American Institute of Mathematical Sciences

Advanced
April  2015, 8(2): 323-340. doi: 10.3934/dcdss.2015.8.323

Diffusive transport in two-dimensional nematics

Ibrahim Fatkullin 1, and  Valeriy Slastikov 2,

1. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721, United States
2. 

Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Received  June 2013 Revised  November 2013 Published  July 2014

We discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the Doi-Smoluchowski kinetic equation and vortex dynamics in two-dimensional systems. We also discuss moment closures for the kinetic equation and Landau-de Gennes-type free energy dissipation.
Keywords: diffusive transport, Liquid crystals, Doi-Smoluchowski, nematics, vortex motion..
Mathematics Subject Classification: Primary: 76A15, 82B21, 82D30; Secondary: 35Q82, 35R0.
Citation: Ibrahim Fatkullin, Valeriy Slastikov. Diffusive transport in two-dimensional nematics. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 323-340. doi: 10.3934/dcdss.2015.8.323
References:
[1]

R. Alicandro and M. Ponsiglione, Ginzburg-Landau functionals and renormalized energy: A revised $\Gamma$-convergence approach,, J. Funct. Anal., 266 (2014), 4890.  doi: 10.1016/j.jfa.2014.01.024.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics, (2005).   Google Scholar

[3]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure,, Oxford University Press, (1994).   Google Scholar

[4]

F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices,, Progress in Nonlinear Differential Equations and their Applications, (1994).  doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[5]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, Clarendon Press, (1995).   Google Scholar

[6]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics,, Clarendon Press, (1999).   Google Scholar

[7]

W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity,, Physica D, 77 (1994), 383.  doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

[8]

W. E and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit,, Methods and Applications of Analysis, 13 (2006), 181.  doi: 10.4310/MAA.2006.v13.n2.a5.  Google Scholar

[9]

J. L. Ericksen, Conservation laws for liquid crystals,, Journal of Rheology, 5 (1961), 23.  doi: 10.1122/1.548883.  Google Scholar

[10]

I. Fatkullin and V. Slastikov, On spatial variations of nematic ordering,, Physica D, 237 (2008), 2577.  doi: 10.1016/j.physd.2008.03.048.  Google Scholar

[11]

I. Fatkullin and V. Slastikov, Vortices in two-dimensional nematics,, Communications in Mathematical Sciences, 7 (2009), 917.  doi: 10.4310/CMS.2009.v7.n4.a6.  Google Scholar

[12]

F. M. Leslie, Some constitutive equations for liquid crystals,, Archive for Rational Mechanics and Analysis, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[13]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices,, Communications on Pure and Applied Mathematics, 49 (1996), 323.  doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.  Google Scholar

[14]

J. C. Neu, Vortices in complex scalar fields,, Physica D, 43 (1990), 385.  doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[15]

E. Sandier, Lower bounds for the energy of unit vector fields and applications,, Journal of Functional Analysis, 152 (1998), 379.  doi: 10.1006/jfan.1997.3170.  Google Scholar

[16]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Communications on Pure and Applied Mathematics, 57 (2004), 1627.  doi: 10.1002/cpa.20046.  Google Scholar

[17]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (1992).   Google Scholar

[18]

P. Zhang W. Wang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation,, , (2013).   Google Scholar

show all references

References:
[1]

R. Alicandro and M. Ponsiglione, Ginzburg-Landau functionals and renormalized energy: A revised $\Gamma$-convergence approach,, J. Funct. Anal., 266 (2014), 4890.  doi: 10.1016/j.jfa.2014.01.024.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics, (2005).   Google Scholar

[3]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure,, Oxford University Press, (1994).   Google Scholar

[4]

F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices,, Progress in Nonlinear Differential Equations and their Applications, (1994).  doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[5]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, Clarendon Press, (1995).   Google Scholar

[6]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics,, Clarendon Press, (1999).   Google Scholar

[7]

W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity,, Physica D, 77 (1994), 383.  doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

[8]

W. E and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit,, Methods and Applications of Analysis, 13 (2006), 181.  doi: 10.4310/MAA.2006.v13.n2.a5.  Google Scholar

[9]

J. L. Ericksen, Conservation laws for liquid crystals,, Journal of Rheology, 5 (1961), 23.  doi: 10.1122/1.548883.  Google Scholar

[10]

I. Fatkullin and V. Slastikov, On spatial variations of nematic ordering,, Physica D, 237 (2008), 2577.  doi: 10.1016/j.physd.2008.03.048.  Google Scholar

[11]

I. Fatkullin and V. Slastikov, Vortices in two-dimensional nematics,, Communications in Mathematical Sciences, 7 (2009), 917.  doi: 10.4310/CMS.2009.v7.n4.a6.  Google Scholar

[12]

F. M. Leslie, Some constitutive equations for liquid crystals,, Archive for Rational Mechanics and Analysis, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[13]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices,, Communications on Pure and Applied Mathematics, 49 (1996), 323.  doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.  Google Scholar

[14]

J. C. Neu, Vortices in complex scalar fields,, Physica D, 43 (1990), 385.  doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[15]

E. Sandier, Lower bounds for the energy of unit vector fields and applications,, Journal of Functional Analysis, 152 (1998), 379.  doi: 10.1006/jfan.1997.3170.  Google Scholar

[16]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Communications on Pure and Applied Mathematics, 57 (2004), 1627.  doi: 10.1002/cpa.20046.  Google Scholar

[17]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (1992).   Google Scholar

[18]

P. Zhang W. Wang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation,, , (2013).   Google Scholar

[1]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
[2]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[3]

Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983
[4]

Jinhae Park, Feng Chen, Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1419-1440. doi: 10.3934/dcds.2010.26.1419
[5]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
[6]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[7]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[8]

Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445
[9]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757
[10]

Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499
[11]

Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243
[12]

Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[13]

Xian-Gao Liu, Jianzhong Min, Kui Wang, Xiaotao Zhang. Serrin's regularity results for the incompressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5579-5594. doi: 10.3934/dcds.2016045
[14]

Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1
[15]

Tiziana Giorgi, Feras Yousef. Analysis of a model for bent-core liquid crystals columnar phases. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2001-2026. doi: 10.3934/dcdsb.2015.20.2001
[16]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[17]

Shijin Ding, Junyu Lin, Changyou Wang, Huanyao Wen. Compressible hydrodynamic flow of liquid crystals in 1-D. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 539-563. doi: 10.3934/dcds.2012.32.539
[18]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
[19]

Zdzisław Brzeźniak, Erika Hausenblas, Paul André Razafimandimby. A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5785-5802. doi: 10.3934/dcdsb.2019106
[20]

Lei Wu. Diffusive limit with geometric correction of unsteady neutron transport equation. Kinetic & Related Models, 2017, 10 (4) : 1163-1203. doi: 10.3934/krm.2017045

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences