules in the cell overcome the energy barrier put up by the surface anchoring successfully and switch between the two bistable states freely. This free switching between the two bistable states contributes another fluctuation mode to the system. However, instead of overdamped, it is a periodically underdamped mode.
Furthermore, the fitting result in Table 3-2-2 also tell us that the frequency of this underdamped oscillation is 73.05 rad/ sec., which approximately equals 11.63 Hz.
q
The doping level we used in the nc-ZnO-doped SSFLC cell was around 1 wt%, this leads to an estimation of nFLC ≈15000 ferroelectric LC molecules surrounding one ZnO nanoparticle in a spherical space of Rmax 14 nm [29]. Because of this wide gap of ratio, not all of the molecules in the cell are affected severely by the nc-ZnO, which is illustrated in the following figure.
P
2 mμ
PZnO
P
Fig. 3-2-6 In the upper figure, the left two circles represent the orbits traversed by the ends of the FLC molecules. One of them is in the unaffected region while the other one is in the balanced region; the left figure show the conic motion of the LC molecules in the balanced region.
Fig. 3-2-5 The three fluctuation modes observed in the measurement. q represents the scattering vector.
Two mechanisms competed in affecting LC molecular dynamical behavior. As shown in this figure, the behavior of the LC molecules in the unaffected region far from the nc-ZnO was still dominated by the effect of the surface anchoring, so the bistable states were preferred in this region and the overdamped fluctuation survived;
in the balanced region, the existence of the nc-ZnO, on account of highly spatial order coming from positional order and doping-induced orientational order, made the LC molecules get rid of the influence of the surface anchoring and recover their primitive conic motions. In the course of conic motion the LC molecular polarization were somewhat affected by that of the nc-ZnO. This effect is illustrated in the bottom figure of Fig. 3-2-6, when a LC molecule tended to rotate conically, say at position (4) for example, its polarization felt the existence of the polarization of the nc-ZnO so that it speeded up toward the position at which its polarization could be parallel to that of the nc-ZnO. Therefore, an accelerated pathway (4)→(5)→(6)→(1) was possibly set up. Likewise, a decelerated pathway (1)→(2)→(3)→(4) was also expected. Once a motion in such pathways happened, it can bring about a periodic motion between the two bistable states. This picture gives a direct insight into the existence of the oscillating behavior observed in the autocorrelation function measured from the nc-ZnO-doped SSFLC cell.
Ch C h ap a pt te er r 4 4. . Co C on n cl c lu us s io i on n s s
To summarize, this thesis study confirmed several facts that have been well developed in some other ways, provided several intuitively physical pictures for several complex theoretical notions, and observed a dynamical phenomenon never found before. Recapitulating as follows,
(1) The hydrodynamic diameters of the nanoparticles suspending in the solution can be determined accurately by the dynamic light scattering technique.
(2) The theoretical derivations of the relationships between the measured correlation time and the cell’s anchoring strength were achieved.
(3) The fact that the anchoring strength of the photoalignment cell is smaller than that of the mechanical rubbing cell was confirmed.
(4) Liquid crystal polymers do improve the surface alignment quality.
(5) The autocorrelation functions measured from the SSFLC were observed successfully; the fitting results are in accordance with the intuitive physical models.
(6) A transformation of fluctuation mode in the measurement of the nc-ZnO-doped SSFLC cell from completely overdamped to partially overdamped and partially underdamped was discovered.
(7) A model responsible for the explanation of the transformation in (6) was presented.
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