Cholesteric Phase in Virus SuspensionsZvonimir Dogic* and Seth FradenThe Complex Fluids Group, Martin Fisher School of Physics, Brandeis University,Waltham, Massachusetts 02454Received March 23, 2000. In Final Form: June 12, 2000We report measurements of the cholesteric pitch and twist elastic constant (K22) in monodispersesuspensionsoftherodlikevirusfilamentousbacteriophagefd.Measurementsweretakenforconcentrationsspanning the entire cholesteric region at several ionic strengths and temperatures. In the limit of highionic strength the cholesteric pitch (P0) scales with concentration (c)asP0∝ c-1.66. As the ionic strengthdecreases, the scaling exponent systematically changes to lower values.IntroductionThesystemwiththesimplestintermolecularinteractionknown to exhibit all the essential features of the nematicstate is that of a hard rod suspension.1,2Because of itsinherent simplicity, much effort has been put intounderstanding the relationship between the microscopicparametersofhardrodsandtheresultingliquidcrystallinebehavior at the macroscopic level. In nature it oftenhappens that a symmetry of the nematic phase can bereduced to form a cholesteric phase, where the nematicdirectorfollowsahelicalpathinspace.Formationofsuchaphaseatthemacroscopiclevelisusuallyassociatedwithchirality of molecules at the molecular level. Details ofhow a simple change of a few atomic positions, requiredtomakeamoleculechiral,causesadrasticchangeinself-organization at the macroscopic level remain unknown.However, in the continuum limit, where details of themicroscopic interactions are ignored, formation of thecholestericphase isunderstoodas acompetition betweentwo elastic energies. On one hand, the free energy of achiralnematicisloweredinatwistedstatebecauseofthetorque a chiral molecule exerts on its neighbor. Such acontribution to the free energy is characterized by the“twist” constant Kt. On the other hand, creation of anelasticallydistortedstatecharacterizedbytheusualtwistelasticconstantK22raisesthefreeenergy.3Itfollowsthatthe wavelength of the cholesteric pitch is proportional tothe ratio of Kt/K22. At present, the challenge lies incalculatingthevalue ofthe“twist” constantKtforagivenmolecule with known microscopic interactions.Inspired by work of Onsager, Straley made the firstattempttoexplainthemicroscopicoriginofthecholestericphase.4,5He considered rods with threads of definitehandiness and for the first time obtained an expressionfor Ktas a function of the microscopic parameters of athreaded rod. As in the case of Onsager’s hard rods,Straley’s cholesteric phase is purely entropy driven. Thenonzerovalueofthechiral“twist”constantKtisassociatedwith the free volume gained as threaded rods approacheach other at an well-defined angle. This work was laterextended to account for flexibility of rods.6,7Itisnotcleariftherearelyotropicliquidcrystalswherethe cholesteric phase is purely entropy driven andtherefore most of the predictions of the Straley modelremain untested. An alternative proposal for the originofanonzero Ktconstantinvolveschiral attractionsofvander Waals origin.8It is likely that for almost all experi-mental systems both entropic temperature-independentinteractions and attractive temperature-dependent in-teractions contribute to the cholesteric twist, furthercomplicatingthe problem. Harrisand co-workersnoticedthat ifthe threaded rod is allowed to rotate freely aroundits long axis, chirality will be effectively averaged awayand proposed that short-ranged biaxial correlations arecritical for formation of a cholesteric phase.9,10Theimplication of their work is that all mean-field theories,like the one of Straley, do not capture the essence ofcholesteric phase since they ignore all correlations.Bacteriophagefdisachiral,monodisperserodlikecolloidthat forms a cholesteric phase with a characteristic“fingerprint”textureshowninFigure1a.11fdhasacontourlength of 880 nm,12persistance length of 2200 nm,13andalinearchargedensityof2e-/ÅatpH8.1.14However,Pf1,achiralviruswithastructureextremelysimilartofd,15,16doesnotshowanyevidenceofformingacholestericliquidcrystalas shownby the absenceof a“fingerprint” texturein Figure 1b. This sets the lower limit of the cholestericpitchofPf1virustothesizeofthecapillary.Thetheoreticalchallenge is to explain why two such similar chiralmoleculeshaveextremely differentvaluesof thecholestericpitch.The concentrations of the coexisting isotropic andcholesteric phases are quantitatively explained by theOnsager theory establishing fd as an ideal model of hardrods.17,18Onthe other hand, the origin and mechanism of(1) Onsager, L. Ann. N.Y. Acad. Sci. 1949, 51, 627-659.(2) Vroege, G. J.; Lekkerkerker, H. N. W. Rep. Prog. Phys. 1992, 8,1241-1309.(3) DeGennes, P. G.;Prost, J. In The Physics of Liquid Crystals, 2nded.; Clarendon Press: Oxford, 1992.(4) Straley, J. P. Phys. Rev. A 1973, 8, 2181-2183.(5) Straley, J. P. Phys. Rev. A 1976, 14, 1835-1841.(6) Odijk, T. J. Phys. Chem. 1987, 91, 6060-6062.(7) Pelcovits, R. A. Liq. Cryst. 1996, 21, 361-364.(8) Issaenko, S. A.;Harris, S. A.;Lubensky, T. C.Phys. Rev. E 1999,60, 578-597.(9) Harris, A. B.; Kamien R. D.; Lubensky, T. C. Phys. Rev. Lett.1997, 78, 1476-1480.(10) Harris, A. B.; Kamien R. D.; Lubensky, T. C. Rev. Mod. Phys.1999, 71, 1745-1757.(11) Lapointe,J.;Marvin,D.A.Mol.Crys. Liq. Cryst. 1973,19,269-278.(12) Day, L. A.; Marzec, C. J.; Reisberg, S. A.; Casadevall, A. Annu.Rev. Biophys. Chem. 1988, 17, 509.(13) Song,L.;Kim,U.;Wilcoxon,J.;Schurr,J.M.Biopolymers 1991,31, 547-567.(14) Zimmernann, K.; Hogedorn, H.; Heuck, C. C.; Hinrichsen, M.;Ludwig, H. J. Biol. Chem. 1986, 261, 1653-1655.(15) Caspar, D. L. D.; Makowski, L. J. Mol. Biol. 1981, 145, 611-617.(16) Marvin, D. A. Curr. Opin. Struct. Biol. 1998, 8, 150-158.(17) Tang, J.; Fraden S. Liq. Cryst. 1995, 19, 459-467.7820 Langmuir 2000, 16, 7820-782410.1021/la000446t CCC: $19.00 © 2000 American Chemical SocietyPublished on Web 08/31/2000the formation of the cholesteric structure in liquidcrystallinefd remainsa challenge. Itis importantto notethat the Onsager theory predicts equally well the con-centration dependence of the isotropic-nematic andisotropic-cholesteric first-order phase transition. Thereason for this is that the free energy difference betweenthe isotropic and nematic phase is much larger than