The measured results are shown below with an order of a SSFLC cell with different doping percentage of nc-ZnO from zero to five percent. In order to observe
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the fluctuation caused by thermal, each of the cells was held at a series temperature from 35 to 65℃ and every other 10℃ in between, which falls on the smectic-C*
phase temperature range of the liquid crystals we used. Let us first see the autocorrelation functions of pure SSFLC under different thermal excitation
1E-3 0.01 0.1 1 10
1.000 1.001 1.002 1.003 1.004 1.005
Autocorrelation
delay time (sec)
pure FLC_350C pure FLC_450C pure FLC_550C pure FLC_650C
Fig. 3-3-1. The measured autocorrelation function of dynamic light scattering intensity from pure SSFLC at different temperature.
Let us first see the autocorrelation functions of pure SSFLC under different thermal excitation shown in Fig. 3.3.1. There are two distinct trends: for one thing, the autocorrelation function decays rapidly when temperature rises up. This phenomenon is reasonable and consistent with our previous analysis in chapter two, which also implies that we should gradually increase the parameter of random force c in
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simulation as the temperature rises up. Therefore, the autocorrelation function drops faster to 1, reflecting the state of completely uncorrelated. To be more specific, the more random the system gets, the time duration from correlated to uncorrelated becomes shorter. The other characteristic we can easily observe is the autocorrelation value, which could be explained theoretically by using the eqn. (2.3.1) - (2.3.3) and is given by the following deduction. First, let us begin with eqn. (2.2.21)
).
We could therefore calculate the correlation value at delay time zero ACF(τ =0)by calculating first
.
And the normalization factor
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Intuitively, when a system gets more chaotic, the signals become noisier and result in a lower correlation value [27]. This is also in agreement with our deduction in eqn.
(3.3.5), which shows that the correlation value is inverse proportional to chaos c.
However, eqn. (3.3.5) involves with at least four parameters. Without the precise value of them, it would be hard to evaluate the autocorrelation value. Fortunately, we could obtain these values by simulation. The parameters we would need for
simulation are random force c, K~
and η. Though we may not know the precise value of c, we do know its trends when temperature rises up. Besides that, the material properties η of Felix017/100 is extremely sensitive to temperature as indicated in Table 3-2-2. It almost drops to half of its original value every 10℃ increasing. By substitution the material property into eqn. (3.3.5) and supposing the elastic constant
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K does not vary abruptly with temperature, we would obtain the simulated autocorrelation values listed in table 3-3-1, where one could see the autocorrelation values gradually increase just as those of experiment data with increasing temperature.
Furthermore, we use eqn. (3.3.1) as a fitting model for dynamic light scattering from pure SSFLC. The simulation results and related parameters used are given below
35 ℃ 45 ℃
Fig. 3-3-2 The autocorrelation functions at a series temperature. The red curves represent the experiment data and the black lines are simulated from eqn. (3.3.1). The
related parameters used in simulation are listed in table 3-3-1.
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Temperature
(°K) Rotational viscosity
(10-3Pas．s) (N/m2) (s-1)
C (dimesionless)
G(0) (dimesionless)
308°K 105 0.17 0.61 1.5 1.0699
318°K 60 0.15 0.4 2.1 1.46371
328°K 35 0.12 0.292 3.5 1.51
338°K 20 0.1 0.2 6 1.52
2
2 )
d θ(π Ksin
K =~ =
K η~ τ
Table 3-3-1 The fitting parameters used in simulation.
The simulations in Fig. 3-3-2 do match the experiment results and the trend of parameter c that we expect to increase while temperature rises up is also in agreement with our experiment results.
To yield a better understanding, we further compared the four different doping levels of nc-ZnO under the same temperature. The corresponding autocorrelation functions acquired by the digital correlator hardware Flex02-01D are shown in the following figure.
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Fig. 3-3-2 The autocorrelation function of various doping level of nc-ZnO at 35, 45, 55, 65℃. The comparison at temperature 45℃ does not show clearly the variation between SSFLC cells of doping level 0, 0.5, 5% because of the scaling problem, and its zooming
figure will be shown later in Fig. 3.3.3. The black, red, blue and green lines are the measured data of pure SSFLC, 0.5% doped nc-ZnO, 1% doped nc-ZnO and 5% doped
nc-ZnO, respectively.
Fig. 3-3-3 The zooming picture of SSFLC cells with 0%, 0.5% and 5% doped nc-ZnO at 45℃. The black, red and green lines represent the measured raw data of 0%, 0.5%
and 5% doped nc-ZnO, respectively.
1E-3 0.01 0.1 1 10 100 0.5% doped nc_ZnO 1% doped nc_ZnO 5% doped nc_ZnO
1E-4 1E-3 0.01 0.1 1 10 100 0.5% doped nc_ZnO 0.5% doped nc_ZnO 0.5% doped nc_ZnO
1E-3 0.01 0.1 1 10 100 0.5% doped nc_ZnO 1% doped nc_ZnO 5% doped nc_ZnO
1E-3 0.01 0.1 1 10 0.5% doped nc-ZnO 5% doped nc-ZnO
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The comparisons give some surprising results: (1) The autocorrelation functions of SSFLC with doping level 0%, 0.5%, 5% of nc-ZnO have two things in common－
when the temperature rises up (a) the time duration from correlated to uncorrelated shorten. (b) The correlation value at small delay time increases. These two features are attributed to the abruptly changes in viscosity η as is deduced previously. (2) The measured raw data of doping level of 1% nc-ZnO have showed a very unique behavior from the others. As one could easily tell that at temperature 45, 55, 65℃
this SSFLC cell exhibits an oscillation in the autocorrelation function while none of the other cells does. (3) Except for the data of SSFLC with doping level of 5%
nc-ZnO (the green lines), the other data with doping nc-ZnO (i.e., the red and blue lines) have larger autocorrelation values at small delay time and longer time duration from correlated to uncorrelated than those of pure SSFLC’s (black lines) measured data. (4) The measured raw data of SSFLC with doping level of 5% nc-ZnO are very close to those of pure SSFLC. That is to say, the black lines at temperature 35, 45, 55 and 65℃ are almost identical to the green ones in magnitude and trends. Combining the fact with eqn. (2.2.23) and eqn. (3.3.5), we would discover that doping 5%
nc-ZnO into SSFLC probably does not affect much the macroscopic property of SSFLC such as viscosity η and the elastic constant K so that the autocorrelation
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values and the duration time from correlated to uncorrelated stay unchanged.
Let us now try to explain these observations presented above. Recall that in section 1.3, we described the smectic phase to have more spatial order than nematic phase does, which is revealed by a fact that the smectic phases has somewhat positional order while the nematic does not. This fact strongly suggests 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 solid crystal, molecular fluctuations in the ferroelectric liquid crystals are still overdamped as proved in the autocorrelation function measured with the undoped SSFLC cell shown in Fig. 3.3.1. However, something interesting happened right after we doped 1% nc-ZnO into the pure SSFLC cell which then exhibit oscillation in its autocorrelation. With the knowledge in mind, we turn ourselves back to eqn. (2.2.13) and analyze it with
thermal
Again, we employed the separation of variables to extract the dynamical part of the equation
Defining the angular velocity ω and
2 2
2 Ksin ( )
d θ π
α = ρ , we derive [28]
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The characteristic equation of the matrix
⎟⎟
By using the theory of the harmonic oscillator, we distinguish three cases [28, 29]:
1. overdamped
From eqn. (3.3.11), we now understand that the dynamics of SSFLC from overdamped to underdamped can occur with the changes of the material properties.
Furthermore, with the help of eqn. (3.3.5) and eqn. (2.2.25) we can recover the following three characteristic features that correspond to the experimental observations (2) and (3) by gradually increasing the parameters K:
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(1) the smaller c parameter resulting in the longer time duration from correlated to uncorrelated;
(2) higher value of autocorrelation at small delay time; and (3) the dynamical transition from overdamped to underdamped.
10-3 10-2 10-1 100 101 102
0 0.2 0.4 0.6 0.8 1
delay time(s)
autocorrelation
Fig. 3-3-4 A simulation result for underdamped dynamics. Note that the oscillation behavior begins at delay time 0.02 seconds, which equals to the experiment result of
SSFLC doped with 1% ZnO at 45℃.
An underdamped simulation based on eqn. (3.3.7) is given above. Overall speaking, the doping with nc-ZnO somehow changes the material properties such as K and η especially the former one which, on the other hand, is related to the order parameter [30]. To construct a possible mechanism of the doping effect, it might be better to begin with the interaction between the liquid crystals and ZnO crystals. Experimental and theoretical studies on ZnO crystals have all revealed a presence of a giant
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permanent dipole moment that is about 4800 nC/cm2 [31], which serves as a local electric field that would force the surrounding ferroelectric liquid crystals to align along its electric lines of force. As a consequence, the SSFLC doped with nc-ZnO have better spatial order [21, 30]. However, if the liquid crystals surrounded by a moderate amount of nc-ZnO (i.e., a higher doping level of nc-ZnO) would not know how to align but to follow its nature’s lead since the dipolar fields generated from each nc-ZnO are very likely to cancel one another. In short, it is because of the unchanged alignment that results in the unchanged material property and lead to similar results as those of pure SSFLC.
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Chapter 4
Probing into the Influence of the Stochastic Processes on the Field-Driven Motion in SSFLC
4.1 Introduction
Many of the physical properties of mesomorphic materials, such as birefringence, optical activity, viscosity and thermal conductivity could be affected by varying external stimulations. Electric fields, magnetic fields, thermal and acoustical excitation can be invoked to induce material responses [32]. As illustrating in Figure 4.1 for the case of liquid crystal, efforts are focused on the electro-optic effect because of the ease and efficiency of LC excitation with an applied voltage as compared with other means of stimulation. However, many more possibilities can be explored further.
Figure 4-1-1 Responsivity of LC that can be coupled via LC orientational order.
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Among these responsivities of mesomorphic materials, not many of them have been explored about how the spontaneous fluctuation interferes with a deterministic motion of the constituent molecules. Based on the kinetic theory, such a stochastic motion should always exist and accompany all kinds of activities within the substances at temperature above the absolute zero. It is also inevitable in a practical application, which may cause an unexpected result or error.
In this chapter, we shall first discuss how a random force perturbs deterministic excitation modes of FLC molecules excited by an external electric field. We will show the simulation results and compare them with the experimental investigation. Finally, we will examine how the coupling of the field-driven motion in SSFLC with the random fluctuation affected by doping of nc-ZnO.