Lattice Boltzmann simulations of liquid crystalline fluids: active gels and blue phases

M. E. Cates, O. Henrich, D. Marenduzzo, K. Stratford

Research output: Contribution to journalArticle

65 Citations (Scopus)

Abstract

Lattice Boltzmann simulations have become the method of choice to solve the hydrodynamic equations of motion of a number of complex fluids. Here we review some recent applications of lattice Boltzmann to study the hydrodynamics of liquid crystalline materials. In particular, we focus on the study of (a) the exotic blue phases of cholesteric liquid crystals, and (b) active gels - a model system for actin plus myosin solutions or bacterial suspensions. In both cases lattice Boltzmann studies have proved useful to provide new insights into these complex materials.

LanguageEnglish
Pages3791-3800
Number of pages10
JournalSoft Matter
Volume5
Issue number20
Early online date26 Aug 2009
DOIs
Publication statusPublished - 21 Oct 2009

Fingerprint

Crystal lattices
Hydrodynamics
Gels
gels
Crystalline materials
Cholesteric liquid crystals
Liquid Crystals
Fluids
fluids
Liquids
Myosins
liquids
Equations of motion
myosins
Actins
Suspensions
simulation
hydrodynamic equations
equations of motion
liquid crystals

Keywords

  • lattice Boltzmann simulations
  • liquid crystals
  • blue phases

Cite this

Cates, M. E. ; Henrich, O. ; Marenduzzo, D. ; Stratford, K. / Lattice Boltzmann simulations of liquid crystalline fluids : active gels and blue phases. In: Soft Matter. 2009 ; Vol. 5, No. 20. pp. 3791-3800.
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note = "[1] Succi, S. (2001) The Lattice Boltzmann Equation for Fluid Dynamics and beyond. Oxford University Press [2] Cates, M.E., Clegg, P.S. Bijels: a new class of soft materials (2008) Soft Matter, 4 (11), pp. 2132-2138. doi: 10.1039/b807312k [3] Swift, M.R., Orlandini, E., Osborn, W.R., Yeomans, J.M. Lattice Boltzmann simulations of liquid-gas and binary fluid systems (1996) Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 54 (5), pp. 5041-5052. [4] Gonnella, G., Orlandini, E., Yeomans, J.M. Spinodal decomposition to a lamellar phase: Effects of hydrodynamic flow (1997) Physical Review Letters, 78 (9), pp. 1695-1698. [5] Denniston, C., Marenduzzo, D., Orlandini, E., Yeomans, J.M. Lattice Boltzmann algorithm for three-dimensional liquid-crystal hydrodynamics (2004) Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 362 (1821), pp. 1745-1754. http://rsta.royalsocietypublishing.org/ doi: 10.1098/rsta.2004.1416 [6] Marenduzzo, D., Orlandini, E., Yeomans, J.M. Hydrodynamics and rheology of active liquid crystals: A numerical investigation (2007) Physical Review Letters, 98 (11), art. no. 118102. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.98.118102&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.98.118102 [7] Cates, M.E., Fielding, S.M., Marenduzzo, D., Orlandini, E., Yeomans, J.M. Shearing active gels close to the isotropic-nematic transition (2008) Physical Review Letters, 101 (6), art. no. 068102. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.101.068102&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.101.068102 [8] Stratford, K., Adhikari, R., Pagonabarraga, I., Desplat, J.-C., Cates, M.E. Chemistry: Colloidal jamming at interfaces: A route to fluid-bicontinuous gels (2005) Science, 309 (5744), pp. 2198-2201. doi: 10.1126/science.1116589 [9] Beris, A.N., Edwards, B.J. (1994) Thermodynamics of Flowing Systems. Oxford University Press, Oxford [10] Wright, D.C., Mermin, N.D. Crystalline liquids: The blue phases (1989) Reviews of Modern Physics, 61 (2), pp. 385-432. doi: 10.1103/RevModPhys.61.385 [11] Cates, M.E., Desplat, J.-C., Stansell, P., Wagner, A.J., Stratford, K., Adhikari, R., Pagonabarraga, I. Physical and computational scaling issues in lattice Boltzmann simulations of binary fluid mixtures (2005) Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 363 (1833), pp. 1917-1935. http://rsta.royalsocietypublishing.org/ doi: 10.1098/rsta.2005.1619 [12] Tiribocchi, A., Stella, N., Gonnella, G., Lamura, A. Hybrid lattice Boltzmann model for binary fluid mixtures (2009) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 80 (2), art. no. 026701. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.80.026701&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.80.026701 [13] Marenduzzo, D., Orlandini, E., Cates, M.E., Yeomans, J.M. Steady-state hydrodynamic instabilities of active liquid crystals: Hybrid lattice Boltzmann simulations (2007) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 76 (3), art. no. 031921. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.76.031921&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.76.031921 [14] Faller, R. Coarse-grained modeling of soft condensed matter (2009) Physical Chemistry Chemical Physics, 11 (12), pp. 1867-1868. doi: 10.1039/b903299c [15] Hughes, Z.E., Stimson, L.M., Slim, H., Lintuvuori, J.S., Ilnytskyi, J.M., Wilson, M.R. An investigation of soft-core potentials for the simulation of mesogenic molecules and molecules composed of rigid and flexible segments (2008) Computer Physics Communications, 178 (10), pp. 724-731. doi: 10.1016/j.cpc.2008.01.047 [16] Memmer, R. Computer simulation of chiral liquid crystal phases VIII. Blue phases of the chiral Gay-Berne fluid (2000) Liquid Crystals, 27 (4), pp. 533-546. doi: 10.1080/026782900202723 [17] Hatwalne, Y., Ramaswamy, S., Rao, M., Simha, R.A. Rheology of active-particle suspensions (2004) Physical review letters, 92 (11), p. 118101. [18] Narayan, V., Ramaswamy, S., Menon, N. Long-lived giant number fluctuations in a swarming granular nematic (2007) Science, 317 (5834), pp. 105-108. doi: 10.1126/science.1140414 [19] Harshey, R.M. Bees aren't the only ones: swarming in Gram‐negative bacteria (1994) Molecular Microbiology, 13 (3), pp. 389-394. doi: 10.1111/j.1365-2958.1994.tb00433.x [20] Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R.E., Kessler, J.O. Self-concentration and large-scale coherence in bacterial dynamics (2004) Physical Review Letters, 93 (9), pp. 098103-1-098103-4. doi: 10.1103/PhysRevLett.93.098103 [21] Sokolov, A., Aranson, I.S., Kessler, J.O., Goldstein, R.E. Concentration dependence of the collective dynamics of swimming bacteria (2007) Physical Review Letters, 98 (15), art. no. 158102. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.98.158102&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.98.158102 [22] Cisneros, L.H., Cortez, R., Dombrowski, C., Goldstein, R.E., Kessler, J.O. Fluid dynamics of self-propelled microorganisms, from individuals to concentrated populations (2007) Experiments in Fluids, 43 (5), pp. 737-753. doi: 10.1007/s00348-007-0387-y [23] Bray, D. (2000) Cell Movements: From Molecules to Motility. Garland Publishing, New York [24] Morozov, A.N., Van Saarloos, W. Subcritical finite-amplitude solutions for plane couette flow of viscoelastic fluids (2005) Physical Review Letters, 95 (2), art. no. 024501. http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix=oai_apsmeta_2&set=journal:PRL:95 doi: 10.1103/PhysRevLett.95.024501 [25] Larson, R.G. Fluid dynamics: Turbulence without inertia (2000) Nature, 405 (6782), pp. 27-28. doi: 10.1038/35011172 [26] Kruse, K., Joanny, J.F., J{\"u}licher, F., Prost, J., Sekimoto, K. Asters, Vortices, and Rotating Spirals in Active Gels of Polar Filaments (2004) Physical Review Letters, 92 (7), pp. 781011-781014. [27] Liverpool, T.B., Marchetti, M.C. Bridging the microscopic and the hydrodynamic in active filament solutions (2005) Europhysics Letters, 69 (5), pp. 846-852. doi: 10.1209/epl/i2004-10414-0 [28] Simha, R.A., Ramaswamy, S. Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles (2002) Physical Review Letters, 89 (5), pp. 058101/1-058101/4. [29] Ishikawa, T., Pedley, T.J. Coherent structures in monolayers of swimming particles (2008) Physical Review Letters, 100 (8), art. no. 088103. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.100.088103&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.100.088103 [30] Chakrabarti, B., Das, M., Dasgupta, C., Ramaswamy, S., Sood, A.K. Spatiotemporal Rheochaos in Nematic Hydrodynamics (2004) Physical Review Letters, 92 (5), pp. 555011-555014. [31] Meiboom, S., Sethna, J.P., Anderson, P.W., Brinkman, W.F. Theory of the blue phase of cholesteric liquid crystals (1981) Physical Review Letters, 46 (18), pp. 1216-1219. doi: 10.1103/PhysRevLett.46.1216 [32] Kikuchi, H., Yokota, M., Hisakado, Y., Yang, H., Kajiyama, T. Polymer-stabilized liquid crystal blue phases (2002) Nature Materials, 1 (1), pp. 64-68. doi: 10.1038/nmat712 [33] Coles, H.J., Pivnenko, M.N. Liquid crystal 'blue phases' with a wide temperature range (2005) Nature, 436 (7053), pp. 997-1000. doi: 10.1038/nature03932 [34] Grebel, H., Hornreich, R.M., Shtrikman, S. Landau theory of cholesteric blue phases: The role of higher harmonics (1984) Physical Review A, 30 (6), pp. 3264-3278. doi: 10.1103/PhysRevA.30.3264 [35] Dupuis, A., Marenduzzo, D., Yeomans, J.M. Numerical calculations of the phase diagram of cubic blue phases in cholesteric liquid crystals (2005) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 71 (1), art. no. 011703. http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PLEEE8000071000001011703000001&idtype=cvips doi: 10.1103/PhysRevE.71.011703 [36] Alexander, G.P., Marenduzzo, D. (2008) Europhys. Lett., 81, p. 66004. [37] Yang, D.K., Crooker, P.P. Chiral-racemic phase diagrams of blue-phase liquid crystals (1987) Physical Review A, 35 (10), pp. 4419-4423. doi: 10.1103/PhysRevA.35.4419 [38] Alexander, G.P., Yeomans, J.M. Stabilizing the blue phases (2006) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 74 (6), art. no. 061706. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.74.061706&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.74.061706 [39] Dupuis, A., Marenduzzo, D., Orlandini, E., Yeomans, J.M. Rheology of Cholesteric blue phases (2005) Physical Review Letters, 95 (9), art. no. 097801. http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix=oai_apsmeta_2&set=journal:PRL:95 doi: 10.1103/PhysRevLett.95.097801 [40] Marenduzzo, D., Orlandini, E., Yeomans, J.M. Permeative flows in cholesteric liquid crystals (2004) Physical review letters, 92 (18), p. 188301. [41] Henrich, O., Marenduzzo, D., Stratford, K., Cates, M.E. (2009) Comput. Math. with Appl arXiv:0901.3293;., accepted for publication [42] Llopis, I., Pagonabarraga, I. Dynamic regimes of hydrodynamically coupled self-propelling particles (2006) Europhysics Letters, 75 (6), pp. 999-1005. doi: 10.1209/epl/i2006-10201-y [43] Nash, R.W., Adhikari, R., Cates, M.E. Singular forces and pointlike colloids in lattice Boltzmann hydrodynamics (2008) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 77 (2), art. no. 026709. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.77.026709&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.77.026709 [44] Giomi, L., Marchetti, M.C., Liverpool, T.B. Complex spontaneous flows and concentration banding in active polar films (2008) Physical Review Letters, 101 (19), art. no. 198101. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.101.198101&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.101.198101",
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Lattice Boltzmann simulations of liquid crystalline fluids : active gels and blue phases. / Cates, M. E.; Henrich, O.; Marenduzzo, D.; Stratford, K.

In: Soft Matter, Vol. 5, No. 20, 21.10.2009, p. 3791-3800.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Lattice Boltzmann simulations of liquid crystalline fluids

T2 - Soft Matter

AU - Cates, M. E.

AU - Henrich, O.

AU - Marenduzzo, D.

AU - Stratford, K.

N1 - [1] Succi, S. (2001) The Lattice Boltzmann Equation for Fluid Dynamics and beyond. Oxford University Press [2] Cates, M.E., Clegg, P.S. Bijels: a new class of soft materials (2008) Soft Matter, 4 (11), pp. 2132-2138. doi: 10.1039/b807312k [3] Swift, M.R., Orlandini, E., Osborn, W.R., Yeomans, J.M. Lattice Boltzmann simulations of liquid-gas and binary fluid systems (1996) Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 54 (5), pp. 5041-5052. [4] Gonnella, G., Orlandini, E., Yeomans, J.M. Spinodal decomposition to a lamellar phase: Effects of hydrodynamic flow (1997) Physical Review Letters, 78 (9), pp. 1695-1698. [5] Denniston, C., Marenduzzo, D., Orlandini, E., Yeomans, J.M. Lattice Boltzmann algorithm for three-dimensional liquid-crystal hydrodynamics (2004) Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 362 (1821), pp. 1745-1754. http://rsta.royalsocietypublishing.org/ doi: 10.1098/rsta.2004.1416 [6] Marenduzzo, D., Orlandini, E., Yeomans, J.M. Hydrodynamics and rheology of active liquid crystals: A numerical investigation (2007) Physical Review Letters, 98 (11), art. no. 118102. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.98.118102&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.98.118102 [7] Cates, M.E., Fielding, S.M., Marenduzzo, D., Orlandini, E., Yeomans, J.M. Shearing active gels close to the isotropic-nematic transition (2008) Physical Review Letters, 101 (6), art. no. 068102. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.101.068102&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.101.068102 [8] Stratford, K., Adhikari, R., Pagonabarraga, I., Desplat, J.-C., Cates, M.E. Chemistry: Colloidal jamming at interfaces: A route to fluid-bicontinuous gels (2005) Science, 309 (5744), pp. 2198-2201. doi: 10.1126/science.1116589 [9] Beris, A.N., Edwards, B.J. (1994) Thermodynamics of Flowing Systems. Oxford University Press, Oxford [10] Wright, D.C., Mermin, N.D. Crystalline liquids: The blue phases (1989) Reviews of Modern Physics, 61 (2), pp. 385-432. doi: 10.1103/RevModPhys.61.385 [11] Cates, M.E., Desplat, J.-C., Stansell, P., Wagner, A.J., Stratford, K., Adhikari, R., Pagonabarraga, I. Physical and computational scaling issues in lattice Boltzmann simulations of binary fluid mixtures (2005) Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 363 (1833), pp. 1917-1935. http://rsta.royalsocietypublishing.org/ doi: 10.1098/rsta.2005.1619 [12] Tiribocchi, A., Stella, N., Gonnella, G., Lamura, A. Hybrid lattice Boltzmann model for binary fluid mixtures (2009) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 80 (2), art. no. 026701. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.80.026701&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.80.026701 [13] Marenduzzo, D., Orlandini, E., Cates, M.E., Yeomans, J.M. Steady-state hydrodynamic instabilities of active liquid crystals: Hybrid lattice Boltzmann simulations (2007) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 76 (3), art. no. 031921. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.76.031921&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.76.031921 [14] Faller, R. Coarse-grained modeling of soft condensed matter (2009) Physical Chemistry Chemical Physics, 11 (12), pp. 1867-1868. doi: 10.1039/b903299c [15] Hughes, Z.E., Stimson, L.M., Slim, H., Lintuvuori, J.S., Ilnytskyi, J.M., Wilson, M.R. An investigation of soft-core potentials for the simulation of mesogenic molecules and molecules composed of rigid and flexible segments (2008) Computer Physics Communications, 178 (10), pp. 724-731. doi: 10.1016/j.cpc.2008.01.047 [16] Memmer, R. Computer simulation of chiral liquid crystal phases VIII. Blue phases of the chiral Gay-Berne fluid (2000) Liquid Crystals, 27 (4), pp. 533-546. doi: 10.1080/026782900202723 [17] Hatwalne, Y., Ramaswamy, S., Rao, M., Simha, R.A. Rheology of active-particle suspensions (2004) Physical review letters, 92 (11), p. 118101. [18] Narayan, V., Ramaswamy, S., Menon, N. Long-lived giant number fluctuations in a swarming granular nematic (2007) Science, 317 (5834), pp. 105-108. doi: 10.1126/science.1140414 [19] Harshey, R.M. Bees aren't the only ones: swarming in Gram‐negative bacteria (1994) Molecular Microbiology, 13 (3), pp. 389-394. doi: 10.1111/j.1365-2958.1994.tb00433.x [20] Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R.E., Kessler, J.O. Self-concentration and large-scale coherence in bacterial dynamics (2004) Physical Review Letters, 93 (9), pp. 098103-1-098103-4. doi: 10.1103/PhysRevLett.93.098103 [21] Sokolov, A., Aranson, I.S., Kessler, J.O., Goldstein, R.E. Concentration dependence of the collective dynamics of swimming bacteria (2007) Physical Review Letters, 98 (15), art. no. 158102. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.98.158102&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.98.158102 [22] Cisneros, L.H., Cortez, R., Dombrowski, C., Goldstein, R.E., Kessler, J.O. Fluid dynamics of self-propelled microorganisms, from individuals to concentrated populations (2007) Experiments in Fluids, 43 (5), pp. 737-753. doi: 10.1007/s00348-007-0387-y [23] Bray, D. (2000) Cell Movements: From Molecules to Motility. Garland Publishing, New York [24] Morozov, A.N., Van Saarloos, W. Subcritical finite-amplitude solutions for plane couette flow of viscoelastic fluids (2005) Physical Review Letters, 95 (2), art. no. 024501. http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix=oai_apsmeta_2&set=journal:PRL:95 doi: 10.1103/PhysRevLett.95.024501 [25] Larson, R.G. Fluid dynamics: Turbulence without inertia (2000) Nature, 405 (6782), pp. 27-28. doi: 10.1038/35011172 [26] Kruse, K., Joanny, J.F., Jülicher, F., Prost, J., Sekimoto, K. Asters, Vortices, and Rotating Spirals in Active Gels of Polar Filaments (2004) Physical Review Letters, 92 (7), pp. 781011-781014. [27] Liverpool, T.B., Marchetti, M.C. Bridging the microscopic and the hydrodynamic in active filament solutions (2005) Europhysics Letters, 69 (5), pp. 846-852. doi: 10.1209/epl/i2004-10414-0 [28] Simha, R.A., Ramaswamy, S. Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles (2002) Physical Review Letters, 89 (5), pp. 058101/1-058101/4. [29] Ishikawa, T., Pedley, T.J. Coherent structures in monolayers of swimming particles (2008) Physical Review Letters, 100 (8), art. no. 088103. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.100.088103&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.100.088103 [30] Chakrabarti, B., Das, M., Dasgupta, C., Ramaswamy, S., Sood, A.K. Spatiotemporal Rheochaos in Nematic Hydrodynamics (2004) Physical Review Letters, 92 (5), pp. 555011-555014. [31] Meiboom, S., Sethna, J.P., Anderson, P.W., Brinkman, W.F. Theory of the blue phase of cholesteric liquid crystals (1981) Physical Review Letters, 46 (18), pp. 1216-1219. doi: 10.1103/PhysRevLett.46.1216 [32] Kikuchi, H., Yokota, M., Hisakado, Y., Yang, H., Kajiyama, T. Polymer-stabilized liquid crystal blue phases (2002) Nature Materials, 1 (1), pp. 64-68. doi: 10.1038/nmat712 [33] Coles, H.J., Pivnenko, M.N. Liquid crystal 'blue phases' with a wide temperature range (2005) Nature, 436 (7053), pp. 997-1000. doi: 10.1038/nature03932 [34] Grebel, H., Hornreich, R.M., Shtrikman, S. Landau theory of cholesteric blue phases: The role of higher harmonics (1984) Physical Review A, 30 (6), pp. 3264-3278. doi: 10.1103/PhysRevA.30.3264 [35] Dupuis, A., Marenduzzo, D., Yeomans, J.M. Numerical calculations of the phase diagram of cubic blue phases in cholesteric liquid crystals (2005) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 71 (1), art. no. 011703. http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PLEEE8000071000001011703000001&idtype=cvips doi: 10.1103/PhysRevE.71.011703 [36] Alexander, G.P., Marenduzzo, D. (2008) Europhys. Lett., 81, p. 66004. [37] Yang, D.K., Crooker, P.P. Chiral-racemic phase diagrams of blue-phase liquid crystals (1987) Physical Review A, 35 (10), pp. 4419-4423. doi: 10.1103/PhysRevA.35.4419 [38] Alexander, G.P., Yeomans, J.M. Stabilizing the blue phases (2006) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 74 (6), art. no. 061706. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.74.061706&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.74.061706 [39] Dupuis, A., Marenduzzo, D., Orlandini, E., Yeomans, J.M. Rheology of Cholesteric blue phases (2005) Physical Review Letters, 95 (9), art. no. 097801. http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix=oai_apsmeta_2&set=journal:PRL:95 doi: 10.1103/PhysRevLett.95.097801 [40] Marenduzzo, D., Orlandini, E., Yeomans, J.M. Permeative flows in cholesteric liquid crystals (2004) Physical review letters, 92 (18), p. 188301. [41] Henrich, O., Marenduzzo, D., Stratford, K., Cates, M.E. (2009) Comput. Math. with Appl arXiv:0901.3293;., accepted for publication [42] Llopis, I., Pagonabarraga, I. Dynamic regimes of hydrodynamically coupled self-propelling particles (2006) Europhysics Letters, 75 (6), pp. 999-1005. doi: 10.1209/epl/i2006-10201-y [43] Nash, R.W., Adhikari, R., Cates, M.E. Singular forces and pointlike colloids in lattice Boltzmann hydrodynamics (2008) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 77 (2), art. no. 026709. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevE.77.026709&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevE.77.026709 [44] Giomi, L., Marchetti, M.C., Liverpool, T.B. Complex spontaneous flows and concentration banding in active polar films (2008) Physical Review Letters, 101 (19), art. no. 198101. http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org:PhysRevLett.101.198101&metadataPrefix=oai_apsmeta_2 doi: 10.1103/PhysRevLett.101.198101

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