Permeation flows in cholesteric liquid crystal polymers under oscillatory shear

Previous Article
Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays
DCDS-B Home
This Issue
Next Article
A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity

January 2011, 15(1): 45-60. doi: 10.3934/dcdsb.2011.15.45

Permeation flows in cholesteric liquid crystal polymers under oscillatory shear

Zhenlu Cui ^1, and Qi Wang ^2,

1.	Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC 28301, United States
2.	Department of Mathematics & NanoCenter, University of South Carolina, Columbia, SC 29208, United States

Received November 2009 Revised July 2010 Published October 2010

Abstract
Related Papers

We investigate the permeation flow of cholesteric liquid crystal polymers (CLCPs) subject to a small amplitude oscillatory shear using a tensor theory developed by the authors [8]. We model the material system by the Stokes hydrodynamic equations coupled with the orientational dynamics. At low frequencies, the steady permeation modes are recovered and the director rotates in phase with the applied shear. At high frequencies, the out of phase component dominates the dynamics. The asymptotic formulas for the loss modulus ($G''$) and storage modulus ($G^{'}$) are obtained at both low and high frequencies. In the low frequency limit, both the loss modulus and the storage modulus are shown to exhibit a classical frequency $\omega$ dependence ($G^{''} \propto \omega$, $G^{'} \propto \omega^2$ ) with the proportionality of order $O(Er)$ and $O(q)$, respectively, where $\frac{2\pi}{q}$ defines the pitch of the chiral liquid crystal and $Er$ is the Ericksen number of the liquid crystal polymer system. The magnitudes of dimensionless complex flow rate and complex viscosity are calculated. They are shown to have two Newtonian plateaus at low and high frequencies while a power-law response at intermediate frequencies.

Keywords: Cholesteric liquid crystal polymers, asymptotic expansion, permeation flows, oscillatory shear, viscoelasticity..

Mathematics Subject Classification: 76A15, 76M4.

Citation: Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45

References:

[1]	B. Bird, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids," Vol 1, John Wiley and Sons, New York, 1987.
[2]	W. R. Burghardt, Oscillatory shear flow of nematic liquid crystals, J. Rheol., 35 (1991), 49-62. doi: doi:10.1122/1.550208.
[3]	E. Choate, Z. Cui and M. G. Forest, Effects of strong anchoring on the dynamics moduli of heterogeneous nematic polymers, Rheological Acta., 47 (2008), 223-236. doi: doi:10.1007/s00397-007-0235-2.
[4]	Chandrasekhar, "Liquid Crystals," 2nd ed., Cambridge University Press, Cambridge, 1992.
[5]	Z. Cui, Small amplitude oscillatory shear permeation flow of cholesteric liquid crystal polymers, Communications in Mathematical Sciences, 8 (2010), 943-963.
[6]	Z. Cui, M. C. Calderer and Q. Wang, Mesostructures in flows of weakly sheared chiral liquid crystalline polymers, Discrete and Continuous Dynamical Systems-Series B, 6 (2006), 291-310.
[7]	Z. Cui, et al., On weak plane shear and Poiseuille flows of rigid rod and platelet ensembles, SIAM J. Appl. Math., 66 (2006), 1227-1260. doi: doi:10.1137/04061934x.
[8]	Z. Cui and Q. Wang, A continuum model for flows of chiral liquid crystal polymers and permeation flows, J. Non-Newtonian Fluid Mech., 138 (2006), 44-61. doi: doi:10.1016/j.jnnfm.2006.04.005.
[9]	L. R. P. de Andrade Lima and A. D. Rey, Superposition and universality in the linear viscoelasticity of Leslie-Ericken liquid crystals, J Rheol., 48 (2004), 1067-1084. doi: doi:10.1122/1.1773784.
[10]	L. R. P. de Andrade Lima and A. D. Rey, Assessing flow alignment of nematic liquid crystals through linear viscoelasticity, Phys. Rev. E, 70 (2004), 011701. doi: doi:10.1103/PhysRevE.70.011701.
[11]	L. R. P. de Andrade Lima and A. D. Rey, Superposition principles for small amplitude oscillatory shearing of nematic mesophases, Rheol. Acta., 45 (2006), 591-600.
[12]	P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Oxford University Press, 1993.
[13]	G. Derfel, Shear flow induced cholesteric-nematic transition, Mol. Cryst. Liq. Cryst., 92 (1983), 41-47. doi: doi:10.1080/01406568308084517.
[14]	W. Helfrich, Capillary flow of cholesteric and semctic liquid crystals, Phys. Rev. Lett., 23 (1969), 372. doi: doi:10.1103/PhysRevLett.23.372.
[15]	W. Helfrich, Capillary viscometry of cholesteric liquid crystals, in "Proceedings Am. Chem. Soc.," 2nd ed., (1970), 405-418.
[16]	U. D. Kini, G. S. Ranganath and S. Chandrasekhar, Flow of cholesteric liquid crystals-I: Flow along the helical axis, Pramana, 5 (1975), 101-106. doi: doi:10.1007/BF02846036.
[17]	U. D. Kini, Shear flow of cholesterics normal to the helical axis, J. Phys. (France), 40 (1979), 62-65.
[18]	R. G. Larson, "The Structure and Rheology of Complex Fluids," Oxford University Press (1999).
[19]	F. M. Leslie, Continuum theory of cholesteric liquid crystals, Mol. Cryst. Liq. Cryst., 7 (1969), 407-420. doi: doi:10.1080/15421406908084887.
[20]	F. M. Leslie, The theory of flow phenomena in liquid crystals, Adv. Liq. Cryst. 4 (1979), 1-81.
[21]	T. Lubenski, Hydrodynamics of cholesteric liquid crystals, Phys. Rev. A, 6 (1969), 452-461. doi: doi:10.1103/PhysRevA.6.452.
[22]	D. Marenduzzo, E. Orlandini and J. M. Yeomans, Permeative flows in cholesteric liquid crystals, Physical Review Letters, 92 (2004), 188301. doi: doi:10.1103/PhysRevLett.92.188301.
[23]	D. Marenduzzo, E. Rlandini and J. M. Yeomans, Interplay between shear flow and elastic deformations in liquid crystals, J. Chem. Phys. 121 (2004), 582-591. doi: doi:10.1063/1.1757441.
[24]	J. Prost, Y. Pomeau and E. Guyon, Stability of permeative flows in 1 dimensionally ordered systems, J. Phys. II, 1 (1991), 289-309. doi: doi:10.1051/jp2:1991169.
[25]	A. D. Rey, Flow-alignment in the helix uncoiling of cholesteric liquid crystals, Phys. Rev. E, 53 (1986), 4198-4201. doi: doi:10.1103/PhysRevE.53.4198.
[26]	A. D. Rey, Structural transformations and viscoelastic response of shear fingerprint cholesteric textures, J. Non-Newt. Fluid Mech., 64 (1996), 207-227. doi: doi:10.1016/0377-0257(96)01434-6.
[27]	A. D. Rey, Helix uncoiling modes of sheared cholesteric liquid crystals, J. Chem. Phys., 104 (1996), 789-792. doi: doi:10.1063/1.471184.
[28]	A. D. Rey, Theory of linear viscoelasticity for chiral liquid crystals, Rheol. Acta, 35 (1996), 400-409. doi: doi:10.1007/BF00368991.
[29]	A. D. Rey, Theory of linear viscoelasticity in cholesteric liquid crystals, J. Rheol., 44 (2000), 855-869. doi: doi:10.1122/1.551112.
[30]	A. D. Rey, Generalized cholesteric permeation flows, Phys. Rev. E, 65 (2002), 022701. doi: doi:10.1103/PhysRevE.65.022701.
[31]	A. D. Rey, Simple shear and small amplitude oscillatory rectilinear shear permeation flows of cholesteric liquid crystals, J. Rheol., 46 (2002), 225-240. doi: doi:10.1122/1.1428317.
[32]	N. Scaramuzza, F. Simoni and R. Bartolino, Permeative flow in cholesteric liquid crystals, Phys. Rev. Lett., 53 (1984), 2246-2249. doi: doi:10.1103/PhysRevLett.53.2246.

show all references

References:

[1]	B. Bird, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids," Vol 1, John Wiley and Sons, New York, 1987.
[2]	W. R. Burghardt, Oscillatory shear flow of nematic liquid crystals, J. Rheol., 35 (1991), 49-62. doi: doi:10.1122/1.550208.
[3]	E. Choate, Z. Cui and M. G. Forest, Effects of strong anchoring on the dynamics moduli of heterogeneous nematic polymers, Rheological Acta., 47 (2008), 223-236. doi: doi:10.1007/s00397-007-0235-2.
[4]	Chandrasekhar, "Liquid Crystals," 2nd ed., Cambridge University Press, Cambridge, 1992.
[5]	Z. Cui, Small amplitude oscillatory shear permeation flow of cholesteric liquid crystal polymers, Communications in Mathematical Sciences, 8 (2010), 943-963.
[6]	Z. Cui, M. C. Calderer and Q. Wang, Mesostructures in flows of weakly sheared chiral liquid crystalline polymers, Discrete and Continuous Dynamical Systems-Series B, 6 (2006), 291-310.
[7]	Z. Cui, et al., On weak plane shear and Poiseuille flows of rigid rod and platelet ensembles, SIAM J. Appl. Math., 66 (2006), 1227-1260. doi: doi:10.1137/04061934x.
[8]	Z. Cui and Q. Wang, A continuum model for flows of chiral liquid crystal polymers and permeation flows, J. Non-Newtonian Fluid Mech., 138 (2006), 44-61. doi: doi:10.1016/j.jnnfm.2006.04.005.
[9]	L. R. P. de Andrade Lima and A. D. Rey, Superposition and universality in the linear viscoelasticity of Leslie-Ericken liquid crystals, J Rheol., 48 (2004), 1067-1084. doi: doi:10.1122/1.1773784.
[10]	L. R. P. de Andrade Lima and A. D. Rey, Assessing flow alignment of nematic liquid crystals through linear viscoelasticity, Phys. Rev. E, 70 (2004), 011701. doi: doi:10.1103/PhysRevE.70.011701.
[11]	L. R. P. de Andrade Lima and A. D. Rey, Superposition principles for small amplitude oscillatory shearing of nematic mesophases, Rheol. Acta., 45 (2006), 591-600.
[12]	P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Oxford University Press, 1993.
[13]	G. Derfel, Shear flow induced cholesteric-nematic transition, Mol. Cryst. Liq. Cryst., 92 (1983), 41-47. doi: doi:10.1080/01406568308084517.
[14]	W. Helfrich, Capillary flow of cholesteric and semctic liquid crystals, Phys. Rev. Lett., 23 (1969), 372. doi: doi:10.1103/PhysRevLett.23.372.
[15]	W. Helfrich, Capillary viscometry of cholesteric liquid crystals, in "Proceedings Am. Chem. Soc.," 2nd ed., (1970), 405-418.
[16]	U. D. Kini, G. S. Ranganath and S. Chandrasekhar, Flow of cholesteric liquid crystals-I: Flow along the helical axis, Pramana, 5 (1975), 101-106. doi: doi:10.1007/BF02846036.
[17]	U. D. Kini, Shear flow of cholesterics normal to the helical axis, J. Phys. (France), 40 (1979), 62-65.
[18]	R. G. Larson, "The Structure and Rheology of Complex Fluids," Oxford University Press (1999).
[19]	F. M. Leslie, Continuum theory of cholesteric liquid crystals, Mol. Cryst. Liq. Cryst., 7 (1969), 407-420. doi: doi:10.1080/15421406908084887.
[20]	F. M. Leslie, The theory of flow phenomena in liquid crystals, Adv. Liq. Cryst. 4 (1979), 1-81.
[21]	T. Lubenski, Hydrodynamics of cholesteric liquid crystals, Phys. Rev. A, 6 (1969), 452-461. doi: doi:10.1103/PhysRevA.6.452.
[22]	D. Marenduzzo, E. Orlandini and J. M. Yeomans, Permeative flows in cholesteric liquid crystals, Physical Review Letters, 92 (2004), 188301. doi: doi:10.1103/PhysRevLett.92.188301.
[23]	D. Marenduzzo, E. Rlandini and J. M. Yeomans, Interplay between shear flow and elastic deformations in liquid crystals, J. Chem. Phys. 121 (2004), 582-591. doi: doi:10.1063/1.1757441.
[24]	J. Prost, Y. Pomeau and E. Guyon, Stability of permeative flows in 1 dimensionally ordered systems, J. Phys. II, 1 (1991), 289-309. doi: doi:10.1051/jp2:1991169.
[25]	A. D. Rey, Flow-alignment in the helix uncoiling of cholesteric liquid crystals, Phys. Rev. E, 53 (1986), 4198-4201. doi: doi:10.1103/PhysRevE.53.4198.
[26]	A. D. Rey, Structural transformations and viscoelastic response of shear fingerprint cholesteric textures, J. Non-Newt. Fluid Mech., 64 (1996), 207-227. doi: doi:10.1016/0377-0257(96)01434-6.
[27]	A. D. Rey, Helix uncoiling modes of sheared cholesteric liquid crystals, J. Chem. Phys., 104 (1996), 789-792. doi: doi:10.1063/1.471184.
[28]	A. D. Rey, Theory of linear viscoelasticity for chiral liquid crystals, Rheol. Acta, 35 (1996), 400-409. doi: doi:10.1007/BF00368991.
[29]	A. D. Rey, Theory of linear viscoelasticity in cholesteric liquid crystals, J. Rheol., 44 (2000), 855-869. doi: doi:10.1122/1.551112.
[30]	A. D. Rey, Generalized cholesteric permeation flows, Phys. Rev. E, 65 (2002), 022701. doi: doi:10.1103/PhysRevE.65.022701.
[31]	A. D. Rey, Simple shear and small amplitude oscillatory rectilinear shear permeation flows of cholesteric liquid crystals, J. Rheol., 46 (2002), 225-240. doi: doi:10.1122/1.1428317.
[32]	N. Scaramuzza, F. Simoni and R. Bartolino, Permeative flow in cholesteric liquid crystals, Phys. Rev. Lett., 53 (1984), 2246-2249. doi: doi:10.1103/PhysRevLett.53.2246.

[1]	Zhenlu Cui, M. Carme Calderer, Qi Wang. Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291
[2]	Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739
[3]	Marissa Condon, Jing Gao, Arieh Iserles. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4813-4837. doi: 10.3934/dcds.2016008
[4]	Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691
[5]	Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[6]	Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455
[7]	Chun Liu, Jie Shen. On liquid crystal flows with free-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 307-318. doi: 10.3934/dcds.2001.7.307
[8]	Xiaoli Li, Boling Guo. Well-posedness for the three-dimensional compressible liquid crystal flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1913-1937. doi: 10.3934/dcdss.2016078
[9]	Hong Zhou, M. Gregory Forest, Qi Wang. Anchoring-induced texture & shear banding of nematic polymers in shear cells. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 707-733. doi: 10.3934/dcdsb.2007.8.707
[10]	Yang Liu, Xin Zhong. On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5219-5238. doi: 10.3934/cpaa.2020234
[11]	Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229
[12]	Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure and Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73
[13]	Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165
[14]	Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations and Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007
[15]	Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[16]	Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[17]	Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2631-2648. doi: 10.3934/dcdsb.2016065
[18]	Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
[19]	Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2125-2146. doi: 10.3934/cpaa.2012.11.2125
[20]	Zhaoyang Qiu, Yixuan Wang. Martingale solution for stochastic active liquid crystal system. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2227-2268. doi: 10.3934/dcds.2020360

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Zhenlu Cui Qi Wang
on Google Scholar
Zhenlu Cui Qi Wang

[Back to Top]