1. Slow Dynamics in Binary Liquids : Microscopic Theory and Computer Simulation Studies of Diffusion, Density Relaxation, Solvation and Composition Fluctuation Biman Bagchi SSCU, IISc, Bangalore. Decemmber 2003

2. Outline Introduction Local composition fluctuations in strongly nonideal binary mixtures Diffusion of small light particles in a solvent of large massive molecules Pair dynamics in a glass-forming binary mixture Diffusion and viscosity in a highly supercooled polydisperse system Conclusion Solvation Dynamics in Binary Mixture.

3. Polarization Relaxation in Binary Dipolar Mixture Molecular Hydrodynamic Theory of Chandra and Bagchi. (1990,1991) The theory uses density functional theory to describe the equilibrium aspect of solvation in a binary mixture.

4. Definition of Non-ideality Raoult’s Law

11. Dynamics of Solvation

15. Local composition fluctuations in strongly nonideal binary mixtures Spontaneous local fluctuations  rich and complex behavior in many- body system R  V (R) What is the probability of finding exactly n particle centers within  V (R) ? In one component liquid  local density fluctuations are Gaussian Binary mixtures that are highly nonideal, play an important role in industry

16. N P T simulations of Nonideal Binary Mixtures Study of Composition Fluctuations N P T simulations of Nonideal Binary Mixtures Study of Composition Fluctuations m A = m B = m parameters  AA  BB  AB AAAA BBBB ABAB Equal size model 1.0 0.52.0 Kob- Andersen model 1.00.880.801.00.51.5 x A = 0.8 x B = 0.2 Two model binary mixtures : Kob-Andersen model (glass-forming mixture) Equal size model

17. Probability Distributions of Composition Fluctuation Kob-Andersen Model R = 2.0  AA T * = 1.0 P * = 2.0 Gaussian distribution  N A  = 27.3  A = 1.995 Both A and B fluctuations are large  N B  = 6.74  B = 1.995 System is indeed locally heterogeneous

18. Joint Probability Distribution Function Kob-Andersen Model R = 2.0  AA Nearly Gaussian Corr[  N A,  N B ] = - 0.203  Fluctuations in A and B are anticorrelated

19. tcf  AA0.190.47 BB4.300.48 AB3.000.41 tcf  AA0.200.42 BB8.070.58 AB6.500.60 R = 2.0  AA P * = 2.0 R = 2.0  AA P * = 4.0 Dynamical Correlations in Composition Fluctuation : Kob-Andersen Model Non-exponential decay Distribution of relaxation times Stretched exponential fit Slow Dynamics

20. Diffusion of small light particles in a solvent of large massive molecules Isolated small light particles in a solvent of large heavy particles can mimic concentrated solution of polysaccharide in water, motion of water in clay The coexistence of both hopping and continuous diffusive motion

21. — M R =5 — M R =25 — M R =50 — M R =250 Relaxation of Solute and Solvent : The Self-intermediate Scattering Function F s (k,t) Solute Solvent k * =k  11 ~2  F s (k,t) begins to stretch at long time for higher solvent mass ! Sum of two stretched exponential function No stretching at long times ! Exponential decay Solute probes progressively more local heterogeneous environment

22. Non-Gaussian Parameter — M R =5 — M R =25 — M R =50 — M R =250 Solute The peak height increases  heterogeneity probed by the solute increases with solvent mass No such increase for the solvent

23. 50.0820.70 250.0800.960.490.67 500.0830.940.590.64 2500.0850.911.010.63 The Self-intermediate Scattering Function of the Solute k * ~2  12 Two stretched exponential separated by a power law type plateau, often observed in deeply supercooled liquids Separation of time scale between binary interaction and solvent density mode — increases with solvent mass

24. The Velocity Autocorrelation Function of the Solute Particles — M R =25 — M R =50 — M R =250 Development of an increasingly negative dip followed by pronounced oscillations at longer times  “dynamic cage” formation in which the solute particle executes a damped oscillatory motion : observed in supercooled liquid

25. Generalized self-consistent scheme Generalized self-consistent scheme

26. 523.6513.95 2525.4033.80 5025.4550.20 25025.5899.80 Self-consistent scheme : overestimates diffusion (faster decay of F s (k,t)) The relative contribution of the binary term decreases with solvent mass Contribution of the density mode increases ! Gaussian approximation is poor

27. Comparison of MCT prediction with simulations 50.1350.1065 250.1080.0675 500.1010.0530 2500.08050.0320 For larger mass ratio, MCT breaks down more severely ! Overestimates the friction contribution from the density mode Solute probes almost quenched system  breakdown of MCT can be connected to its similar breakdown near the glass transition temperature  hopping mode plays the dominant role in the diffusion process

28. 4 : Pair dynamics in a glass-forming binary mixture Dynamics in supercooled liquids has been investigated solely in terms of single particle dynamics The relative motion of the atoms that involve higher-order (two-body) correlations can provide much broader insight into the anomalous dynamics of supercooled liquids

29. Radial Part of the Time Dependent Pair Distribution Function (TDPDF) The TDPDF, g 2 (r o,r;t), is the conditional probability that two particles are separated by r at time t if that pair were separated by r o at time t = 0, thus measures the relative motion of a pair of atoms Nearest neighbor AA pair  t=500  Jump motions are the dominant diffusive mode by which the separation between pairs of atoms evolves in time

30. Angular Part of the Time Dependent Pair Distribution Function (TDPDF) AA pairAB pair BB pair Nearest neighbor pair Compared to AA pair, the approach to the uniform value is faster in case of AB pair Relaxation of BB pair is relatively slower at short times as compared to the AB pair

31. Relative Diffusion : Mean-Square Relative Displacement (MSRD) Nearest neighbor pair Faster approach of the diffusive limit of BB pair separation Time scale needed to reach the diffusive limit is shorter for the AB pair than that for the AA pair Relative diffusion coefficients

32. The Non-Gaussian Parameter for the Relative Motion Single particle dynamics Pair dynamics for nearest neighbor pair B particles probe a much more heterogeneous environment than the A particles The dynamics explored by the BB pair is less heterogeneous than the AA and AB pairs

33. Theoretical Analysis Mean-field Smoluchowski equation Potential of mean force Nonlinear time

34. Comparison Between Theory and Simulation AA pairAB pair BB pair Nearest neighbor pair Mean-field model successfully describes the dynamics of the AA and AB pairs Relative diffusion considered as over- damped motion in an effective potential, occurs mainly via hopping The agreement for the BB pair is less satisfactory !

35. Nearest neighbor BB pair executes large scale anharmonic motions in a weak effective potential The fluctuations about the mean-force field experienced by the BB pair are large and important ! Next nearest neighbor BB pair Better agreement compared to nearest neighbor pair

36. Angell´s ‘strong’ and ‘fragile’ classification 5 : Diffusion and viscosity in a highly supercooled polydisperse system Fragile liquid: Super-Arrhenius  follows VFT equation Accompanied by Stretched exponential relaxation Small D  more fragile Progressive decoupling between D T and  (D T   - ,  < 1), in contrast to the high T behavior (  = 1 ; SE relation) :

37. Temperature Dependence of Viscosity Super-Arrhenius behavior of viscosity VFT fit Critical temperature for viscosity T o  = 0.57 Within the temperature range investigated, Angell´s fragility index, D  1.42  A very fragile liquid More fragile than Kob-Andersen Binary mixture, D  2.45 Arrhenius plot

38. Temperature Dependence of Diffusion Coefficients Arrhenius plots Diffusion shows a super-Arrhenius T dependence Particles are categorized into different subsets of width VFT law Critical glass transition temperature for diffusion Critical temperature depends on the size of the particles ! DlDl DsDs

39. Critical Glass Transition Temperature for Diffusion : Particle Size Dependence increases with size of the particles Size only Size + mass  Near the glass transition the diffusion is partly decoupled from the viscosity, and for smaller particles the degree of decoupling is more The increase of critical temperature with size is not an effect of mass polydispersity  related to the dynamical heterogeneity induced by geometrical frustration

40. Size Dependence of Diffusion Coefficient : Breakdown of Stokes-Einstien Relation T * = 0.67 SE relation A marked deviation from Stokesian behavior at low T  A highly nonlinear size dependence of the diffusion For the smallest size particles, D s   -0.5 At low T, the observed nonlinear dependence of diffusion on size may be related to the increase in dynamic heterogeneity in a polydisperse system

41. Self-part of the van Hove correlation function Smallest particle Largest particle T * = 0.67 The gradual development of a second peak at r  1.0 indicates single particle hopping For the larger particles hopping takes place at relatively longer times

42. The Self-intermediate Scattering Function Smallest Largest T * = 0.67, The long time decay of F s (k,t) is well fitted by the Kohlrausch-Williams-Watts (KWW) stretched exponential form : Particle ii ii Smallest2420.49 Largest7170.64 The enhanced stretching (  s   l ) is due to the greater heterogeneity probed by the smaller size particles

43. 6: Hetergeneous relaxation in supercooled liquids: A density functional theory analysis Recent time domain experiments,  het  2-3 nm Spatially heterogeneous dynamics in highly supercooled liquids Near T g, dynamics differ by 1-5 orders of magnitude between the fastest and slowest regions Why do these heterogeneities arise ?

44. R I =4.0  R I =2.5  R I =1.5  Hard sphere liquid Large free energy cost to create larger inhomogeneous region S(k) is nearly zero for small k, density fluctuation only in intermediate k Unlikely to sustain inhomogeneity, l f  5 

45. Rotational Dynamics in Relaxing Inhomogeneous Domains VFT form Orientational correlation function Av. Rotational correlation time R I =2.5  The decay is nonexponential and av. correlation time is increased by a factor 1.8 Increase in , slower regions become slower at a faster rate

46. 7 : Isomerization dynamics in highly viscous liquids Isomerization reactions involve large amplitude motion of a bulky group  Strongly coupled to the enviornment For barrier frequency,  b  10 13 s -1, the situation is not starightforward  reactive motion probes mainly the elastic (high frequency) response of the medium At high viscosities, experiments and simulations predict

47. Enskog friction Frequency-Dependent Friction from Mode-Coupling Theory In the high frequency regime the  total (z) is much less than  E and is dominated entirely by  B (z)   E always overestimate  total (z) for continuous potential  * =0.85 T * =0.85

48. Frequency Dependent Viscosity MCT Maxwell relation Maxwell viscoelastic model fails to describe higher frequency peak : even poorly describe low-frequency peak The two-peak structure is a clear indication of the bimodal response of a dense liquid  * =0.85 T * =0.73 Maxwell relation Viscoelastic relaxation time

49. Transmission coefficient Barrier Crossing Rate  * =0.6-1.05, T * =0.85  b (s -1 )  3  10 12 0.72 5  10 12 0.58 10 13 0.22 2  10 13 0.045  strongly depends on  b  b  2  10 13 s -1,   0  TST result The values of the exponent appear to be in very good agreement with many experimental results

50. ACKNOWLEDGEMENT Dr. Rajesh Murarka (Berkeley) Dr. Sarika Bhattacharyya (CalTech) Dr. Goundla Srinivas (UPenn) DST CSIR