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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

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 & 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. Google Scholar

[2]

W. R. Burghardt, Oscillatory shear flow of nematic liquid crystals, J. Rheol., 35 (1991), 49-62. doi: doi:10.1122/1.550208.  Google Scholar

[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.  Google Scholar

[4]

Chandrasekhar, "Liquid Crystals," 2nd ed., Cambridge University Press, Cambridge, 1992. Google Scholar

[5]

Z. Cui, Small amplitude oscillatory shear permeation flow of cholesteric liquid crystal polymers, Communications in Mathematical Sciences, 8 (2010), 943-963. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Oxford University Press, 1993. Google Scholar

[13]

G. Derfel, Shear flow induced cholesteric-nematic transition, Mol. Cryst. Liq. Cryst., 92 (1983), 41-47. doi: doi:10.1080/01406568308084517.  Google Scholar

[14]

W. Helfrich, Capillary flow of cholesteric and semctic liquid crystals, Phys. Rev. Lett., 23 (1969), 372. doi: doi:10.1103/PhysRevLett.23.372.  Google Scholar

[15]

W. Helfrich, Capillary viscometry of cholesteric liquid crystals, in "Proceedings Am. Chem. Soc.," 2nd ed., (1970), 405-418. Google Scholar

[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.  Google Scholar

[17]

U. D. Kini, Shear flow of cholesterics normal to the helical axis, J. Phys. (France), 40 (1979), 62-65. Google Scholar

[18]

R. G. Larson, "The Structure and Rheology of Complex Fluids," Oxford University Press (1999). Google Scholar

[19]

F. M. Leslie, Continuum theory of cholesteric liquid crystals, Mol. Cryst. Liq. Cryst., 7 (1969), 407-420. doi: doi:10.1080/15421406908084887.  Google Scholar

[20]

F. M. Leslie, The theory of flow phenomena in liquid crystals, Adv. Liq. Cryst. 4 (1979), 1-81. Google Scholar

[21]

T. Lubenski, Hydrodynamics of cholesteric liquid crystals, Phys. Rev. A, 6 (1969), 452-461. doi: doi:10.1103/PhysRevA.6.452.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. D. Rey, Theory of linear viscoelasticity for chiral liquid crystals, Rheol. Acta, 35 (1996), 400-409. doi: doi:10.1007/BF00368991.  Google Scholar

[29]

A. D. Rey, Theory of linear viscoelasticity in cholesteric liquid crystals, J. Rheol., 44 (2000), 855-869. doi: doi:10.1122/1.551112.  Google Scholar

[30]

A. D. Rey, Generalized cholesteric permeation flows, Phys. Rev. E, 65 (2002), 022701. doi: doi:10.1103/PhysRevE.65.022701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[2]

W. R. Burghardt, Oscillatory shear flow of nematic liquid crystals, J. Rheol., 35 (1991), 49-62. doi: doi:10.1122/1.550208.  Google Scholar

[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.  Google Scholar

[4]

Chandrasekhar, "Liquid Crystals," 2nd ed., Cambridge University Press, Cambridge, 1992. Google Scholar

[5]

Z. Cui, Small amplitude oscillatory shear permeation flow of cholesteric liquid crystal polymers, Communications in Mathematical Sciences, 8 (2010), 943-963. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Oxford University Press, 1993. Google Scholar

[13]

G. Derfel, Shear flow induced cholesteric-nematic transition, Mol. Cryst. Liq. Cryst., 92 (1983), 41-47. doi: doi:10.1080/01406568308084517.  Google Scholar

[14]

W. Helfrich, Capillary flow of cholesteric and semctic liquid crystals, Phys. Rev. Lett., 23 (1969), 372. doi: doi:10.1103/PhysRevLett.23.372.  Google Scholar

[15]

W. Helfrich, Capillary viscometry of cholesteric liquid crystals, in "Proceedings Am. Chem. Soc.," 2nd ed., (1970), 405-418. Google Scholar

[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.  Google Scholar

[17]

U. D. Kini, Shear flow of cholesterics normal to the helical axis, J. Phys. (France), 40 (1979), 62-65. Google Scholar

[18]

R. G. Larson, "The Structure and Rheology of Complex Fluids," Oxford University Press (1999). Google Scholar

[19]

F. M. Leslie, Continuum theory of cholesteric liquid crystals, Mol. Cryst. Liq. Cryst., 7 (1969), 407-420. doi: doi:10.1080/15421406908084887.  Google Scholar

[20]

F. M. Leslie, The theory of flow phenomena in liquid crystals, Adv. Liq. Cryst. 4 (1979), 1-81. Google Scholar

[21]

T. Lubenski, Hydrodynamics of cholesteric liquid crystals, Phys. Rev. A, 6 (1969), 452-461. doi: doi:10.1103/PhysRevA.6.452.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. D. Rey, Theory of linear viscoelasticity for chiral liquid crystals, Rheol. Acta, 35 (1996), 400-409. doi: doi:10.1007/BF00368991.  Google Scholar

[29]

A. D. Rey, Theory of linear viscoelasticity in cholesteric liquid crystals, J. Rheol., 44 (2000), 855-869. doi: doi:10.1122/1.551112.  Google Scholar

[30]

A. D. Rey, Generalized cholesteric permeation flows, Phys. Rev. E, 65 (2002), 022701. doi: doi:10.1103/PhysRevE.65.022701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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