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Rods, Rotations, Gels Soft-Matter Experiment and Theory from Penn. 2/23/06 Texas Colloquium The Penn Team • Experiment – – – – – – – – – – – • Theory – – – – – TCL Randy Kamien Andy Lau Fangfu Ye Ranjan Mukhopadhyay – David Lacoste – Leo Radzihovsky – X.J. Xing 2/23/06 Texas Colloquium Arjun Yodh Dennis Discher Jerry Gollub Mohammad Islam Ahmed Alsayed Zvonimir Dogic Jian Zhang Mauricio Nobili Yilong Han Paul Dalhaimer Paul Janmey Outline • Some rods • Rotational and Translational Diffusion of a rod - 100th Anniversary of Einstein’s 1906 paper • Chiral granular gas • Semi-flexible Polymers in a nematic solvent • Nematic Phase • Carbon Nanotube Nematic Gels 2/23/06 Texas Colloquium Comments • Experimental advances in microsocpy and imaging: real space visualization of fluctuating phenomena in colloidal systems • Wonderful “playground” to for interaction between theory and experiment to test what we know and to discover new effects • Simple theories - great experiments 2/23/06 Texas Colloquium Rods PMMA Ellipsoid fd Virus 1 mm 900 nm ~1 nm Carbon Nanotube 100 nm – 10,000 nm 2/23/06 Texas Colloquium Einstein – Brownian Motion 1. “On the Movement of Small Particles Suspended in Stationary Liquid Required by the Molecular Kinetic Theory of Heat”, Annalen der Physik 17, 549 (1905); x&= Gf ; G = (6pha )- 1 2 (D x ) = 2Dt ; D = k BT G a = particle radius; h = viscosity; f = force 2/23/06 Texas Colloquium Langevin Oscillator Dynamics p&= - kx - gv + V p2 1 2 H = + kx 2m 2 p = mv = mx& V(t )V(t ') = 2gT d(t - t ') g = 6pha Ignore inertial terms: x&= - Gkx + x; G = g- 1 x 2 x(t )x(t ') = 2T Gd(t - t ') | x ( w) | 2 2T Gk = k w2 + G2k 2 P. Langevin, Comptes Rendues 146, 530 (1908) Uhlenbeck and Ornstein, Phys. Rev. 36, 823 (1930) 2/23/06 dw kBT 2 ( ) = ò |x w | = 2p k Dynamics must retrieve equilibrium static fluctuations: sets scale of noise fluctuations to be 2TG Texas Colloquium Diffusion with no potential k ® 0: x&= x Gaussian probability distribution [x (t ) - x (0)]2 P (x , t ) = d w 2k BT G - i wt = ò (1 - e ) 2 æ x2 ö p w 1 ÷ exp çç÷ çè 4Dt ÷ = 2k BT Gt = 2Dt 4p Dt ø kBT 1 RT D= = 6pha N AVG 6pha Measurement of D gives Avogrado’s number R = Gas constant 2/23/06 Texas Colloquium Density Diffusion n (x , t ) = å d (x - x a (t )) a x&a = xa ; x a (t + D t ) = x a (t ) + å n (x , t + D t ) = ò t + Dt xa (t ')dt ' t d (x - x a (t + D t )) a = å d (x - x a (t )) - ¶ x å a 1 2 + ¶x 2 x&a d (x - x a (t )) a å a d (x - x a (t )) ò t + Dt t ¶ t n (x , t ) = - ¶ x nv + D ¶ x2 n (x , t ) ¶ t n ( r, t ) = - Ñ ×n v + D Ñ 2n ( r, t ) 2/23/06 Texas Colloquium ò t t + Dt xa (t 1 )xa (t 2 ) dt 1dt 2 J. Perrin Expts. (1908) n (r , t ) = æ r2 ö N0 ÷ exp çç÷ 3/ 2 ç è 4Dt ø÷ (4p Dt ) NAVG=7.05 x 1023 2/23/06 Texas Colloquium Rotational Diffusion “On the Theory of Brownian Motion,” ibid. 19, 371 (1906): Idea of Brownian motion of arbitrary variable, application torotational diffusion of a spherical particle. 3 - 1 & q = G t ; G = (8pha ) q q (D q)2 = 2Dqt ; Dq = kBT Gq t = torque P. Zeeman and Houdyk, Proc. Acad. Amsterdam, 28, 52 (1925) W. Gerlach, Naturwiss 15,15 (1927) G.E. Uhlenbeck and S. Goudsmit, Phys. Rev. 34,145 (1929) F. Perrin, Ann. de Physique 12, 169 (1929) W.A. Furry, “Isotropic Brownian Motion”, Phys. Rev. 107, 7 (1957) 2/23/06 Texas Colloquium Diffusion of Anisotropic Particles 1. Brownian motion of an anisotropic particle: F. Perrin, J. de Phys. et Rad. V, 497 (1934); VII, 1 (1936). Interaction of Rotational and Translational Diffusion: Stephen Prager, J. of Chem. Physics 23, 12 (1955) 2/23/06 Texas Colloquium Diffusion of a rod Han, Alsayed,Nobili,Zhang,TCL,Yodh 2/23/06 Texas Colloquium Defining trajectories Can extract lab- and body-frame displacements, trajectories at fixed initial angle and averaged over initial angles. 2/23/06 Texas Colloquium Rotational Langevin (2d) ¶H & & Iq = - Gqq&+ Gq xq = p&q ¶q ¶H & q = - Gq + xq ® q&= xq ¶q xq (t )xq (t ') = 2k BT Gqd(t - t ') 2 [q(t ) - q(0) ] º [D q] = 2Dqt Dq = 2 [D q ]2 2t 2 cos[n D q ] = exp[- n Dqt ] 2/23/06 Texas Colloquium Translation and Rotation Anisotropic friction coefficients x& P = - Ga fP; x&^ = - Gb f^ Lab-frame equations ¶H x&i = - Gij (q) + xi ¶xj Gij (q) = Gan i (q)n j (q) + Gb[dij - n i (q)n j (q)] xi (t )xj (t ') q0 = 2kBT Gij (q(t ))d(t - t ') In lab-frame, noise depends on angle: expect . anisotropic crossover 2/23/06 Texas Colloquium F. Perrin, J. de Phys. et Rad. V, 497 (1934); VII, 1 (1936). Body-frame equations ædx%ö çç ÷ ÷= ççdy%÷ è ø÷ æ cos q sin q öædx ö ÷ çç çç ÷ ÷ ÷ ÷ çç- sin q cos q÷ ç ÷ d y è øèç ø÷ x%i (t ) = å dx%ni n ¶ t x%i = x% ; i Body frame : x%and y%are are independent; simple Langevin equations with æGa % = çç G ij çç 0 è 2 ö 0÷ ÷ ÷ Gb ÷ ø [D x%] = 2Dat ; constant diffusion. 2/23/06 %(t ') = 2k T G % x% ( t ) x i i ij B Texas Colloquium 2 [D y%] = 2Dbt Anisotropic Crossover D = (Da + Db ) / 2 1 é ù Dij (t , q0 ) = [D x i (t ) ]ëD x j (t )û q 0 2t D D = Da - Db æcos 2q0 sin 2q0 ö DD - nDqt ÷ ç ÷ (1 e ) = D dij + t 4 (t ) çç ÷ ÷ t n (t ) = 2t çèsin 2q0 - cos 2q0 ø nD 1. Diffusion tensor averaged over angles is isotropic. 2. Lab-frame diffusion is anisotropic at short times and isotropic at long times 3. Body-frame diffusion tensor is constant and anistropic at all times 1 Dij = d q0Dij (t , q0 ) = D dij ò 2p 2 2 [D x%] = 2Dat ; [D y%] = 2Dbt 2/23/06 Texas Colloquium q Lab- and body-frame diffusion 2/23/06 Texas Colloquium Gaussian Body Frame Statistics ¶ t x%i = x% ; i %(t ') = 2k T G % x% ( t ) x i i ij B 2/23/06 P (x%i ) = Texas Colloquium æ x%2 ö 1 i ÷ ÷ exp ççç÷ çè 4Dit ÷ 4 p Di t ø Non-Gaussian lab-frame statistics C ( 4) q0 4 2 2 = [D x (t ) ] - 3 [D x (t ) ] = 1 2 p(t ) = + ( t q t 4 (t ) - t q t 16 (t ) - 3t (t )) cos 4q0 ] 0 2 2 3 [D x (t ) ] 2 (D D ) [3( t qt - t q t 4 (t ) - t 42 (t )) 2 4 C q( 4) (t ) 2 ® (Da - Db ) 2 2 (Da + Db ) Fixed q0: Gaussian at short times – same as body frame; Gaussian at long times – central limit theorem. Average q0: nonGaussian at short times, Gaussian at long times 2/23/06 Texas Colloquium Non-Gaussian Distribution. Probability distribution at fixed angle and small t is Gaussian. The average over initial angles is not 2/23/06 Texas Colloquium Rattleback gas Tsai, J.-C., Ye, Fangfu , Rodriguez, Juan , Gollub, J.P., and Lubensky, T.C., A Chiral Granular Gas, Phys. Rev. Lett. 94, 214301 (2005). Chiral Rattlebacks spin in a preferred direction; Achiral ones do not. Chiral wires spin in a preferred direction on a vibrating substrate 2/23/06 Texas Colloquium Rattleback gas II 2/23/06 Texas Colloquium Spin angular momentum dynamics l = nI W= Spin angular momentum W= Spin angular frequency l( x, t ) = å pqa d (x - x a (t )) a &= I q& ¶H - Gqq&+ Gq xq = p&q ¶q å ¶ t l = - n GqW+ ¶ i ¶ j Dij (qa )pqa d( x - x a (t ) a = - n GqW+ D Ñ 2l = - ¶ j (lv j ) - GW+ DWÑ 2W 2/23/06 Texas Colloquium Rattleback gas III l = I W= Spin angular momentum gi = r vi = Center-of-mass mometum W= Spin angular frequency w = (Ñ ´ v)z / 2 = CM angular frequency ¶ tl = - ¶ j (lv j ) - GWW- G(W- w) + DWÑ 2W+ t Substrate friction Spin-vorticity coupling ¶ t gi = - ¶ j (giv j ) - ¶ i p + hÑ vi - G vi + 2 B.C.: W(R ) = w(R ) = v(R ) / R ; 2/23/06 v ¶ r v = - l - 1v Texas Colloquium Vibrational torque 1 2 eij ¶ j G(W- w) Nematic phase Order Qij = S (n in j Parameter F = 1 2 ò d x {K 1 (Ñ ×n ) Twist + K 2 [n ×(Ñ ´ n )] 2 + K 3 [n ´ (Ñ ´ n )] } Frank free energy Texas Colloquium 1 3 dij n ( n) Bend n ( n) 2 2/23/06 dij ) = ni n j - Splay n 2 3 1 3 n =Frank director Isotropic-to-Nematic Transition increasing concentration (q)orientational distribution functions D - rod diameter L – rod length I N - rod concentration f I- N 2/23/06 order parameter S : at I - N phase transition D L = 4 when ? 1 L D S = 1 2 2 dq sin q f ( q )(3 cos q - 1) ò Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949) Texas Colloquium Semi-flexible biopolymers DNA Neurofilament 16 micron length 2 nm in diameter 40 nm persistence length 5 - 20 micron length 12 nm in diameter ~ 220 nm persistence length Wormlike Micelle Actin ( polybutadiene-polyethyleneoxide ) 10 – 50 micron length ~ 15 nm in diameter ~ 500 nm persistence length 2/23/06 2 – 30 micron length 7-8 nm in diameter ~ 16 micron persistence length Texas Colloquium Semi-flexible Polymer in Aligning Field h= kB T G/ t(s) R(s) R dR = t(s ) ds 2/23/06 | t(s ) |= 1; t(s ) = ( t ^ (s ), 1- | t ^ (s ) |2 ) Texas Colloquium Fluctuations é ædt ö2 ù 2 ê ú ÷ ç H = ds l G | t | z ÷ ú 2 ò êëP çèds ø û é ædt ö2 ù kBT 2 êl ç ^ ÷ ú » ds + G | t | ÷ ^ ÷ ú 2 ò êëP ççè ds ø û k BT lp = k / kBT = Persistence length G = h / kBT = Alignment parameter l = lp G = Odijk deflection length G=0 át(s ) ×t(0)ñ= e 2/23/06 - s / 2lp Texas Colloquium t x (z )t x (0) = G>0 l - |z |/ l e 2lp Polymers in fd-virus suspensions Isotropic Nematic Actin in Nematic Fd Actin 16 mm Wormlike Micelle 500 nm Neurofilament 200 nm DNA 50 nm Hairpin defects 10 mm 2/23/06 10 mm Dogic Z, Zhang J, Lau AWC, Aranda-Espinoza H, Dalhaimer P, Discher DE, Janmey PA, Kamien RD, Lubensky TC, Yodh AG, Phys. Rev. Lett. 92 (12): 2004 Texas Colloquium Tangent-tangent correlations t(s’) t(s’+s) h(s) actin in nematic phase Isotropic phase – quasi 2D át(s ) ×t(0)ñ= e - s / 2lp Orientational correlations decay exponentially lp – persistence length of actin in isotropic phase 2/23/06 Texas Colloquium Polymer in nematic solvent Bending energy F/T = lp 2 ò L 0 ædt ^ dz çç çè dz Coupling energy Elastic energy 2 L ö 2 G 1 2 3 ÷ é ù + dz t ( z ) d n (0, z ) + K d x Ñ n ( ) ÷ ë^ û 2 ò ÷ ø 2 ò0 lp = Persistence length G = Coupling constant l = lP G = Odijk length K = Elastic constant l - z/ l t x (z )t x (z + z ¢) = e 2l p 1 + 4p 2K l 2/23/06 ¥ cos(xz / l ) log(1 + D 2 / x 2 ) ò dx (1 + x 2 )[1 + x 2 + a x 2 log(1 + D 2 / x 2 )] 0 Texas Colloquium