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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

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.
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.

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