Superposition and universality in the linear viscoelasticity of Leslie–Ericksen liquid crystals L. R. P. de Andrade Lima and A. D. Reya) Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, Quebec H3A 2B2, Canada (Received 23 December 2003; final revision received 19 May 2004)
Synopsis The linear viscoleasticity of seven lyotropic and thermotropic liquid crystalline polymers is characterized using the Leslie–Ericksen equations of defect-free nematodynamics for small amplitude oscillatory capillary Poiseuille flow, using analytical, numerical, and scaling methods. The experimentally measured seven data sets correspond to shear flow-aligning and shear nonaligning materials. The predicted equivalent rheological responses between these two classes of polymers demonstrate the universality of nematodynamics. Principles of superposition are developed, applied, and shown to be accurate in collapsing the data sets for aligning and non-aligning polymers. The scaled resonance peak in the loss tangent (G ⬙ /G ⬘ ) is shown to be a universal constant for monodomain nematics. © 2004 The Society of Rheology. 关DOI: 10.1122/1.1773784兴
I. INTRODUCTION A common rheological classification scheme for nematic liquid crystal polymers 共LCPs兲 is based on their shear flow aligning characteristics, which is set by the sign and magnitude of the reactive order parameter 关de Gennes and Prost 共1993兲; Larson 共1999兲; Rey and Denn 共2002兲兴; for the aligning regime 共 ⬎ 1兲, the average molecular orientation or director n, is close to the streamline, while the nonalignment regime 共0 ⭐ ⬍ 1兲, the steady state orientation is nonplanar and nonhomogeneous 关Han and Rey 共1995兲; Rey and Denn 共2002兲兴. Lyotropic LCPs are usually nonaligning at low shear rates and aligning at high shear rates 关Larson 共1999兲兴. On the other hand, main-chain thermotropic LCPs with flexible spacers are likely to be flow aligning at all shear rates, while main-chain thermotropic LCPs without flexible spacers are apparently non-aligning 关Mather et al. 共1996; 2000兲; Romo-Uribe and Windle 共1996兲; Ugaz and Burghardt 共1998兲; Larson 共1999兲; Zhou et al. 共2001兲; Grecov and Rey 共2003兲兴. A very complete data set of viscoelastic parameters for seven thermotropic and lyotropic LCPs has been presented and analyzed by Martins 共2001兲. The data of Martins 共2001兲 lends credence to the rheological differentiation between lyotropic and thermotropic LCPs. A common experiment to show shear flow alignment is the response to start-up shear flow 关Larson 共1999兲兴. For flow-aligning materials no significant oscillatory overshoots are observed. On the other hand, for nonaligning materials, distinct overshoots arise. The oscillatory response can then be used to evaluate the magnitude of the reactive parameter 关Larson 共1999兲兴. The rheological classification of LCPs is a useful framework to interpret a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
© 2004 by The Society of Rheology, Inc. J. Rheol. 48共5兲, 1067-1084 September/October 共2004兲
0148-6055/2004/48共5兲/1067/18/$25.00
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experiments, measure properties, and design processes. On the other hand, rheologically universal features of LCPs have not been identified in the literature. In this paper we present a flow geometry for which the predicted linear viscoelasticity of aligning and nonaligning nematic polymer materials is equivalent. The predicted equivalence is based on the fact that the viscoelastic response to many flow process is a function of a dimensionless number P ⫽ (1⫺) 2 , which represent the characteristic ratio of viscous torque to rotational dissipation coefficients. Since P() ⫽ P(2⫺), identical responses for aligning and nonaligning materials is expected. Another example of universality for LCPs is the texture scaling under shear. Grecov and Rey 共2003兲 have recently shown that their computed texture scaling for thermotropic flow-aligning LCPs reproduces the texture scaling predicted 关Marrucci and Greco 共1993兲兴 and measured 关Larson and Mead 共1993兲兴 for lyotropic nonaligning LCPs. Superposition is a useful framework to characterize the rheological response of LCPs. For lyotropic nonaligning polymers it has been shown that proper scaling of shear stress amplitude and elapsed time lead to superposition 关Larson 共1999兲兴. For example, overshoots in start-up shear flows at different shear rates are usually superposable when plotted as a function of strain. In this paper we show that the predicted linear viscoelastic response of seven different defect free materials can be superposed, when the proper material viscoelastic parameters are absorbed in the amplitude-frequency axes. Superposition of the rheological response of aligning and nonaligning materials is another unifying principle in nematodynamics. The Leslie–Ericksen liquid crystal 共LELC兲 is a mathematical material model for rodor disk-like uniaxial incompressible isothermal nematic liquid crystals 关Rey and Denn 共2002兲兴 widely used to describe anisotropic viscoelastic behavior. The elasticity described by the Leslie–Ericksen theory is due to director gradients, and is know as Frank elasticity. This type of gradient orientation elasticity is macroscopic, since the length scale of director gradients are typically in the micron range. The orientation process involved in Frank elasticity has a characteristic time for orientation relaxation denoted by o . Under a shear flow of characteristic shear rate ␥˙ , the magnitude of the Ericksen number E ⫽ ␥˙ o determines whether flow-induced orientation occurs: for E ⬍ 1 there is no flowinduced orientation, and for E ⬎ 1, flow overcomes Frank elasticity and orients the liquid crystal. Liquid crystalline polymers in addition to gradient orientation elasticity also exhibit molecular elasticity due to deviations in the scalar order parameter 共i.e., molecular alignment along the average orientation兲 from its equilibrium value. This type of elasticity is not included in the LELC model. The molecular alignment process involved in the homogeneous molecular elasticity has a characteristic relaxation time denoted by m . Under a shear flow of characteristic shear rate ␥˙ , the magnitude of the Deborah number De ⫽ ␥˙ m determines whether flow-induced molecular alignment changes occurs: for De ⬍ 1 there are no flow-induced molecular alignment deviations form equilibrium values, and for De ⬎ 1, the flow results in deviations of molecular alignment from equilibrium values. It is well known 关de Gennes and Porst 共1993兲; Rey and Denn 共2002兲兴 that the orientation process is much slower than the molecular alignment process: E ⬎ De. Thus, molecular elasticity in a given shear flow and a given nematic liquid crystal can be neglected if E ⬍ o / m , and under these conditions the LELC model should be applicable. In a cone and plate or capillary rheometer, the orientation time scale associated with the process is o ⫽ L 2 /K, where K is a characteristic Frank elastic constant, is a characteristic viscosity, and L is a characteristic system length scale. For polymeric nematics, the typical values for the characteristic length scale, elastic constant and characteristic viscosity are L ⫽ 0.01 cm, K ⫽ 10⫺11 N, and ⬇ 10 Pa s, respectively, that gives for orientation time o ⬇ 1000 s 关Larson and Mead 共1992兲兴.
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For polymeric nematics the molecular time scales ( m ) typical are in the range of 0.001–10 s 关Larson 共1999兲兴; for poly共␥-benzyl-glutamate兲 solutions in m-cresol 共PBLG兲, for instance, the relaxation time is ⬇ 0.05 s 关Larson and Mead 共1993兲兴. The ratio of orientation-to-molecular times scales is in the range of 103 – 106 , and therefore the frequency region of predominance of the Frank elasticity over the molecular elasticity has a magnitude range of 103 – 106 . Under an oscillatory flow, the magnitude of Deborah number is given by De ⫽ m ˜ / o , where ˜ is the dimensionless frequency ( ˜ ⫽ o ); for polymeric nematics the effect of molecular orientation is negligible typically when ˜ Ⰶ o / m ⫽ 102 – 106 ; for PBLG the experimental data show that the limiting dimensionless frequency is ˜ ⫽ o ⬇ 2⫻105 关Larson and Mead 共1993兲兴. In this paper we neglected the molecular elasticity for all nematic polymers characterized by Martins 共2001兲, and based on available data 关Larson and Mead 共1993兲兴 we assume that the LELC model can describe their rheological response. Small amplitude oscillatory flows are a main rheological tool used to characterize viscoelasticity 关Barnes et al. 共1989兲; Bird et al. 共1989兲兴 in terms of the storage G ⬘ ( ,T) and loss G ⬙ ( ,T) moduli as a function of frequency 共兲 and temperature 共T兲. Although simple shear is commonly used, pressure driven flows, as considered in this paper, are also equally useful 关Pasechnik et al. 共2001兲; Pikin 共1991兲兴. Previous theoretical work on small-amplitude oscillatory shear of liquid crystals in a parallel plate geometry has been presented for rod-like nematics liquid crystals 关Burghardt 共1991兲兴, nematic liquid crystal mixtures 关Rey 共1996兲兴, chiral nematic liquid crystal 关Rey 共2000兲兴, and side-chain nematic polymers 关Rey 共2002兲兴. In this paper we use small-amplitude pressure driven Poiseuille capillary flow, and characterize the linear viscoelasticity of seven LCPs 关Martins 共2001兲兴 using the Leslie–Ericksen equations of defect-free nematodynamics 关de Gennes and Prost 共1993兲; Larson 共1999兲兴. The objectives of this paper are: 共1兲 to demonstrate that flow-aligning and nonaligning materials can exhibit identical viscoelastic responses, thus showing features of universality in the nematodynamics of lyotropic and thermotropic materials; and 共2兲 to demonstrate that superposition of rheological linear viscoleastic responses are expected by proper moduli-frequency scaling. This paper is organized as follows. Section II presents the governing equations and auxiliary data to describe the nematic liquid crystals oscillatory capillary Poiseuille flow. Section III presents the material viscoelastic properties and functions. Section IV presents the linear viscoelasticity of seven distinct LCPs. Section V characterizes the superposition and universality. Section VI presents the conclusions. II. THEORY AND GOVERNING EQUATIONS The governing equations of the Ericksen and Leslie 共EL兲 theory consists of the linear momentum balance, director torque balance, and constitutive equations for the stresses, viscous and elastic torques, that takes into account external forces that distort the spatially uniform equilibrium configurations of liquid crystals 关Chandrasekhar 共1992兲; de Gennes and Prost 共1993兲; Ericksen 共1961兲; Leslie 共1968, 1979兲兴. Uniaxial nematic liquid crystals 共NLCs兲 are characterized by an average molecular orientation represented by the director vector n collinear with the average molecular orientation direction. For small-amplitude oscillatory Poiseuille capillary flow 共SAOPF兲 of a nematic liquid crystals, driven by pressure drop oscillations of infinitesimal amplitude, the flow is described by an axisymmetric oscillatory planar director field 兵 n(r,t)
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⫽ 关 sin (r,t),0,cos (r,t)兴其 and a purely axial oscillatory velocity field 兵 v (r,t) ⫽ 关 0,0,v(r,t) 兴 其 with finite velocity gradient at the centerline. Linearizing the orientation equation, resulting from the angular moment balance, around the axial direction 共i.e., sin ⬵ , cos ⬵ 1兲, the dimensionless governing equations for the director tilt angle (r˜ ,t˜ ) and the axial velocity v˜ (r˜ ,t˜ ) simplify to 关Atkin 共1970兲; de Andrade Lima and Rey 共2003a兲兴:
冋
册
1 ␣˜ 3 ⫽ Er˜, 共r˜兲 ⫹ ˜t r˜ r˜ r˜ 2˜ 1
共1a兲
Er˜ v˜ ⫽⫺ ⫹B˜, r˜ 2˜ 1
共1b兲
␣˜ 3 , ˜ 1 ˜t
共2兲
␥1 ⫽ ␣3⫺␣2 ,
共3a兲
␥2 ⫽ ␣6⫺␣5 ⫽ ␣3⫹␣2 ,
共3b兲
B˜ ⫽ ⫺
⫽⫺
␥2 ␥1
⫽⫺
␣6⫺␣5 ␣3⫺␣2
⫽⫺
␣3⫹␣2 ␣3⫺␣2
,
共3c兲
where ␣˜ i are the dimensionless Leslie viscosities ( ␣˜ i ⫽ ␣ i / splay), splay is the splay viscosity ( splay ⫽ ␥ 1 ⫺ ␣ 23 / 1 ), 1 is the Miesowicz viscosity when the director is parallel to the velocity direction 关 1 ⫽ ( ␣ 3 ⫹ ␣ 4 ⫹ ␣ 6 )/2兴 , E( ˜ ˜t ) ⫽ R 3 ⫻ 关 ⫺dp( ˜ ˜t )/dz 兴 /K 11 is the ratio of viscous flow effects to long-range elasticity effects known as the Ericksen number, r˜ ⫽ r/R is the dimensionless radius, R is the capillary radius, ˜t ⫽ K 11t/(R 2 splay) is the dimensionless time. v˜ ⫽ splayR v /K 11 is the scaled axial velocity, ⫺dp( ˜ ˜t )/dz is the given small amplitude oscillatory pressure drop in the capillary per unit length, ˜ ⫽ (R 2 splay)/K 11 is the dimensionless frequency, K 11 is the splay Frank elastic constants, ␥ 1 is the rotational viscosity, and ␥ 2 is the irrotational torque coefficient and B˜ is the dimensionless backflow 关Lacerda Santos et al. 共1985兲兴. The boundary conditions for the director orientation angle represent strong planar anchoring, (0,t˜ ) ⫽ (1,t˜ ) ⫽ 0, and for the axial velocity the no slip condition at the bounding surface is used, v˜ (1,t˜ ) ⫽ 0. The director oscillates around the velocity 共z兲 direction, and the undistorted director field is: no ⫽ (0,0,1). For the small amplitude oscillatory capillary Poiseuille flow considered in this paper, the Ericksen number 共i.e., dimensionless pressure drop兲 oscillates as follows: E ⫽ E0 sin ˜ ˜t,
共4兲
where E 0 is the infinitesimal dimensionless amplitude. Note that the frequency is scaled with the orientation time scale o ⫽ (R 2 splay)/K 11 and the maximum elastic storage is expected for frequencies close to the reciprocal of this value. III. MATERIAL PROPERTIES AND FUNCTIONS The viscoelastic material properties needed to characterize the small amplitude oscillatory Poiseuille flow of NLCs aligned along the capillary axis include the Miesowicz
LINEAR VISCOELASTICITY OF LIQUID CRYSTALS
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viscosities 1 , the reactive parameter , the torque coefficient ␣ 3 , and the reorientation viscosity splay 关de Gennes and Prost 共1993兲; Leslie 共1979兲兴. The Miesowicz shear viscosities that characterize viscous anisotropy are measured in a steady simple shear flow between parallel plates with fixed director orientations along three characteristic orthogonal directions: 1 ⫽ ( ␣ 3 ⫹ ␣ 4 ⫹ ␣ 6 )/2 when the director is parallel to the velocity direction, 2 ⫽ (⫺ ␣ 2 ⫹ ␣ 4 ⫹ ␣ 5 )/2 when it is parallel to the velocity gradient, and 3 ⫽ ␣ 4 /2 when it is parallel to the vorticity axis; the measured Miesowicz shear viscosities for aligning nematics usually follow the ordering:
2 ⬎ 3 ⬎ 1 .
共5兲
In the present flow 1 is the relevant steady shear viscosity, while the splay viscosity splay is the appropriate transient viscosity. The shear flow alignment of rod-like NLCs is governed by the magnitude of the reactive parameter (T). According to Eq. 共3c兲, ⫽ f ( ␣ 2 , ␣ 3 ), and for rods the inequality ␣ 2 ⬍ 0 holds at all temperatures, but ␣ 3 may change sign. For rod-like molecules, when ⬎ 1 ( ␣ 3 ⬍ 0) the material is known as shear flow aligning, and the director aligns within the shear plane, at an angle L , known as the flow-alignment Leslie angle, given by Leslie 共1979兲:
冉冊
1 1 L ⫽ cos⫺1 . 2
共6兲
In a steady simple shear flow when the director is aligned along L the viscous torques are zero. The Leslie angle can be measured using optical methods and the reactive parameter can be evaluated directly using Eq. 共6兲; however, when ⬍ 1 ( ␣ 3 ⬎ 0), nonaligning behavior arises and Eq. 共6兲 does not hold 关Beens and de Jeu 共1985兲兴. When the director angle is in the plane of shear and close to zero the viscous torque v ⌫ around the azimuthal direction is ⌫v ⫽ ⫺␣3␥˙ ⫽ ⫺
␥1
共1⫺兲 ␥˙ , 共1⫺兲␥˙ ⫽ ␣2 2 共1⫹兲
共7兲
where ␥˙ is the characteristic shear rate, and were we used the definitions, Eqs. 共3a兲 and 共3c兲. In SAOPF the viscous torque is balanced by the elastic torque; as a result, the measure of the linear viscoelastic storage and loss moduli yields ␣ 3 . Since for rod-like NLCs the Leslie coefficient ␣ 2 is always negative, the sign of ␣ 3 determines whether is greater or less than 1. Thus, flow alignment can be determined using simple and purely mechanical measurements. At the nonaligning/aligning transition, the viscous torque vanv ishes: ⌫ ⫽ 0. The director reorientation is a viscoelastic process, and the reorientation viscosities associated with splay, twist, and bend deformations are defined by 关de Gennes and Prost 共1993兲; Lacerda Santos et al. 共1985兲兴:
twist ⫽ ␥ 1 , splay ⫽ ␥ 1 ⫺ bend ⫽ ␥ 1 ⫺
␣ 23
1 ␣ 22
2
冋 冋
⫽ ␥ 1 1⫺ ⫽ ␥ 1 1⫺
␥1 41
␥1 42
册 册
共8a兲
共 1⫺ 兲 2 ,
共8b兲
共 1⫹ 兲 2 .
共8c兲
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These viscosities are given by the rotational viscosity ( ␥ 1 ) decreased by a factor introduced by the backflow effect. Backflow is reorientation driven flow and is essentially the reverse effect to flow-induced orientation. The general expression for the reorientation viscosities can be rewritten as ␣ ⫽ ␥ 1 ⫺(TCi ) 2 / i , where i denotes the corresponding Miesowicz viscosity and TCi the corresponding torque coefficient. Since twist is the only mode that creates no backflow then twist ⫽ ␥ 1 . For a bend distortion the backflow is normal to n, for this reason the torque coefficient is ␣ 2 , and the Miesowicz viscosity is 2 . On the other hand, for a splay distortion the backflow is parallel to n and, hence, the torque coefficient is ␣ 3 , and the Miesowicz viscosity is 1 . In the capillary Poiseuille flow the relevant re-orientation viscosity is splay . In this paper we use the viscoelastic material parameters of aligning 共 ⬎ 1兲, neutral 共 ⫽ 1兲 and nonaligning 共 ⬍ 1兲 LCPs 关Martins 共2001兲兴: 共i兲
共ii兲 共iii兲
Aligning LCPs 共 ⬎ 1兲:
PSi4 共poly关共2,3,5,6-tetradeuterio-4-methoxyphentl-4⬘-butanoxybenzoate兲-methylsiloxane兴兲, AZA9 共poly共4,4⬘-dioxy-2,2⬘-dimethylazoxybenzene-dodeccanediyl兲兲, and DDA9 共poly共4,4⬘-dioxy-2,2⬘-dimethylazoxybenzene-dodeccanediyl兲兲.
Neutral LCP 共 ⫽ 1兲:
TPB10 共poly关1,10-decylene-1-共4-hydroxy-4⬘-biphenylyl兲-2-共4-hydroxyphenyl兲 butane兴兲.
Nonaligning LCPs 共0 ⬍ ⬍ 1兲: PBLG 共poly共␥-benzyl-L-glutamate兲兲 17% in m-cresol, PPTA 8.8% 共poly共p-phenylene terephthalamide兲兲 in SO4 D2 , and PBLG 12% in m-cresol. The measured values are shown in Table I. It is highly likely that the neutral LCP 共TBP10兲 has a value of ␣ 3 that is beyond experimental resolution.
IV. LINEAR VISCOELASTICITY Imposing pressure oscillations on the NLCs will produce spatially nonhomogeneous director oscillations. Since NLCs are viscoelastic materials, the director oscillations will not be in-phase with the applied pressure drop. Thus the total director angle (r˜ ,t˜ , ˜ ) is given by the sum of the following in-phase and out-phase components 共see the Appendix兲:
共r˜,t˜,˜ 兲 ⫽ i共r˜,˜ 兲sin共˜ ˜t兲⫹o共r˜,˜ 兲cos共˜ ˜t兲.
共9兲
Since the director field n is coupled to the velocity field v, imposing an oscillatory pressure drop to the NLC will produce a velocity field with in-phase and out-of-phase components. Thus, the total dimensionless velocity field v˜ (r˜ ,t˜ , ˜ ) is given by the sum of the following in-phase and out-phase components 共see the Appendix兲: ˜ 兲 ⫽ v˜ i共r˜,˜ 兲sin共˜ ˜t兲⫹v˜ o共r˜,˜ 兲cos共˜ ˜t兲. v˜ 共r˜,t˜,
共10兲
The orientation and velocity are obtained by solving Eqs. 共1a兲 and 共1b兲 using separation of variables. The symbolic and numerical calculations presented in the next sections were performed using the software MAPLE release 7 by Waterloo Maple Inc. and MATLAB version 6.5 by Math Works Inc. A. Analytical results ˜ ) and the dimensionless apparent For Poiseuille flow the dimensionless flow rate (Q viscosity ( ˜ ) are given by the follow relations 关de Andrade Lima and Rey 共2003b兲兴:
0.6308 1.775 0.8245
1.007 1.000 0.3554
17.02 3.678⫻10⫺2 6.262⫻10⫺1
˜ 1 ˜ 2 ˜ 3
˜ twist ˜ splay ˜ bend
M M1 M2
5.337 2.029⫻10⫺3 1.083⫻10⫺2
1.024 1.000 0.01035
0.01109 1.068 0.06812
1.024 ⫺1.056
⫺0.8609 ⫺1.040 ⫺0.01630 0.1362 0.9588 ⫺0.09777
360
TPB10
4
⫺1.793⫻105 ⫺1.810⫻105 0.000 7.838⫻103 1.752⫻105 ⫺5.838⫻103 Reactive parameter
10.50 1.134⫻10⫺3 1.191⫻10⫺2
⬁ 0 5.525⫻10⫺3
1.012 1.000 1.000 1.000 0.01152 0.005598 Viscoelastic parameter ratiosb
0.01205 0.005525 1.048 1.006 0.04822 0.02165 Dimensionless reorientation viscosities
1.012 1.000 ⫺1.036 ⫺1.000 Dimensionless Miesowicz viscosities
b
10.41 7.215⫻10⫺5 7.509⫻10⫺4
1.012 1.000 0.0006402
0.0007599 1.007 0.04320
1.012 ⫺1.006
⫺0.9759 ⫺0.9906 ⫺0.9210 ⫺1.024 ⫺1.000 ⫺1.009 ⫺0.01205 0.000 0.003040 0.09644 0.04330 0.08640 0.9759 0.9680 0.9179 ⫺0.06030 ⫺0.03225 ⫺0.08792 Dimensionless rotational viscosity and irrotational torque coefficient
0.9940
0.3408 7.994⫻10⫺3 2.725⫻10⫺3
1.579 1.000 0.002870
0.004302 1.4834 0.08644
1.579 ⫺1.479
⫺0.8426 ⫺1.529 0.04991 0.1729 1.265 ⫺0.2142
0.9368
⫺1.177⫻106 ⫺2.136⫻106 6.972⫻104 2.415⫻105 1.767⫻106 ⫺2.992⫻105
300
0.6042 3.832⫻10⫺2 2.315⫻10⫺2
1.261 1.000 0.02622
0.02919 1.116 0.09002
1.261 ⫺1.086
⫺0.5808 ⫺1.174 0.08727 0.1800 0.8775 ⫺0.2089
0.8616
⫺193.0 ⫺390.0 29.00 59.83 291.6 ⫺69.43
302
PBLG 12% in m-cresol
PBLG 17% in m-cresol 302
7
6 PPTA 8.8% in SO4 D2
5
⫺1212 ⫺1328 4.000 113.7 1208 ⫺115.7
1.0238 1.0000 ˜ i ⫽ ␣ i / splay) a Dimensionless Leslie viscosities coefficients ( ␣
⫺162.0 ⫺170.0 ⫺2.000 16.01 162.0 ⫺10.01
Leslie viscosities coefficients 共Pa s兲 关Martins 共2001兲兴
394
DDA9
3
The splay viscosity is defined as: splay ⫽ ␥ 1 ⫺ ␣ 23 / 1 , where ␥ 1 ⫽ ␣ 3 ⫺ ␣ 2 and 1 ⫽ ( ␣ 3 ⫹ ␣ 4 ⫹ ␣ 6 )/2. See definition at Eqs. 共20兲, 共21兲, and 共22兲.
1.007 ⫺1.144
␥˜ 1 ␥˜ 2
a
⫺0.5163 ⫺1.076 ⫺0.06831 1.649 0.8251 ⫺0.3191
␣˜ 1 ␣˜ 2 ␣˜ 3 ␣˜ 4 ␣˜ 5 ␣˜ 6
1.0318
⫺1320 ⫺1595 ⫺25.00 208.9 1470 ⫺149.9
⫺960.0 ⫺2000 ⫺127.0 3066 1534 ⫺593.2
1.1356
393
AZA9
PSi4
348
2
1
␣1 ␣2 ␣3 ␣4 ␣5 ␣6
T ( K)
Set
TABLE I. Parameter values.
LINEAR VISCOELASTICITY OF LIQUID CRYSTALS 1073
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de ANDRADE LIMA and REY
˜ ⫽ 2 Q
冕
1
0
共11a兲
v˜ 共r˜兲r˜dr˜,
E ˜. 8Q
˜ ⫽
共11b兲
Specifying these expressions for SAOPF using the dimensionless oscillatory flow rate given by 共see the Appendix兲: ˜ 共˜ 兲sin ˜ ˜t⫹Q ˜ 共˜ 兲cos ˜ ˜t ˜ *共˜ ,t˜兲 ⫽ Q Q i i
共12兲
and the dimensionless complex viscosity ˜ * ( ˜ ) defined by: ˜ * ( ˜ ) ⫽ ˜ ⬘ ( ˜ ) ˜ ⬙ / ˜ and ˜ ⬙ ⫽ G ˜ ⬘ / ˜ are the dissipative 共loss兲 and elastic ⫺i ˜ ⬙ ( ˜ ), where ˜ ⬘ ⫽ G 共storage兲 components. The following expressions were obtained for the dimensionless ˜ ⬘ ), loss modulus (G ˜ ⬙ ), and loss tangent (tan ␦ ⫽ G ˜ ⬙/G ˜ ⬘) 共see the storage modulus (G Appendix兲: ˜⬘ ⫽ G
M 1关˜ F2⫺1兴
冉 冊冉 冊 冋 册 冉 冊冉 冊 2
F1
F2 1 1⫹ ⫹ ⫺ M M ˜ M
˜ F1
M2
˜⬙ ⫽ G
M
1 F2 1⫹ ⫹ ⫺ M M ˜ M M⫹F1共˜ 兲
tan ␦ ⫽
共13兲
2,
共14兲
⫹˜
2
F1
2,
F2共˜ 兲⫺
1
共15兲
,
˜
where F1共˜ 兲 ⫽
F2共˜ 兲 ⫽
ber1 冑˜ 共 bei0 冑˜ ⫺ber0 冑˜ 兲 ⫺bei1 冑˜ 共 ber0 冑˜ ⫹bei0 冑˜ 兲 2 冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
bei1 冑˜ 共 ber0 冑˜ ⫺bei0 冑˜ 兲 ⫺ber1 冑˜ 共 bei0 冑˜ ⫹ber0 冑˜ 兲 2 冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
⬁
ber x ⫽
兺 k⫽0 ⬁
bei x ⫽
兺
k⫽0
cos
冉
3
⫹ k 4 2
k!⌫共k⫹1⫹兲 sin
冉
3
⫹ k 4 2
k!⌫共k⫹1⫹兲
冊冉 冊 x
x
2
共16兲
,
共17兲
2k⫹
2
冊冉 冊
,
,
共18兲
.
共19兲
2k⫹
LINEAR VISCOELASTICITY OF LIQUID CRYSTALS
1075
The functions bei (x) and ber (x) are the Kelvin functions of order , 关Abramowitz and Stegun 共1972兲兴. The main viscoelastic parameter ratios that controls the dimensionless storage modulus (G ⬘ ), the loss modulus (G ⬙ ) and the loss tangent 共tan ␦兲 are, respectively, M 1 , M 2 , and M, given by: 共20兲
共21兲
共22兲
The text under Eqs. 共20兲–共22兲 indicates the physical meaning of the three-dimensionless ratios, while their numerical values for the seven polymers are shown in Table I. The ˜ ⬘ ) increases with M since as this ratio increases the viscous torques storage modulus (G 1 create more elastic storage and less rotational dissipation; when ⫽ 1 viscous torques ˜ ⬙ ) increases are absent and no elastic storage is possible (G ⬘ ⫽ 0). The loss modulus (G with M 2 since as this ratio increases translation dissipation is larger than rotational dissipation; when ␥˜ 1 → ⬁ rotational dissipation dominates (G ⬙ ⫽ 0). The ratio M is the product of the two dissipation to torque ratios, and can be rewritten in terms of as M⫽
1 ˜ 1
•
1
2 ␥˜ 1 共1⫺兲2
.
共23兲
Newtonian behavior, M → ⬁, is found under: 共i兲 共ii兲 共iii兲
infinite translation viscosity: ˜ 1 → ⬁; inviscid rotations: ␥˜ 1 → 0; and absence of flow torques: ⫽ 1.
The resonance frequency at which maximum elastic storage occurs can be found solving the equation: d tan ␦/d˜ ⫽ 0, using Eq. 共15兲. The resulting nonlinear equation that defines resonance is ber20 冑˜ r ⫹bei20 冑˜ r ber21 冑˜ r ⫹bei21 冑˜ r
⫹
4
˜ r
⫺2 ˜ r F 2 共 ˜ r 兲 2 ⫽ 0
共24兲
and its unique solution is a universal constant independent of material parameters
˜ r ⫽ 18.6522.
共25兲
The EL theory thus predict that for the present flow geometry the resonance frequency is related to the splay orientational diffusivity D splay and material parameters as follows:
1076
de ANDRADE LIMA and REY
r ⫽ 18.6522
Dsplay ⫽
D splay R2
K 11
splay
共26a兲
,
共26b兲
,
where R is the capillary radius. The loss tangent at resonance (tan ␦r) can be calculated replacing Eq. 共25兲 in Eq. 共15兲 giving tan ␦r ⫽ 23.42M ⫺1.87395.
共27兲
Thus, measuring the loss tangent at resonance tan ␦r gives the viscoelastic ratio M. The earlier results show that the general material property dependency is ˜ ⬘共˜ ,M ,M 兲, ˜⬘ ⫽ G G 1
˜⬙ ⫽ G ˜ ⬙共˜ ,M ,M 兲, G 2
tan ␦ ⫽ tan ␦共˜ ,M 兲,
共28兲
while the characterization study presented below shows that for the seven data sets, the material dependence above resonance reduces to ˜⬘ ⫽ G ˜ ⬘共˜ ,M 兲, G 1
˜⬙ ⫽ G ˜ ⬙共˜ ,M 兲, G 2
tan ␦ ⫽ tan ␦共˜ ,M 兲.
共29兲
B. Characterization of viscoleastic functions and discussion For brevity in what follows the moduli for each polymer is identified by its number; ˜ ⬘ (5), while M (1) is ˜ ⬘ 关 ˜ ,(material properties for PBLG 17% in m-cresol) 兴 ⬅ G i.e., G the value of M for polymer 1, and so on. By using the known asymptotic behavior 关see Abramowitz and Stegun 共1972兲兴 of the Kelvin functions the frequency dependence of the viscoelastic moduli for ␣˜ 3 ⫽ 0 is as follows: the loss modulus is always greater than the storage modulus, the low frequency 共terminal兲 regime is classic of a viscous fluid, and the characteristic slopes are ˜ ⬘ ⬃ ˜ 2 , G ˜ ⬙ ⬃ ˜ ; as ˜ → 0, G
˜ ⬘ ⬃ ˜ 1/2, G ˜ ⬙ ⬃ ˜ . as ˜ → ⬁, G
共30兲
It follows from Eq. 共20兲 that for ␣˜ 3 ⫽ 0, M 1 ⫽ 0, as a result, the behavior is Newtonian ˜ ⬘ ⫽ 0. and G ˜ ⬙ ) as a function of dimensionless frequency ( ˜ ) The dimensionless loss modulus (G for the material set 1–7 共PSi4, AZA9, DDA9, TPB10, PBLG 17%, PPTA 8.8%, and PBLG 12%兲 shows that the slope of the seven lines is essentially one at all frequencies, ˜ ⬙ (1). As seen from ˜ ⬙ (5), and the largest is G the smallest value of the loss modulus is G ˜ Eq. 共14兲 the modulus G ⬙ is proportional to M 2 , which is a minimum for PBLG 17% in m-cresol 共set 5兲 and a maximum for PSi4 共set 1兲, see Table I. ˜ ⬘ ) as a function of dimensionless frequency The dimensionless storage modulus (G ˜ ( ˜ ) for the material set 1–3, 5–7; G ⬘ (4) ⫽ 0 because for TPG10 ␣ 3 ⫽ 0, shows that ˜ ⬘ (7) and the the slopes of the six curves follow the scaling 关Eq. 共30兲兴. The largest is G ˜ smallest is G ⬘ (5). The storage modulus increases with increasing M 1 , hence, explaining the ordering in the six curves at large frequency. The crossover involving sets 1 and 6 is due to additional parameters 关see Eq. 共13兲兴 that contribute in the low-frequency range but vanish at larger frequencies. The transition 共resonance兲 frequency ˜ r is inversely proportional to the reorientation splay viscosity ˜ splay . In agreement with the ˜ splay values in Table I, the largest is ˜ r (1) and the smallest is ˜ r (5).
LINEAR VISCOELASTICITY OF LIQUID CRYSTALS
1077
˜ ⬙ ) as a function of the dimensionless storage The dimensionless loss modulus (G ˜ modulus (G ⬘ ) for set 1–3, 5–7 shows that all data sets follow the same asymptotic behavior ˜⬙ ⬃ as ˜ → 0, G
冑G˜ ⬘ ;
˜⬙ ⬃ G ˜ ⬘2 as ˜ → ⬁, G
共31兲
˜ ⬙ (5) in agreement with the scaling shown in Eq. 共30兲. At large frequency the largest is G ˜ because M 1 (5) is smallest, while G ⬙ (7) is smallest because M 1 (7) is smallest. ˜ ⬙/G ˜ ⬘) as a function of the dimensionless frequency ( ˜ ) for The loss tangent (tan ␦ ⫽ G sets 1–3, 5–7 shows that tan ␦共1兲 is largest and tan ␦共6兲 is smallest, according to Eq. 共15兲 tan ␦ is a function of M; also tan ␦共1兲 ⬇ tan ␦共3兲 since M (1) ⬇ M (3). V. SUPERPOSITION AND UNIVERSALITY In this section we establish the origin of scaling, showing how to collapse the seven data sets in master plots. In addition we also show universality features between flowaligning and nonaligning materials, by superposing their responses. In the first part of this section we use order of magnitude analysis to show the origin of scaling and universality, and in the second part we use the seven data sets to assess the degree of scaling using the exact results given by Eqs. 共13兲, 共14兲, and 共15兲. A. Analytical results 1. Superposition
As indicated earlier, due to the properties and asymptotic behavior of the Kelvin functions 关Abramowicz and Stegun 共1972兲兴, at frequencies larger than resonance the dependency on material properties simplifies; as shown earlier, resonance is found at ˜ r ⫽ 18.6522, and the large frequency regime results in this section hold for ˜ ⭓ 10˜ r . The factorization of material properties from frequency dependent functions is obtained by direct order of magnitude analysis. At frequencies larger than the resonance ( ˜ r ), the terms F 1 /M and F 2 /M ⫺1/˜ M in Eqs. 共13兲 and 共14兲 are very small compared with one; in addition in the numerator of Eq. 共14兲 F 2 /M is very small compared with one; therefore, the asymptotic expressions for the viscoelastic functions for frequencies larger than ˜ r are ˜⬘ G M1
⫽ ˜ F2⫺1 ⫽ ⌽⬘共˜ 兲,
˜⬙ G M2 tan ␦ M
⫽ ˜ ⫽ ⌽⬙共˜ 兲,
冋
⫽ F2共˜ 兲⫺
1
˜
册
共32兲
共33兲
⫺1
⫽ ⌽共˜ 兲.
共34兲
These equations show the origin of the vertical 共amplitude兲 scaling of all data sets. The horizontal scaling 共frequency兲 is obtained by plotting the dimensionless storage modulus ˜ ⬙ ), and loss tangent 共tan ␦兲 as functions of ˜ . In addition, the ˜ ⬘ ), loss modulus (G (G functionality between the moduli is ˜⬙ G M2
F2
冉 冊 ˜⬙ G
M2
⫽
˜⬘ G M1
⫹1.
共35兲
1078
de ANDRADE LIMA and REY
˜ ⬘ /M ) as a function of the dimensionless frequency ( ˜ ) for FIG. 1. Scaled dimensionless storage modulus (G 1 PSi4, AZA9, DDA9, PBLG 17%, PPTA 8.8%, and PBLG 12%. The result to TPB10 共4兲 is not shown in the log–log plot, because its value is zero for all frequencies.
˜ ⬙ /M as a function of G ˜ ⬘ /M will be a unique This leads to expect that the plot of G 2 1 curve, especially for high frequencies, and the asymptotic slopes are given in Eq. 共31兲. 2. Universality
The superposition equation in the loss tangent is given in Eq. 共34兲. Combining Eq. 共34兲 and Eqs. 共24兲 and 共25兲, we find that for any nematic liquid crystal in the present flow geometry the universality condition at resonance is tan ␦r⫹1.87395 M
⫽ 23.42.
共36兲
When comparing two materials C and D (M ⫽ M C and M ⫽ M D ) the universality condition 关Eq. 共36兲兴 states that equivalence is present if tan ␦r共 M C兲 ⫽ tan ␦r共 M D兲,
MC ⫽ MD .
共37兲
For two aligning or two nonaligning materials, equivalence at resonance is present when the reactive parameters ( C , D ) are related by
冑冉 冊 冉 冊
C ⫽ 1⫺共1⫺D兲
˜ 1 ␥˜ 1
C
␥˜ 1 ˜ 1
, D
共38兲
LINEAR VISCOELASTICITY OF LIQUID CRYSTALS
1079
˜ ⬙ /M ) as a function of the dimensionless frequency ( ˜ ) for FIG. 2. Scaled dimensionless loss modulus (G 2 PSi4, AZA9, DDA9, TPB10, PBLG 17%, PPTA 8.8%, and PBLG 12%.
where the subscripts on the parenthesis identifies the material, and where the definition of M 关Eq. 共23兲兴 was used. Material equivalence between one aligning ( A ⬎ 1) and one nonaligning ( NA ⬍ 1) at resonance is present when the reactive parameters ( A , NA) are related by NA ⫽ 1⫹ 共 1⫺ A 兲
冑冉 冊 冉 冊 ˜ 1 ␥˜ 1
NA
␥˜ 1 ˜ 1
.
共39兲
A
The loss tangent 关tan共␦兲兴 as a function of the dimensionless frequency ( ˜ ) for polymers 1–3–5 is identical at large frequency, because M (1) ⬇ M (3) ⬇ M (5), which demonstrate the universality features between aligning and nonaligning materials. An additional important universality result that emerges in the high frequency regime is ˜ ⬘共˜ , 兲; 共1⫺ 兲2 ⫽ 共1⫺ 兲2. ˜ ⬘共˜ , 兲 ⫽ G G C D C D
共40兲
When comparing two flow aligning materials or two nonaligning materials the universality condition 共40兲 yields ˜ ⬘共˜ , 兲 ⫽ G ˜ ⬘共˜ , 兲; ⫽ , G C D C D
C ⫽ D ⬎ 1 or C ⫽ D ⬍ 1.
共41兲
On the other hand, when comparing a flow-aligning and a nonaligning material the universality condition given by Eq. 共40兲 becomes ˜ ⬘共˜ , 兲 ⫽ G ˜ ⬘共˜ , 兲 ; G A NA
NA ⫽ 2⫺ A .
共42兲
1080
de ANDRADE LIMA and REY
˜ ⬙ /M ) as a function of the scaled dimensionless storage moduFIG. 3. Scaled dimensionless loss modulus (G 2 ˜ lus (G ⬘ /M 1 ) for PSi4, AZA9, DDA9, PBLG 17%, PPTA 8.8%, and PBLG 12%. The result to TPB10 共4兲 is not shown in the log–log plot, because its value is zero for all frequencies.
Thus, we have shown that for the present flow conditions the storage modulus is independent of flow-alignment characteristics. A plot of dimensionless storage modulus ˜ ⬘ ) as a function of the dimensionless frequency ( ˜ ) shows that for high frequencies (G ˜ ⬘ (7), since 共7兲 ⫽ 0.861 ⬇ 2⫺共1兲 ⫽ 0.864, which demonstrate the uni˜ G ⬘ (1) ⬇ G versality features between aligning and nonaligning materials, predicted by Eq. 共42兲. B. Numerical evaluation of degree of superposition Figure 1 shows the scaled dimensionless storage modulus (G ⬘ /M 1 ) as a function of the dimensionless frequency ( ˜ ) for six data sets 共PSi4, AZA9, DDA9, PBLG 17%, PPTA 8.8%, and PBLG 12%兲. The figure shows a collapse of the curves is essentially perfect for ˜ ⬎ 30˜ r , validating the predictions of Eq. 共32兲. For PTB10 the storage modulus is zero. ˜ ⬙ /M ) as a function of the Figure 2 shows the scaled dimensionless loss modulus (G 2 dimensionless frequency ( ˜ ) for the seven data sets 共PSi4, AZA9, DDA9, PTB10, PBLG 17%, PPTA 8.8%, and PBLG 12%兲. The figure shows a collapse of the curves is essentially perfect for ˜ ⬎ 16˜ r , validating the prediction 关Eq. 共33兲兴. ˜ ⬙ /M ) as a function of the Figure 3 shows the scaled dimensionless loss modulus (G 2 scaled storage modulus (G ⬘ /M 1 ) to sets 1–3, and 5–7 共PSi4, AZA9, DDA9, PBLG 17%, PPTA 8.8%, and PBLG 12%兲. The figure shows a collapse of the curves is essentially perfect for nearly all frequencies, validating the predictions 关Eqs. 共32兲 and 共33兲兴. The slopes follow the scaling 关Eq. 共31兲兴. Figure 4 shows the scaled loss tangent (tan ␦/M) as a function of the dimensionless frequency ( ˜ ) for sets 1–3, and 5–7 共PSi4, AZA9, DDA9, PBLG 17%, PPTA 8.8%, and
LINEAR VISCOELASTICITY OF LIQUID CRYSTALS
1081
FIG. 4. Scaled loss tangent (tan ␦/M) as a function of the dimensionless frequency ( ˜ ) for PSi4, AZA9, DDA9, PBLG 17%, PPTA 8.8%, and PBLG 12%. The result for TPB10 共4兲 is not shown because its value is zero for all frequencies.
PBLG 12%兲. These results show a collapse of the plots is almost perfect especially to high frequencies, validating Eq. 共34兲.
VI. CONCLUSIONS This paper shows through scaling, analytical, and numerical solutions of the Leslie– Ericksen equations that the rheological responses of defect-free liquid crystal polymers subjected to small-amplitude oscillatory pressure-driven Poiseuille flow, show superposition and universality. In general the degree of superposition is almost perfect at frequencies above resonance. Although liquid crystals are usually differentiated into shear aligning and nonaligning materials, this paper shows that under certain flow conditions, linear viscoelasticity only depends on the dimensionless factor (1⫺) 2 , and hence, rheological equivalence between aligning ( A ) and nonaligning ( NA) materials exists if NA ⫽ 2⫺ A . In addition resonance is also shown to be independent of flow alignment.
ACKNOWLEDGMENTS This research was supported by a grant from Engineering Research Center program of the National Science Foundation under award No. EEC 9731680. L.R.P.d.A.L. gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada 共NSERC兲.
1082
de ANDRADE LIMA and REY
APPENDIX The in-phase i (r˜ , ˜ ) and out-of-phase o (r˜ , ˜ ) director components in Eq. 共9兲 are found to be:
i ⫽
o ⫽
冋
册
␣˜ 3E0 ber1 冑˜ r˜ bei1 冑˜ ⫺bei1 冑˜ r˜ ber1 冑˜ , 2˜ 1 ˜ 共 ber2 冑˜ ⫹bei2 冑˜ 兲 1
冋
1
共A1兲
册
␣˜ 3E0 ber1 冑˜ r˜ ber1 冑˜ ⫹bei1 冑˜ r˜ bei1 冑˜ r˜ ⫺ , 2˜ 1 ˜ ˜ 共 ber2 冑˜ ⫹bei2 冑˜ 兲 1
1
共A2兲
where bei (x) and ber (x) are the Kelvin functions of order . The in-phase v˜ i (r˜ , ˜ ) and out-of-phase v˜ o (r˜ , ˜ ) velocity components in Eq. 共10兲 are found to be
˜2
v˜ i ⫽ 共1⫺r 兲
⫹
E0 4˜ 1
冋
冉 冊 冋 1⫹
␣˜ 23
˜ 1
⫹
␣˜ 23E0 ber1 冑˜ 共 ber0 冑˜ r˜ ⫺bei0 冑˜ r˜ ⫺ber0 冑˜ ⫹bei0 冑˜ 兲
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
2˜ 21
␣˜ 23 E 0 bei1 冑˜ 共 bei0 冑˜ r˜ ⫹ber0 冑˜ r˜ ⫺bei0 冑˜ ⫺ber0 冑˜ 兲 2 ˜ 21
v˜ o ⫽ ⫺
⫹
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
冋 冋
册
共A3兲
,
␣˜ 23E0 bei1 冑˜ 共 ber0 冑˜ r˜ ⫺bei0 冑˜ r˜ ⫺ber0 冑˜ ⫹bei0 冑˜ 兲 2˜ 21
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
册
␣˜ 23 E 0 ber1 冑˜ 共 bei0 冑˜ r˜ ⫹ber0 冑˜ r˜ ⫺bei0 冑˜ ⫺ber0 冑˜ 兲 2 ˜ 21
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
册
册
.
共A4兲
˜ ) dimensionless ˜ ) and out-of-phase (Q Using Eqs. 共A3兲, 共A4兲, and 共11a兲 the in-phase (Q i o flow rates components in Eq. 共12兲 are found to be
˜ ⫽ Q i
E0 8˜ 1
⫹
冉 冊 1⫹
␣˜ 23
˜ 1
2 E 0 8 ␣˜ 3
8 ˜ 1 ˜ 1
⫹
冋
2 E0 8␣˜ 3
8˜ 1 ˜ 1
冉
冋
ber1 冑˜
bei1 冑˜ ⫺
冉
1 1 bei1 冑˜ ⫹ bei0 冑˜ ⫺ ber0 冑˜ 2 2 冑2 ˜ 2
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
2
1 1 ber1 冑˜ ⫺ bei0 冑˜ ⫺ ber0 冑˜ 2 2 冑2 ˜
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
冊册
,
冊册 共A5兲
LINEAR VISCOELASTICITY OF LIQUID CRYSTALS
˜ ⫽ Q o
2 E0 8␣˜ 3
8˜ 1 ˜ 1
⫹
冋
冉
bei1 冑˜ ⫺
2 E 0 8 ␣˜ 3
8 ˜ 1 ˜ 1
冋
2
1 1 bei1 冑˜ ⫺ bei0 冑˜ ⫹ ber0 冑˜ 2 2 冑2 ˜
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
冉
ber1 冑˜ ⫺
2
1083
冊册
1 1 ber1 冑˜ ⫺ bei0 冑˜ ⫺ ber0 冑˜ 2 2 冑2 ˜
冑2 ˜ 共 ber21 冑˜ ⫹bei21 冑˜ 兲
Generalizing Eq. 共11b兲 to oscillatory flow we find
冊册
. 共A6兲
共A7兲
Separating real and imaginary components we find
˜ ⬘Q˜o⫺˜ ⬙Q˜i ⫽ 0, ˜ ⬘Q˜i⫹˜ ⬙Q˜o ⫽
共A8a兲
E0
共A8b兲
8
and
˜ ⬘ ⫽ ˜ ⬙ ⫽
˜⬙ G
˜ ˜⬘ G
˜
⫽
⫽
E0
˜ Q i
˜ 2⫹Q ˜ 2兲 8 共Q i o
E0
˜ Q o
˜ 2⫹Q ˜ 2兲 8 共Q i o
,
共A9a兲
,
共A9b兲
˜ ⬘ ⫽ ˜ ⬙ ˜ ) ˜ ⬙ ⫽ ˜ ⬘ ˜ and G where the definitions for the loss and storage modulus (G were used.
References Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions 共Dove, New York, 1972兲. Atkin, R. J., ‘‘Poiseuille flow of liquid crystals of the nematic type,’’ Arch. Ration. Mech. Anal. 38, 225–240 共1970兲. Barnes, H. A., J. F. Hutton, and K. Walters, An Introduction to Rheology 共Elsevier Science, New York, 1989兲. Beens, W. W., and W. H. de Jeu, ‘‘The flow alignment of some nematic liquid crystals,’’ J. Chem. Phys. 82, 3841–3846 共1985兲. Bird, R. B., R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids 共Wiley, New York, 1989兲. Burghardt, W. R., ‘‘Oscillatory shear flow of nematic liquid crystals,’’ J. Rheol. 35, 49– 62 共1991兲. Chandrasekhar, F. R. S., Liquid Crystals, 2nd ed. 共Cambridge University Press, Cambridge, 1992兲. de Andrade Lima, L. R. P., and A. D. Rey, ‘‘Linear and nonlinear viscoelasticity of discotic nematics under transient Poiseuille flows,’’ J. Rheol. 47, 1261–1282 共2003a兲. de Andrade Lima, L. R. P., and A. D. Rey, ‘‘Poiseuille flow of Leslie–Ericksen discotic liquid crystals: solution multiplicity, multistability, and non-Newtonian rheology,’’ J. Non-Newtonian Fluid Mech. 110, 103–142 共2003b兲. de Gennes, P. G., and J. Prost, The Physics of Liquid Crystals, 2nd ed. 共Oxford University Press, London, 1993兲. Ericksen, J. L., ‘‘Conservation laws for liquid crystals,’’ Trans. Soc. Rheol. 5, 23–34 共1961兲.
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Grecov, D., and A. D. Rey, ‘‘Shear-induced textural transitions in flow-aligning liquid crystal polymers,’’ Phys. Rev. E 68, 061704 –1-20 共2003兲. Han, W. H., and A. D. Rey, ‘‘Simulation and validation of temperature effects on the nematorheology of aligning and non-aligning liquid crystals,’’ J. Rheol. 39, 301–322 共1995兲. Lacerda Santos, M. B., Y. Galerne, and G. Durand, ‘‘Orientational diffusivities in a disk-like lyotropic nematic phase: A strong backflow effect,’’ J. Phys. 共Paris兲 46, 933–937 共1985兲. Larson, R. G., and D. W. Mead, ‘‘Development of orientation and texture during shering of liquid-crystalline polymers,’’ Liq. Cryst. 12, 751–768 共1992兲. Larson, R. G., and D. W. Mead, ‘‘The Ericksen number and Deborah number cascades in sheared polymeric nematics,’’ Liq. Cryst. 15, 151–169 共1993兲. Larson, R. G., The Structure and Rheology of Complex Fluids 共Oxford University Press, New York, 1999兲. Leslie, F. M., ‘‘Some constitutive equations for liquid crystals,’’ Arch. Ration. Mech. Anal. 28, 265–283 共1968兲. Leslie, F. M., ‘‘Theory of flow phenomena in liquid crystals,’’ Adv. Liq. Cryst. 4, 1– 81 共1979兲. Marrucci, G., and F. Greco, ‘‘Flow behavior of liquid crystalline polymers,’’ Adv. Chem. Phys. 86, 331– 440 共1993兲. Martins, A. F., ‘‘Measurement of viscoelastic coefficients for nematic mesophases using magnetic resonance,’’ in Physical Properties of Liquid Crystals: Nematics, edited by D. A. Dunmur, A. Fukuda, and G. Luckhurst 共IEE, London, 2001兲, pp. 405– 413. Mather, P. T., D. S. Pearson, and R. G. Larson, ‘‘Flow patterns and disclination-density measurements in sheared nematic liquid crystals I: Flow-aligning 5CB,’’ Liq. Cryst. 20, 527–538 共1996兲. Mather, P. T., H. G. Jeon, C. D. Han, and S. Chang, ‘‘Morphological and rheological responses to shear start-up and flow reversal of thermotropic liquid-crystalline polymers,’’ Macromolecules 29, 7594 –7608 共2000兲. Pasechnik, S. V., V. A. Tsvetkov, A. V. Torchinskaya, and D. O. Karandashov, ‘‘Nematic liquid crystals under decay Poiseuille flow: New possibilites for measurement of shear viscosity,’’ Mol. Cryst. Liq. Cryst. 366, 2017–2023 共2001兲. Pikin, S. A., Structural Transformations in Liquid Crystals 共Gordon and Breach Science, New York, 1991兲. Rey, A. D., ‘‘Theory of linear viscoelasticity for single-phase nematic mixtures,’’ Mol. Cryst. Liq. Cryst. 281, 155–170 共1996兲. Rey, A. D., ‘‘Theory of linear viscoelasticity in cholesteric liquid crystals,’’ J. Rheol. 44, 855– 869 共2000兲. Rey, A. D., ‘‘Simple shear and small amplitude oscillatory rectilinear shear permeation flows of cholesteric liquid crystals,’’ J. Rheol. 46, 225–240 共2002兲. Rey, A. D., and M. M. Denn, ‘‘Dynamical phenomena in liquid-crystalline materials,’’ Annu. Rev. Fluid Mech. 34, 233–266 共2002兲. Romo-Uribe, A., and A. H. Windle, ‘‘ ‘Log-rolling’ alignment in main-chain thermotropic liquid crystalline polymer melts under shear: An in-situ WAXS study,’’ Macromolecules 29, 6246 – 6255 共1996兲. Ugaz, V. M., and W. R. Burghardt, ‘‘In situ x-ray scattering study of a model thermotropic copolyester under shear: Evidence and consequences of flow-aligning behavior,’’ Macromolecules 31, 8474 – 8484 共1998兲. Zhou, W.-J., J. A. Kornfield, and W. R. Burghardt, ‘‘Shear aligning properties of a main-chain thermotropic liquid crystalline polymer,’’ Macromolecules 34, 3654 –3660 共2001兲.
