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Comparative Study of DirectorResponses in IPS and VA ModesBased Upon Physical Propertiesof Practical LC MaterialsShohei Naemura aa Atsugi Technical Center, Liquid Crystals Division,Merck Ltd., Nakatsu, Kanagawa, JapanPublished online: 31 Aug 2006.
To cite this article: Shohei Naemura (2005) Comparative Study of DirectorResponses in IPS and VA Modes Based Upon Physical Properties of PracticalLC Materials, Molecular Crystals and Liquid Crystals, 433:1, 13-27, DOI:10.1080/15421400590955550
To link to this article: http://dx.doi.org/10.1080/15421400590955550
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Comparative Study of Director Responses in IPSand VA Modes Based Upon Physical Propertiesof Practical LC Materials
Shohei NaemuraAtsugi Technical Center, Liquid Crystals Division, Merck Ltd.,Nakatsu, Kanagawa, Japan
The director response times of pure IPS and VA modes were compared by insertingrespective material constant values of practical LC mixtures into the equationsbased upon the Leslie-Ericksen theory. The VA mode was confirmed to be superiorto the IPS mode from this aspect due to the difference of the Frank elastic constantsfor bend and twist. Further superiority was shown quantitatively because of thecontribution of the back-flow effect in the VA mode. These results were supportedby referring to the molecular theory of the physical properties of LC substances.
Keywords: Frank elastic constants; nematic liquid crystal; response time; switching
INTRODUCTION
Both IPS (In-Plane-Switching) and VA (Vertically Aligned) modes areconsidered to be the most promising electro-optic effects of nematicliquid crystals (LCs) for uses in large LC-TV screens with features ofhigh image quality of moving pictures. There still exists the request,however, to improve the image quality further and many effortshave been being made to actualize the quasi-impulse response in LCdisplays (LCDs), by introducing the over-driving method together withthe blinking backlight technology, for example. In order to make this
The author wishes to express his appreciation to colleagues of LC R&D of Merck Ltd.,Japan, and Merck KGaA, Germany, for the formulation of LC mixtures and the mea-surements of their physical properties. Special thanks are to Dr. Tarumi of Merck KGaAfor many helpful discussions and Mr. Nakazono of Merck Ltd., Japan, for the numericalanalysis of the accumulated data.
Address correspondence to Shohei Naemura, Atsugi Technical Center, LiquidCrystals Division, Merck Ltd., 4084 Nakatsu, Aikawa-machi, Aikou-gun, Kanagawa243-0303, Japan. E-mail: [email protected]
Mol. Cryst. Liq. Cryst., Vol. 433, pp. 13–27, 2005
Copyright # Taylor & Francis Inc.
ISSN: 1542-1406 print=1563-5287 online
DOI: 10.1080/15421400590955550
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approach successful, faster electro-optic switching characteristics areessentially needed to LCs.
In this study, some physical properties of practical-use LC mixtureswill be reviewed and will be used for the quantitative comparison ofthe director switching speed both in IPS and VA modes. Here, theswitching times to be considered correspond to the relaxation of thedirector when an electric field is applied or removed. Moreover, whena fluid motion can be caused by the reorientation of the director, therelaxation of the vortices is also taken into account. This additionalcontribution is generally called as the back-flow effect. As the relax-ation time depends on the field strength, the comparison will be madeat a constant field normalized by the threshold of Freedericksztransition, in which IPS and VA modes differ.
From practical application point of view, the comparison is prefer-ably to be made on optical switching and further more on gray-scaleswitching. As this makes the comparison extremely complicated, how-ever, the present study will be limited to the motion of the director andconsequently any optical property of LC materials will not be involved.The expansion of the present study to more practical arguments onoptical switching will be left for further consideration in more sophis-ticated manner.
In the followings, a brief review will be made first on the theoreti-cal investigations of the dynamics of nematic LCs in the basic IPSand VA display panel configurations within the framework of theLeslie-Ericksen theory. This will provide an idea that Frank elasticconstants and Leslie viscous coefficients are the direct contributorsamong the physical properties of LC materials. This will be followedby a review of the measurements of those material constants of prac-tical LC mixtures. In addition to the visco-elastic properties, thedielectric constants are also included in the argument related tothe reduced electric-field. The arguments of these two sessions cover-ing theories of director dynamics and experiments of material proper-ties will provide some qualitative comparison of the switchingcharacteristics of IPS and VA LCDs from a viewpoint of materialproperty. Finally, some discussions will be made on a validity ofthe results based upon the molecular theories of those materialproperties.
DYNAMICS OF DIRECTOR REORIENTATION
The director is considered to be uniformly oriented within a nematicLC slab with the thickness d in the absence of an external electricfield. The nematic slab is contacted with solid substrates at z ¼�d=2
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and z ¼ d=2 and is infinite in the directions perpendicular to thez axis, where a local right-handed system of Cartesian coordinates isadopted.
In the basic IPS-LCD panel configuration, the x axis is selected tocoincide with the direction of the director. The internal electric field,which is assumed to be identical to the externally applied field, isuniform and in the direction at an angle of U ¼ 45� from the x axisin the x–y plane at the origin. The Frank–Oseen theory of curvaturedeformations of nematic LCs teaches us that the threshold field ofthe Freedericksz transition is given by
Eth;IPS ¼ ðp=dÞ ffip ½K22=ðe0jDejÞ�; ð1Þ
assuming strong anchoring at both interfaces. Here K22 is the Frankelastic constant for twist and e0 is the permittivity in vacuum, leadingto the dielectric constant e ¼ ere0, where er stands for the relative dielec-tric constant. The dielectric anisotropy is defined as De ¼ ek � e?, byrepresenting the relative dielectric constant er in the direction parallelto the director by ek and that perpendicular to the director by e?.
When the LC slab is subject to an electric field E above the thresh-old, a twist deformation is created and the director n has azimuthalcomponents
n ¼ ðcos/; sin/; 0Þ ð0 � / � p=4Þ: ð2Þ
The distribution of the azimuthal angle / along the z axis can beanalyzed by considering the balance between the elastic torque andthe electric torque. This is due to so-called field effects.
The dynamic of the Freedericksz transition is governed by theinteraction between the dielectric torque and the mechanical torques.By using a kinematic viscosity g, the switching times between theuniform state with /(z) ¼ 0 (� d=2 � z�d=2) and the twist deformationcan be obtained as follows:
son;IPS ¼ g=½e0jDejE2 � ðp2K22=d2Þ� ð3Þ
soff ;IPS ¼ gd2=ðp2K22Þ: ð4Þ
The switching time son corresponds to the Freedericksz transitiondue to the field application (E > Eth,IPS) and the switching time soffto the relaxation to the field-free state. By referring to Eq. (1), theswitching-on time can be rewritten as
son;IPS ¼ g=½e0jDejðE2 � E2th;IPSÞ�: ð5Þ
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By defining the total switching time s as
s ¼ son þ soff ð6Þ
the following is obtained for the IPS-mode:
sIPS ¼ ðd=pÞ2ðg=K22ÞðE=Eth;IPSÞ2=½ðE=Eth;IPSÞ2 � 1Þ�: ð7Þ
In the VA-LCD panel configuration, the director aligns uniformly inparallel to the z axis in a field-free state. When the electric field isapplied parallel to the z axis, a bend deformation of the director fieldis created providing that the dielectric anisotropy of the LC materialis negative in sign. Assuming that the deformation occurs in oneplane, the director n has polar components, for instance,
n ¼ ðsin h; 0; cos hÞ ð0 � h � p=2Þ: ð8Þ
Similar arguments as the IPS-mode yield the following equations:
son;VA ¼ g=½e0jDejðE2 � E2th;VAÞ� ð9Þ
soff ;VA ¼ gd2=ðp2K33Þ: ð10Þ
Here K33 is the Frank elastic constant for bend and the threshold fieldof the Freedericksz transition in the bend deformation is given by
Eth;VA ¼ ðp=dÞp½K33=ðe0jDejÞ�; ð11Þ
under the strong anchoring boundary-condition. Thus, the totalswitching time of the VA-mode is yielded as
sVA ¼ ðd=pÞ2ðg=K33ÞðE=Eth;VAÞ2=½ðE=Eth;VAÞ2 � 1Þ�: ð12Þ
When neglecting the gravity center movement, the coefficient g char-acterizes the energy dissipation produced by the rotation of director[1], but it is pointed out that the gradient of the angular velocity ofthe director produces a back-flow motion of the fluid [2].
The hydrodynamic theory, known as Leslie-Ericksen theory, pro-vides an insight in the dynamic properties of incompressible nematicLCs by considering an additional torque due to frictional forces tothe dielectric and elastic torques. The viscous torque can be describedby introducing two frictional coefficients c1 and c2, which are related tothe viscosity coefficients ai (i ¼ 1–6) introduced by Leslie as follows
c1 ¼ a3 � a2c2 ¼ a6 � a5
¼ a3 þ a2
ð13Þ
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For the IPS-mode, as far as it is assumed that the deformation is thepure twist mode and the angle is small enough, the torque balanceleads to [1]
K22ð@2/=@z2Þ ¼ c1ð@/=@tÞ; ð14Þ
yielding
/ðz; tÞ ¼ /ðz; 0Þ expð�t=soff Þsoff ¼ ðd2=p2Þðc1=K22Þ:
ð15Þ
That is, the kinematic viscosity g in Eq. (4) corresponding to thedirector rotation in the IPS-mode is understood as the rotationalviscosity c1.
In the bend mode, LC molecules undergo a translatory movementand, in addition to the torque balance, the equation of motion has tobe taken into account. The contribution of this kind of fluid motionto the reorientation of the director, which was named later as aback-flow effect, was first reported as transient effects in the lighttransmission of a twisted nematic (TN) LCD [3] and was analyzedby neglecting the inertia term in the Leslie-Ericksen equations [4,5].
Regarding the back-flow effect in the pure VA-mode in one dimen-sion, only one velocity component vx has to be considered. In this case,the shear torque is to be added to the equation of the torque balance,resulting in the equation under the small angle assumption and thestrong anchoring condition:
K33ð@2h=@z2Þ ¼ c1ð@h=@tÞ þ a2ð@Vx=@zÞ: ð16Þ
The equation of motion is given by
qð@vx=@tÞ ¼ ð@=@zÞ½g1ð@Vx=@zÞ þ a2ð@h=@tÞ�; ð17Þ
by assuming the non-slip condition at the interface and neglectingthe pressure variation, characteristic time of which is considered tobe much longer than the orientation relaxation time. Here q is thedensity and g1 is the Miesowicz viscosity corresponding to (1=2)(a4þ a5 � a2).
The solutions of the system of two linear differential equations (16)and (17) are obtained [6] following the procedure developed byP. G. de Gennes [7] and the Orsay Liquid Crystal Group [8] underthe condition, vx (z, 0) ¼ 0, yielding the relaxation time of the similarform as Eq. (15)
soff ¼ ðd2=p2Þð1=K33Þðc1 � a22=g1Þ; ð18Þ
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by assuming K33=c1� g1=q. Thus, the viscosity coefficient g in Eq. (10)of the VA-mode is approximately given by the rotational viscosity c1corresponding to the uncoupled bend mode and the coupling coefficient(a22=g1) corresponding to the bend mode coupled to the shear wave byneglecting the uncoupled shear mode [9].
To sum up the above arguments, the total switching time of both theIPS and VA modes is given by
stotal ¼ ðd=pÞ2ðg=KiiÞ½A2=ðA2 � 1Þ�; ð19Þ
where
A ¼ E=Eth;
Eth ¼ ðp=dÞp½Kii=ðe0jDejÞ�:
The differences between the IPS and the VA modes are as follows:
for IPS-mode; i ¼ 2 ðKii ¼ K22Þ and g ¼ c1;for VA-mode; i ¼ 3 ðKii ¼ K33Þ and g ¼ c1 � 2a22=ða4 þ a5 � a2Þ:
That is,
sVAtotal=sIPStotal ¼ gVA=IPSðK22=K33Þ; ð20Þ
where
gVA=IPS ¼ ½c1 � 2a22=ða4 þ a5 � a2Þ�=c1¼ 1� 2a22=ða4 þ a5 � a2Þða3 � a2Þ
Furthermore, De > 0 for IPS-mode and De < 0 for VA-mode.
MATERIAL CONSTANT VALUES OF LC MIXTURES
Physical properties of more than 300 LC mixtures were summarizedand analyzed from various aspects as shown in the following figures.The material constants of main interest in this study are, as shownin Eqs. (19) and (20), Frank elastic constants K22 and K33, the rota-tional viscosity c1, and the dielectric constants ek and e?. All the valuesused in the following arguments are those measured at 20�C.
Figure 1 shows the correlation between K22 and K33 values. Thetwist elastic constant K22 is related to the switching characteristicsof the IPS-mode and the bend elastic constant K33 is to that of theVA-mode. As is shown in Figure 1, the K22 values are distributedwithin a window of 4�9 pN and the K33 values 10�26 pN. Thereclearly exists an inequality K33 > K22 and the ratio K33=K22 is
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2.55 � 0.47. This can provide an idea that the VA-mode is superior tothe IPS-mode in terms of the contribution of the elastic propertyof LC materials. This is for the total response time as the sum of theswitching-on time and the switching-off time under the condition thatthe applied field is same for both modes with regard to its reducedvalue normalized by the threshold field. As a larger elastic constantinduces a higher threshold, the absolute value of the applied field,assumed in the present comparison, is higher for the VA-mode.
The threshold field also depends on the absolute value of the dielec-tric anisotropy jDej and the sign must be negative for VA-mode useLC mixtures and positive for mixtures for uses in practical IPS-LCDs.Figure 2 shows both the relative dielectric constant ek parallel to thedirector and e? perpendicular to the director. The solid line in Figure2 corresponds to the LC mixtures satisfying ek ¼ e?, that is De ¼ 0, andthe normal distance from the line ek ¼ e? represents the De value. Thatis, the LC mixtures in the region above this line are those with positiveDe and mixtures below this line are those with negative De. Figure 2suggests that an increase of jDej value in a negative sign is not easyfor LC mixtures compared to increasing a positive De value. This alsomeans that the threshold field of VA-mode is higher than IPS-modefrom a viewpoint of the dielectric anisotropy of practical LC mixtures.
The LC mixtures, whose K22 and K33 are plotted in Figure 1, con-tain those with positive-De and negative-De. Therefore, for moreprecise arguments, it is desirable to compare the K22 value of
FIGURE 1 Correlation between Frank elastic constants of LC mixtures K22
for twist and K33 for bend.
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positive-De mixtures for uses in IPS LCDs and the K33 value ofnegative-De VA use mixtures. Figure 3 shows K22 and K33 of positive-De mixtures together with their De values. No significant differencescan be seen in the corresponding figure for negative-De mixtures. Thatis, the inequality K33 > K22 is valid independent from the sign of Deand the elastic constant values of K22 ¼ 4�9 pN and K33 ¼10�26 pN can be obtained independent from its absolute value jDej.
The total switching time also depends on the viscous coefficient g,which corresponds to the rotational viscosity c1 for the IPS-modeand to the combination of c1 and a coupling coefficient representingthe contribution of the back-flow effect in the case of the VA-mode.First, the pure rotational viscosity term is discussed. The question iswhether it is possible to design the rotational viscosity of LC mixturesindependent from their elastic constants. In Figure 4, plots are madefor the c1 values of LC mixtures against its elastic constants K22
(symbols&) and K33 (symbols&). The rotational viscosity shows nocorrelation with either of the twist or the bend elastic constant.Moreover, Figure 5 shows that there is no clear correlation betweenthe c1 value and the absolute value of the dielectric anisotropy for bothpositive-De and negative-De LC mixtures. In Figure 5, symbols & arefor the positive-De materials and symbols& for the negative-De ones.
FIGURE 2 Correlation between the relative dielectric constant ek of LCmixtures parallel to the director and e? perpendicular to the director. (Thesolid line corresponds to ek ¼ e?, that is De ¼ 0.)
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FIGURE 3 Frank elastic constants K22 (symbols ^) and K33 (symbols &) ofLC mixtures with positive- De, plotted against their De value.
FIGURE 4 Rotational viscosity c1 of LC mixtures, plotted against their Frankelastic constants K22 (symbols ^) and K33 (symbols &).
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Figures 4 and 5 teach us that the viscous property, as far as the rota-tional viscosity is concerned, can be designed for LC mixtures with Dein both signs independent from their absolute value as well astheir elastic constant value. Actual c1 values of the practical use LCmixtures are down to around 60mPa � s.
The data are summarized in Figure 6 in such a manner to comparethe visco-elastic property (c1=Kii) of the positive-De materials (i ¼ 2)and the negative-De materials (i ¼ 3) against the absolute value oftheir dielectric anisotropy jDej. The low end of the visco-elastic pro-perty is around 6� 109 Pa � s=N for c1=K33 of negative-De materials(symbols &; VA-mode) and around 12� 109 Pa � s=N for c1=K22 ofpositive-De materials (symbols &; IPS-mode). As can be seen inEq. (19), this visco-elastic property is the direct measure of the switch-ing property of LC mixtures for uses in IPS and VA modes as the firstapproximation by neglecting the back-flow effect. The main differenceof the (c1=Kii) values comes from the difference of the elastic constant,that is K22 for IPS-mode LC mixtures and K33 for VA-mode.
Finally, the contribution of the back-flow effect is considered. Thenumber of measured values is quite limited with regard to a set ofthe Leslie viscous coefficient of LC mixtures. Nevertheless, if thereported values for a typical LC mixture ZLI-2293 from Merck
FIGURE 5 Rotational viscosity c1 of LC mixtures plotted against their absol-ute value of the dielectric anisotropy jDej (symbols &; positive-De LC mixtures,symbols ^; negative- D e LC mixtures).
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KGaA [10],
a2 ¼ �0:151 ðPa � sÞa3 ¼ �0:0015 ðPa � sÞa4 ¼ 0:0809 ðPa � sÞa5 ¼ 0:1084 ðPa � sÞ
are referred, the contribution of the back-flow effect can be estimated as
gVA=IPS ¼ 1� 2a22=ða4 þ a5 � a2Þða3 � a2Þ¼ 0:104:
This ratio together with the average (K22=K33) value of 1=2.55 yieldsthe value (sVAtotal=s
IPStotal) as low as 0.04.
It is reported that the reduction of the viscosity in an infinitemedium due to the back-flow is, for instance, around 0.75 in the caseof a typical LC substance MBBA [11], providing an expectation of thevalue gVA=IPS around 0.25. The above ratio gVA=gIPS of 0.104 can be abit overestimated, but the general understanding of the superiorityof the VA-mode to the IPS-mode in terms of the display response timeis quantitatively supported from the viewpoint of the physical proper-ties of practical use LC mixtures. As a conclusion, the contribution ofthe back-flow effect can be essential as is also emphasized in a recent
FIGURE 6 Comparison of the visco-elastic property (c1=Kii) of positive-De LCmixtures (i ¼ 2, symbols &) and negative-De LC mixtures (i ¼ 3, symbols ^)against their absolute value of the dielectric anisotropy jD ej .
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publication [12]. Further quantitative investigation is under way inthe responsible properties of LC materials. Furthermore, precisenumerical simulation of the back-flow effect in VA-LCD is to bereported [13] and a theoretical study will be published separately.
DISCUSSIONS
As far as the director switching under a normalized field application isconcerned, the VA-mode was shown to be superior to the IPS-modewith regard to the contributions of LC material properties. In this sec-tion, discussions are made on the responsible properties of LC mix-tures by referring to the molecular theories of nematic LC substances.
First, the relation K33=K22 ¼ 2.55 observed in average between theFrank elastic constants K22 for twist and K33 for bend is reviewed.According to R.G. Priest [14], the following equations hold for Frankelastic constants including K11 for spray.
K22=K ¼ 1� 2D� D0hP4i=hP2i þ � � �
K33=K ¼ 1þ Dþ 4D0hP4i=hP2i þ � � � ;
ð21Þ
where K ¼ (K11þK22þK33)=3, hP4i ¼ (35 < cos4 b >�30 < cos2 b >þ 3)=8, and hP2i ¼ (3 < cos2b > � 1)=2, as b the angle between themolecular long axis and the director. The coefficients D and D0 arethe constants depending on the molecular interaction potential. Bytaking the three terms in the right-side of Eq. (21) into account, thefollowing relation is obtained:
K33=K22 ¼ ½ð1þ DÞhP2i þ 4D0hP4i�=½ð1� 2DÞhP2i � D0hP4i�: ð22Þ
By introducing a parameter R ¼ L=D, which represents a LC molecu-lar structure using the sphero-cylinder model with the width D andthe length LþD, Eq. (21) can be rewritten as
K22=K ¼ ½3R2 þ 24þ 27ðhP4i=hP2iÞð�R2=16þ 1=6Þ�=ð7R2 þ 20Þ
K33=K ¼ ½9R2 þ 18þ 27ðhP4i=hP2iÞðR2=4� 2=3Þ�=ð7R2 þ 20Þ:
ð23Þ
Thus, D and D0 can be explicitly evaluated as
D ¼ 2ðR2 � 1Þ=ð7R2 þ 20ÞD0 ¼ 27ðR2=16� 1=6Þ=ð7R2 þ 20Þ:
ð24Þ
When R ¼ L=D is large enough, Eqs. (22) and (24) yield
K33=K22 ¼ 4½12þ 9ðhP4i=hP2iÞ�=½16� 9ðhP4i=hP2iÞ�: ð25Þ
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It can be deduced that, in the ultimate case of hP4i << hP2i, that is, inthe case when hP2i is infinitely small, K33 is equal to 3K22. This willallow us to conclude that the average value K33=K22 of 2.6 obtainedfor practical LC mixtures can be reasonably used in a general compari-son of the switching time of IPS and VA modes in a more or lessquantitative manner.
Further considerations can be made on the estimation of practicalvalues of the relative order hP4i=hP2i and the molecular aspect ratioR governing the elastic constant as indicated in Eq. (23). Equation(23) together with the set of measured elastic constant values (K11,K22, K33) provides both values of hP4i=hP2i and R for each LC mixture.The average values are calculated as hP4i=hP2i ¼ 0:577� 0:363.Assuming roughly that the equation hP2i ¼ ð1� T=TNIÞ0:220 [15] holdsfor all LC mixtures in the present study, the value hP2i can be calcu-lated by using measured TNI and T ¼ 293K. The average value of hP2iis 0.676� 0.020 providing the average value of hP4i as 0.386� 0.236.All these values seem to be within an acceptable range as for thepresent rough estimation.
The average of calculated R values is also derived as 3.00� 0.62.The aspect ratio of the spheroid equivalent to a molecular volumecan also be calculated by a MO method and, for instance, is found tobe 3.63 for a typical two ring structure and 4.40 for a three ring struc-ture. The obtained R value of around 3.00 in average also seems to bereasonable and the relatively small value for mixtures can be due tothe dimer formation of polar molecules. That is, the elastic propertyof LC mixtures, especially those with a large positive-De, is considerednot to be additive. By inserting the R value into Eq. (24), the values ofD and D0 are also calculated for each LC mixture and derived asD ¼ 0.187 � 0.027 and D0 ¼ 0.122� 0.033 in average.
Next, the viscous properties are taken up for discussions. Accordingto M.A. Osipov and E.M. Terentjev [16,17], the Leslie coefficients canbe expressed in terms of molecular parameters as follows
a1 ¼ �qk½ðp2 � 1Þ=ðp2 þ 1Þ�hP4ia2 ¼ �ðqk=2ÞhP2i � ð1=2Þc1a3 ¼ �ðqk=2ÞhP2i � ð1=2Þc1a4 ¼ ðqk=35Þ½7� 5hP2i � 2hP4i�a5 ¼ ðqk=2Þfð1=7Þ½ðp2 � 1Þ=ðp2 þ 1Þ�ð3hP2i þ 4hP4iÞ þ hP2iga6 ¼ ðqk=2Þfð1=7Þ½ðp2 � 1Þ=ðp2 þ 1Þ�ð3hP2i þ 4hP4iÞ � hP2ig: ð26Þ
Here, q is the number density, p is the molecular length-to-widthratio, and k is the microscopic friction constant.
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The inequality ja3j << ja2j, which can also be seen in the abovevalues for ZLI-2293, yields ðqkÞ c1=hP2i providing the followings:
a2 c1a4 ð1=35Þ½7=hP2i � 5� 2hP4i=hP2i�c1a5 ð1=2Þfð1=7Þ½ðp2 � 1Þ=ðp2 þ 1Þ�ð3þ 4hP4i=hP2iÞ þ 1gc1:
ð27Þ
Equation (27) under the assumption p ¼ R together with the esti-mated average values of R ¼ 3.00, hP2i ¼ 0:676, and hP4i ¼ 0:386allows a rough estimation of the coupling coefficient ða22=g1Þ for theVA mode as
a22=g1 ¼ 2a22=ða4 þ a5 � a2Þ 0:8c1;
and the coefficient
gVA=IPS ¼ ðc1 � a22=g1Þ=c1 0:2:
This can be said to agree rather well to the above estimation ofgVA=IPS ¼ 0:10 � 0:25 using the measured Leslie coefficients.
Finally, the dielectric property is touched briefly. The well-knownexpression of the dielectric anisotropy given by W. Maier and G. Meieris the following:
De ¼ ðNhF=e0Þ ½ðak � a?Þ þ Fðl2=2kBTÞð3 cos2 /� 1Þ� hP2i: ð28Þ
Here, N is the number density of LC molecules, h and F are the cavityfield factors, ak and a? are the electronic polarizability parallel andperpendicular to the director, and kB is the Boltzmann constant. Itis also well known that the absolute value jDej is mainly subject tothe molecular dipole moment l. Practical LC molecules are designedto have a large dipole l parallel (/ ¼ 0) to the molecular axis to obtaina large De value in a positive sign, and perpendicular (/ ¼ p=2) to themolecular axis in a negative sign. Assuming the dipole l is same in itsvalue, however, the term (3 cos2 /� 1) is equal to 2 in the case / ¼ 0and �1 in the case / ¼ p=2. This means that the contribution of themolecular dipole to the De is two times larger when it is put in parallelto the molecular axis (/ ¼ 0 and De > 0) than the case when put inperpendicular to the molecular axis (/ ¼ p=2 and De < 0). This can alsobe reflected on the De value of LC mixtures and is clearly shown inFigure 2.
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REFERENCES
[1] Labrunie, G. & Robert, J. (1973). J. Appl. Phys., 44, 4869.[2] Brochard, F., Guyon, E., & Pieranski, P. (1972). Phys. Rev. Lett., 28, 1681.[3] Gerritsma, C. J., Van Doorn, C. Z., & Van Zanten, P. (1974). Phys. Lett., 48A, 263.[4] Van Doorn, C. Z. (1975). J. de Physique Colloq. C1, 36, 261.[5] Clark, M. G. & Leslie, F. M. (1978). Proc. R. Doc. Lond. A, 361, 463.[6] Saupe, A. (1973). Ann. Rev. Phys. Chem., 24, 441.[7] De Gennes, P. G. (1969). Mol. Cryst. Liq. Cryst., 7, 325.[8] Group d’Etude des Cristaux Liquides (Orsay). (1969). J. Chem. Phys., 51, 816.[9] De Gennes, P. G. (1974). The Physics of Liquid Crystals, Clarendon Press: Oxford,
UK, 183.[10] Tarumi, K., Jacob, T., & Graf, H.-H. (1995). Mol. Cryst. Liq. Cryst., 263, 459.[11] Brochard, F., Guyon, E., & Pieranski, P. (1972). Phys. Rev. Lett., 28, 1681.[12] Pasechnik, S. V., Chigrinov, V.G., Shmeliova,D. V., Tsvetkov,V. A., &Voronov,A.N.
(2004). Liq. Cryst., 31, 585.[13] Iwata, Y., Naito, H., Inoue, M., Ichinose, H., Klasen-Mammer, M., & Tarumi, K., to
be presented at 20th ILCC, (2004).[14] Priest, R. G. (1973). Phys. Rev. A, 7, 720.[15] Haller, I. (1975). Prog. Solid State Chem., 10, 103.[16] Osipov, M. A. & Terentjev, E. M. (1989). Z. Naturforsch., 44a, 785.[17] Osipov, M. A. & Terentjev, E. M. (1990). Nuovo Cimento, 12D, 1223.
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