Biman Bagchi

1. Relaxation in Glassy Liquids Biman Bagchi Indian Institute of Science, Bangalore, India. Dr. Sarika Bhattacharyya Mr. Arnab Mukherjee Mr. Dwaipayan Chakrabarti Dr. Rajesh Murarka Adv. Chem. Phys. Vol. 116 (2001)

2. Plan of the talk • Introduction (summary of some experimental results). • Theoretical description (de Gennes narrowing, self-consistent mode coupling theory), Applications of MCT (Stokes-Einstein relation, Liquid Crystals) • Some computer simulation results. • More MCT (MCT of glass transition)

3. Basic Features • Rapid rise of viscosity with lowering of T in the deeply supercooled liquid near the glass transition temperature. This rapid rise starts typically 30-50 deg C above Tg. • This increase in viscosity can be described by Vogel-Fulcher expression. The same set of parameters can describe the rise for 4-5 orders of magnitude.

4. Basic Features (continued) • Relaxation functions (that is, relevant time correlation functions) become markedly non-exponential in this temperature range. Stretched exponential (KWW) form provides good fit with a low value of the exponent (typically 0.5). • Computer simulation studies on binary mixtures have shown that at high pressure and low temperature, hopping becomes the effective mode of mass transport and orientational relaxation. The emergence of hopping is rather gradual, that is, it coexists with continuous (Brownian) mode of diffusion until some low temperature where the latter ceases to contribute to diffusion.

5. Stretched-exponential stress relaxation with decreasing temperature P*= at constant 10.0 NPT Simulations of Binary Mixture AM + BB, JCP (2002) ln Cs(t) ln(t) strong Angel Plot

6. Anomalous observations on dynamics Orientational relaxation remains coupled to viscosity [Ediger, JCP 1996] Structural relaxation (diffusion) decouples from mechanical relaxation(viscosity)- fragility of a liquid [C. A. Angell, Science 1995] Translational diffusion is decoupled from orientational diffusion [Sillescu, JCP 1996, Ediger]

7. Theories of Slow Relaxation • Mode Coupling Theory • Adam-Gibbs-DiMarzio Entropy Crisis Theory --- Concept of Cooperatively Rearranging Region (CRR). • Energy Landscape Picture • Random first order theory (RFOT)

8. Small displacements via structural relaxation and transverse current How do molecules move in normal liquid? For many liquids D  T/  Stokes-Einstein relation

9. How to describe the continuos diffusion? But it fails for small molecules which is due to its failure to describe molecular length scale processes  Extended hydrodynamics and Remormalized kinetic theory  Mode Coupling Theory (MCT) Conventional descriptions: (A) Kinetic theory extended by Enskog (B) Navier-Stokes Hydrodynamics –Stick/Slip boundary condition

10. Single Particle diffusion Note that Fick’s Law is phenomenological --- D is obtained by Green -Kubo relation. Exponential decay of wave- Number dependent density fluctuation

11. IS THE INCOHERENT DYNAMIC STRUCTURE FACTOR MEASURED BY NEUTRON SCATTERING EXPERIMENTS. Coupling between Single particle and collective variables missing in hydrodynamic description.

12. Collective dynamics Variables are the conserved Quantities : density, Momentum and energy --- but these are coupled To each other. The simplest description Of coupled equations is Given by Navier-Stokes Hydrodynamic equations. Note that stress-stress tcf gives viscosity Linearization of Hydrodynamic equations

13. The hydrodynamic matrix -- note the decoupling between the transverse and the longitudinal current modes --the determinant (which determines poles and hence the time constants of relaxation) Becomes a prodcut of two.

14. The dynamic structure factor is a sum of three peak Lorentzian known as Rayleigh-Brillouin spectra which can be measured by Light scattering experiments. The central peak is due to density fluctuation and the corresponding hydrodynamic mode is called the “heat mode”. The two branches are due to the sound modes Heat mode Sound mode

15. De Gennes’ narrowing was the first indication that dynamics at small length scales cannot be described by conventional hydrodynamics.

16. The microscopic derivation of de- Gennes’ narrowing is simple. It uses what is known as Smoluchowski- Vlasov equation. This is an equation of motion for singlet density with a Mean-field force term. The basic physics is that at intermediate to large wavenumbers, both momentum and energy relaxation is very fast. At such Small length scales, momentum conservation is no longer a constrain. Instead, local environment controls density relaxation. Note that number density is conserved at all length scales. ****

17. Molecular hydrodynamic Equations of motion Density functional Free energy

18. The eigen-values Of hydrodynamic Equations undergo Sharp changes at Molecular length scales Due to the presence of Intermolecular correlations. Heat mode. Note The near zero value ~ k 2

19. What is MCT? • It is a self-consistent scheme which gets the short time behaviour (nearly) exactly correct (for two point correlation functions) because the lower order moments and satisfied. The long time behaviour is described by a correlator which is expanded in terms of the set of hydrodynamic eigenfunctions. The resulting equations are solved self-consistently. Thus, the long time diffusive limit is described fairly accurately.

20. The mode coupling theory expressions can be derived in several different ways – they all lead to the similar (if not the same) expression. • One of the early elegant applications of the mode coupling theory expression for liquids was made by Gaskel and Miller who derived an expression for the velocity time correlation function of a tagged particle. They argued that a particle moves by coupling to the current mode. So, they projected the propagator on

21. The resulting expression involved wave vector integration over the transverse and longitudinal current correlation functions. The longitudinal function decreases much faster than the transverse time correlation function. The latter decay as When you combine all the factors you recover Stokes-Einstein Relation, with 4. This calculation constitutes the first concrete demonstration of MCT. However, this early success of MCT was based on assumptions which turn out to be untenable.

22. The reason is that in dense liquids (and certainly in the supercooled state) it is the density relaxation of the surrounding solvent that is the slowest relaxation mode. The dramatic slowing down of density relaxation at wave numbers comparable to This is of course de Gennes narrowing. In principle, all the slow modes should be included.

23. Power laws in the orientational relaxation near Isotropic-Nematic phase-transition (INPT)

24. The molecular dynamics simulation is run on a system of 576 Gay-Berne ellipsoids in a Micro-Canonical ensemble. The simulations were run at temperatures T*= 1.0, 1.1, 1.2. The variation of order parameter at different temperatures along the density axis is shown here. Phase diagram of Gay-Berne ellipsoids with aspect ratio 3.

26. New Experimental results (Fayer et al.2002) optical Kerr effect data displaying the time dependence of orientational dynamics of the liquid crystal, 5-OCB at 347 K on a log plot. M. Fayer (1996-2004) Temperature dependent 5-OCB data sets displayed on a log plot.

27. Time Scales involved • Initial exponential decay occurs with a time constant in 1-5 ps range. • The long time Landau-de Gennes exponential decay sets in after 100 ns or so, with a time constant few hundred ns. • There is a big window between 10 ps to few hundred ns when decay is very slow.

28. Temperature dependent 3-CHBT data sets displayed on a log plot. Exponent ≈2/3 The short time portion of the 5-OCB data at 347 K with the exponential contributions removed on a log plot.

29. Comparison with relaxation in glassy liquids The temperature dependence of the time derivative of collective orientational time correlation function is shown here (Cang et al. J. Chem. Phys., 118, 9303 (2003)).

30. Mode coupling theory of orientational relaxation near INPT Origin of the slow down in relaxation can be understood from a mean-field theory which gives the following expression for LdG where S20(k) is the wave number dependent orientational structure factor in the intermolecular frame. DR is the rotational diffusion coefficient. Kerr experiments measure the k= 0 limit of the time derivative of the collective orientational correlation function C2m(k,t). ,

31. Zwanzig - Mori Continued fraction Single Particle Rotation Generalized Langevin equation

32. Mode coupling theory calculation of rotational friction • The baisc idea is that the torque tcf on a tagged ellipsoid slows down due to the marked slow down in orientational density relaxation. • Unlike in supercooled liquid, this happens at small k. • Expression for the torque can be obtained from the DFT free energy functional.

33. MCT expression for rotational friction C20(k) = (20,20) component of the direct correlation function F20(k,t) = (20,20) component of the dynamic structure factor

34. Isotropic-Nematic coexistence Isotropic Nematic

35. The molecular dynamics simulation is run on a system of 576 Gay-Berne ellipsoids in a Micro-Canonical ensemble. The simulations were run at temperatures T*= 1.0, 1.1, 1.2. The variation of order parameter at different temperatures along the density axis is shown here. Phase diagram of Gay-Berne ellipsoids with aspect ratio 3.

36. These coefficients of expansion of angular pair correlation functions can be calculated from simulation using the expression . . The coefficients of the spherical harmonic expansion of pair correlation function tend to diverge when isotropic nematic phase transitions approached along the density axis.

38. (JCP (2002))

39. Collective orientation f20(k) = 1/S20(k)

40. Thus, the leading term in the expansion varies as t-1/2. The above analysis is valid only after the initial short time decay, very close to the INPT. Fayer et al JCP (2002,2003) The time derivative of the theoretical correlation function, C20(t). Also shown is a t -0.63 power law (5-OCB). At short time, the derivative of the theoretical correlation function decays essentially as a power law.

41. Collective orientational correlation function Slow down in the relaxation of collective orientational time correlation function. The regions where power law relaxation is dominant are fitted to the function at density=3.1

42. The Log-log plot of derivative of the collective orientational correlation clearly shows the power law relaxation. Experimental data in the power law region ; Top 4 curves are of liquid crystals and bottom 5 are of supercooled liquids. Data shown on left is for liquid crystal and right is for supercooled liquid (Cang et al. J. Chem. Phys., 118, 9303 (2003)).

43. However, one should add that MCT is quantitatively accurate at normal liquids – far far superior to the Brownian oscillator model (BSO). (AD+DR, JLS, Rabani,Egorof …. Please note that BSO has no diffusive behavior in the proper sense.

44. Basic MCT equations for friction

45. B. Bagchi and S. Bhattacharyya Adv Chem Phys, 116, 67 (2001)

46. Relationship with Stokes-Einstein • It is important to realize that the Stokes-Einstein expression follows strictly from the current term, first derived by Gaskell & Miller.

47. Self-consistent scheme <r2(t)> = 2 d Cv()(t- ) Fs(k,t)=exp(- q2<r2(t)>/6) Cv(z) = kB T / [m(z + (z))]

48. Power law and mass dependence of diffusion D1 and D2 are self-diffusion coefficients of solvent and solute with masses m and M Straight line is the fitting The slope of the line = 0.099. MD simulation studies predict the slope to be 0.1 S.Bhattacharyya and B. Bagchi, PRE,61, 3850 (2000)

49. Microscopic analysis of Stokes-Einstein relation Bhattacharyya and Bagchi, JCP(2001)

50. Projection Operator Formalism