Rotational Brownian Motion and Kerr Effect Relaxation in Electric Fields of Arbitrary Strength. Yuri P. KALMYKOV

Transcription

1 Rotational Brownian Motion and Kerr Effect Relaxation in Electric Fields of Arbitrary Strength Yuri P. KALMYKOV

2 (i) Kerr effect relaxation Related Phenomena (ii) Fluorescence depolarization (iii) Nuclear magnetic and dielectric relaxation of nematic liquid and molecular crystals (iv) Dynamic light scattering (v) Dielectric relaxation of polar fluids (vi) Magnetic relaxation of ferrofluids (colloidal suspensions of single domain ferromagnetic particles) etc.

3 R. Brown ( ) Brownian Motion The first detailed account of Brownian motion was given by the eminent botanist Robert Brown in 87 while studying the plant life of the South Seas. In this study he dealt with the transfer of pollen into the ovulum of a plant. He examined aqueous suspensions of pollen grains of several species under a microscope and found that in all cases the pollen grains were in rapid irregular oscillatory motion. In order to obtain a first interpretation of this phenomenon one had to wait till 95 (78 years!) and the paper of A. Einstein [ Ann. Physik (95)] ½ ½ n- n- n n+ n+ A. Einstein ( ) W t W D x. x /4Dt W( x t) e x. 4 Dt x Dt 3

4 Langevin equation Translational Brownian motion of a particle P. Langevin (87-946) m d dt v vλ() t v v : velocity of the Brownian particle m: masse : friction coefficient : random force λ() t λ() λ() t kt () t P. Langevin Sur la Théorie de Mouvement Brownien C. R. Acad. Sci. Paris (98). 4

5 M. Planck ( ) Fokker-Planck Equation. A.D. Fokker Ann. Physik 43 8 (94).. M.Planck Sitzber. Preuβ. Akad. Wiss. P.34 (97). 3. O. KleinArkiv for Mathematik Astronomi och Fysik 6No. 5 (9) 4. H. A. Kramers Physica (Utrecht) 7 84 (94). Brownian motion in an external potential U d m v v U λ() t dt r W kt v W W U W W t m m v r v r m v v W(rvt): distribution fonction H.A. Kramers (894-95) 5

6 Rotational Brownian motion P. Debye ( ) Peter "Pie" Debye. His first maor scientific contribution was the application of the concept of dipole moment to the charge distribution in asymmetric molecules in 9 developing equations relating dipole moments to temperature dielectric constant Debye relaxation etc. In consequence molecular dipole moments are measured in debyes a unit named in his honor. W W D P. Debye Polar Molecules Chemical Catalog Co. N. Y. 99 t ( x ) Rotational Brownian motion of asymmetric top molecules F. Perrin (9-99) Francis Perrin était fils de Jean Perrin (prix Nobel de Physique 96). En 98 il soutient une première thèse sur le mouvement brownien suivie d une deuxième en 99 sur la fluorescence. Francis Perrin «Mouvement brownien d un ellipsoïd (I). Dispersion diélectrique pour des molécules ellipsoïdales» J. Phys. Radium (934); «Mouvement brownien d un ellipsoïd (II). Rotation libre et dépolarisation des fluorescences. Translation et diffusion de molécules ellipsoïdales» ibid. 7 (936). 6

7 Rotational diffusion model The dynamics of a molecule are described by the Euler-Langevin equation for the angular velocity ω() t written in the body-fixed coordinate system xyz Î λ() t ˆ d I ˆωω I ˆω ˆ ω V λ() t dt is the inertia tensor of the molecule is the white noise driving torque; is the rotational friction tensor kt kt kt kt ˆ ω is the damping torque due to Brownian movement V is the torque acting on the molecule due to an external potential V 7

8 Noninertial limit I /( ) i ii ( when the inertia terms in the Euler-Langevin equation may be neglected) ω() t / t Dˆ λ() t V () t t /( kt) / is the orientation space gradient operator Dˆ ktˆ is the diffusion tensor is an infinitesimal rotation vector so that ω() t / t V( ) t μ E() t E() t ˆ E() t... is the potential energy of the molecule in an external field E(t) 8

9 Noninertial rotational Brownian motion L.D. Favro "Theory of the Rotational Brownian Motion of a Free Rigid Body Phys. Rev (96). W ( t ) W t kt V V W VW W Jˆˆˆ DJ Nonineretial Fokker-Planck equation: Ĵ D D D Dˆ D D D D D D : the angular momentum operator V( ) t μe() t E() t ˆ E() t... xx xy xz xy yy yz xz yz yy : the diffusion tensor X z are the Euler angles Y Z O x N y 9

10 t Noninertial rotational Brownian motion of free asymmetric tops Fokker-Planck equation ˆ ˆ ˆ xx x yy y zz z W D J D J D J W Eigenvalues of the Fokker-Planck operator W( t)~ P ( ) e J mk ˆ ˆ ˆ J J J xx x yy y zz z mk mk mk D J D J D J P P J mk t Rotational motion of free asymmetric tops in quantum mechanics Schrödinger equation i AJ BJ CJ t ˆ ˆ ˆ x y z Eigenvalues of energy mk J / ie t J e mk ( t) ( ) ˆ ˆ ˆ J J J x y z mk mk mk AJ BJ CJ E Dxx Dyy Dzz : Diffusion tensor diagonal components E A BC : J J J J mk mk mk Pmk ( ) ( ) Rotational constants

11 Angular Momentum Operator Ĵ ˆ ˆ ˆ ˆ i ˆ ˆ J J J J J J Jˆ Jˆ x y z ˆ i i ˆ J e cot i J i sin Jˆ ˆ ˆ ˆ J x Jy Jz ˆ J cot cos sin

12 Wigner D functions D km ( ) ˆ Dkm ( ) ( ) Dkm ( ) J J ˆ D ( ) ( ) C D ( ) m km m km ll mm D ( ) D ( ) C C D ( ) l m l m l l m m l l m m C l l l are the Clebsch-Gordan coefficients D ( ) P (cos ) P ( ) z is the Legendre polynomial of the order

13 How to proceed in the low field strength limit 3 () () () () () W W W W... V V V... () i () i J J () i () i J J km km km km W A () td ( ) V v () td ( ) i J () i J () i () i J km km k m D () t D ( ) W ( ) t d ~ A () t () () () W ( t) W ( W - Favro s solution) t W () () () () () () () () ( t ) W. t kt W V V W V W etc ( m). The time-dependent optical anisotropy (birefringence) of asymmetric top molecules nt ()~ ()~ t D () t /6 D () t D () t * * * ( m) ZZ XX m m m m are the spherical components of the polarizability tensor For symmetric top molecules the optical anisotropy nt ()~ P( )() t

14 Applications to the Kerr effect. D. Ridgeway «Transient Electric Birefringence of Suspension of Asymmetric Ellipsoids» J. Am. Chem. Soc (966) ;. R. Pecora «Dispersion of the Electrically Induced Refractive-Index Anisotropy in Nonpolar Fluids» J. Chem. Phys (969) ; 3. V. Rosato and G. Williams «Dynamic Kerr Effect and Dielectric Relaxation of Polarizable Dipolar Molecules» J. Chem. Soc. Faraday Trans (98) ; 4. W. A. Wegener R. M. Dowben and V. J. Koester «Time-Dependent Birefringence Linear Dichroism and Optical Rotation Resulting from Rigid-Body Rotational Diffusion» J. Chem. Phys. 7 6 (979) ; 5. W. A. Wegener «Transient Electric Birefringence of Dilute Rigid-Body Suspension at Low Field Strengths» J. Chem. Phys (986) ; 6. W. A. Wegener «Sinusoidal Electric Birefringence of Dilute Rigid-Body Suspension at Low Field Strengths» J. Chem. Phys (986). 7. etc 4

15 Other Applications of Anisotropic Rotational Diffusion. J. H. Freed «Anisotropic Rotational Diffusion and Electron Spin Resonance Linewidths» J. Chem. Phys (964).. W. T. Huntress «The Study of Anisotropic Rotationof Molecules in Liquids by NMR Quadrupolar Relaxation» Adv. Magn. Reson. 4 (97). 3. H. Brenner and D.W. Condiff «Transport Mechanics in Systems of Orientable Particles. III. Arbitrary Particles» J. Colloid. Interface Sci. 4 8 (97). 4. T. J. Chuang and K. B. Eisenthal «Theory of Fluorescence Depolarization by Anisotropic Rotational Diffusion» J. Chem. Phys (97). 5. B. J. Berne and R. Pecora «Dynamic Light Scattering with Applications to Chemistry Biology and Physics» Wiley New York R. Tarroni and C. Zannoni «On the Rotational Diffusion of Asymmetric Molecules in Liquid Crystals» J. Chem. Phys (99). Etc. 5

16 Example : dielectric (magnetic) relaxation U () t E () t in U () t P () t out : electric (magnetic) field : polarization (magnetization) E(t) P(t) Linear response: P() t ( tt') E( t) dt E() t E i t e t i t P() t ( ) Ee ( ) : dielectric (magnetic) susceptibility Nonlinear AC response to E() t E i t e Polarization P() t ( E) e ( E) e i t i3t 3 Electric birefringence n E E ( ) ( ) e i t 6

17 Transient Nonlinear Response We are interested in the relaxation of the system starting from an equilibrium (stationary) state I with the distribution function W I (x) which evolves under the action of the stimulus to another equilibrium (stationary) state II with the distribution function W II (x). V I A I A (t) V II A II t t The response is characterized by the normalized relaxation function A (t ) of a dynamical variable A: A () t A II A () t t A A I II and the relaxation time defined as: () A A t dt A A n (electric birefringence) 7

18 (i) How to proceed at high field strengths Yu. P. Kalmykov Phys. Rev. E 65 (). Recurrence equation for statistical moments D () t d D () t d D () t dt n m nm n m n m n m X () t AX(). t The general method of solution is effected by successively increasing the size of the system matrix A until convergence is attained. d C (ii) t Q C t t t n n n Q C Q C n n n n dt Tridiagonal vector recurrence equation can always be solved by using matrix continued fractions. In the Kerr effect relaxation the continued fraction approach was used by H. Watanabe and A. Morita Kerr Effect Relaxation in High Electric Fields Adv. Chem. Phys (984). A detailed description of the method one can find in H. Risken The Fokker-Planck Equation (Springer Berlin 984; nd Edition 989); W. T. Coffey Yu. P. Kalmykov and J. T. Waldron. The Langevin Equation nd Ed. (World Scientific Singapore 4). 8 nm

19 Decay transient response t V I ˆ E ε E E μ V I I kt V I Z e W I / ) ( /. d e D Z D kt V m n I m n I / ) ( ) ( ()

20 Step-off transient response with (free rotational diffusion) D d dt D () t D () / t D () t D () t / D () t D () t X () t AX(). t D D () t D () t 34 3/ D () t 3 3 / D () t d D () t 3/ 3 3/ D () t dt 3 / 3 D () t D () t 3/ 34 D () t D () t D D xx D yy D xx Dzz D yy D D xx xx D D yy yy

21 Step-off (decay) transient response nt ()~ ()~ t D () t /6 D () t D () t * * * ( m) ZZ XX m m m m ()~ m( I) m() m nt a t t / m () t e m D D D D D () t D () t D () t D () t eg.. ( t) ( t) etc. () () () () 4 Dxx Dyy Dzz 4 Dyy Dxx Dzz 4Dzz Dyy Dxx D D D D D D D D D yy xx zz xx yy zz yy zz xx D D D D D D D D D yy xx zz xx yy zz yy zz xx

22 V I V II Transient response V t t N kt μ E kt N N N v q q q t D EN N N v u N z v ( ux iuy ) kt u x u y and u z are the components of a unit vector u in the direction of the dipole moment vector ( N I II) μ. m D m m mm dt m d D () t e D () t

23 C () t c c c c c () t () t () t () t () t m () m () m c t D t D C () t II 3 d D C( t) Q( II) C ( t) Q( II) C( t) Q( II) C( t) dt II II C NΔ C() Qk( II) Δk C () k II II Δk ( ) in k k( II) k( II) k( ) k( II) I Q Q Δ Q Recurrence equation for the matrix continued fractions II ( ) Δ k

24 Step-on transient response with C i II II ( II) Δ Δ Q I c c C () t c c c () t () t () t () t () t C i II II II II ( II) ( II) ( II) Δ Q Δ Δ Q Δ Q II II Δk ( ) in k k( II) k( II) k( ) k( II) I Q Q Δ Q m () m () m c t D t D II 4

25 W. A. Wegener «Transient Electric Birefringence of Dilute Rigid-Body Suspension at Low Field Strengths» J. Chem. Phys (986) Step-on transient response with Low field strength II I D m ()~ t II D 3 3/ ux iuy ux iu 3 4 y uz 5 D () t 33 3/ 3 uz ux iuy ux iuy3 uz D () t D () t d D () t II uxiuyuxiuy uz u xiuyux iu y dt 5 D () t D () t 3 3/ 3 D () t uz ux iuy ux iuy3 u z D () t 3/ 3 uz ux iuy34uxiuy 5 D () t 3 4 3/ 3 3 / D () t 3/ 3 3/ D () t 3 / 3 D () t 3/ 34 D () t nt ()~ II bmm() t amm() t m m 5

26 Step-on transient response t / m nt ()~ am( II) e m.4 è Rey w - logwt D 5 x II è. Rey -w - logwt D 5 x II. è Rey w - logwt D 5 x II è.5 Rey -w - logwt D 5 x II è Rey w - logwt D 5 x II 6

27 Conclusions. The transient and ac stationary Kerr effect responses of rigid asymmetric top macromolecules can be evaluated exactly in the context of the anisotropic rotational diffusion model at arbitrary strengths of electric fields.. Two-segmented macromolecules can be treated in a similar manner. 3. The same approach can be applied to other nonlinear phenomena such nonlinear dielectric relaxation etc. 7

28 Acknowledgements. The Organizing Committee and Prof. Tsuneo Okubo Yamagata University (for the invitation).. Prof. Kohzo Ito University of Tokyo (for the financial support towards my conference expenses). 3. My colleagues co-authors and friends (for the inestimable contribution to the present results and encouragement): William Coffey Trinity College Dublin Derrick Crothers Queens University of Belfast Jean-Louis Deardin University of Perpignan Sergey Titov IREE Russian Academy of Sciences Moscow John Waldron Trinity College Dublin 4. The auditorium for the kind attention 8