Biman Bagchi SSCU, IISc, Bangalore. Decemmber 2003

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 Solvation Dynamics in Binary Mixture. 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

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 What is the probability of finding exactly n particle centers within V(R) ? R 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 Two model binary mixtures : Kob-Andersen model (glass-forming mixture) Equal size model xA= 0.8 xB= 0.2 mA = mB = m

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

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

19. Dynamical Correlations in Composition Fluctuation : Kob-Andersen Model Stretched exponential fit R = 2.0AAP* = 2.0 Slow Dynamics R = 2.0AAP* = 4.0 Non-exponential decay Distribution of relaxation times

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

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

23. 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 —MR=25 —MR=50 —MR=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. Generalizedself-consistent scheme

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

27. Comparison of MCT prediction with simulations 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, g2(ro,r;t), is the conditional probability that two particles are separated by r at time t if that pair were separated by ro at time t = 0, thus measures the relative motion of a pair of atoms t=500 Nearest neighbor AA pair 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) Nearest neighbor pair AA pair AB pair BB 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 Relative diffusion coefficients Time scale needed to reach the diffusive limit is shorter for the AB pair than that for the AA pair

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 Nearest neighbor pair AA pair AB pair Mean-field model successfully describes the dynamics of the AA and AB pairs BB pair 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. 5 : Diffusion and viscosity in a highly supercooled polydisperse system Fragile liquid: Super-Arrhenius  follows VFT equation Small D  more fragile Accompanied by Stretched exponential relaxation Angell´s ‘strong’ and ‘fragile’ classification Progressive decoupling between DT and  (DT  -,  < 1), in contrast to the high T behavior ( = 1 ; SE relation) :

37. Temperature Dependence of Viscosity Arrhenius plot Super-Arrhenius behavior of viscosity Critical temperature for viscosity To = 0.57 VFT fit 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

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

39. Critical Glass Transition Temperature for Diffusion : Particle Size Dependence increases with size of the particles Size only Near the glass transition the diffusion is partly decoupled from the viscosity, and for smaller particles the degree of decoupling is more Size + mass 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 A marked deviation from Stokesian behavior at low T T* = 0.67 A highly nonlinear size dependence of the diffusion SE relation For the smallest size particles, Ds -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 T* = 0.67 Smallest particle The gradual development of a second peak at r  1.0 indicates single particle hopping Largest particle For the larger particles hopping takes place at relatively longer times

42. The Self-intermediate Scattering Function T* = 0.67, The long time decay of Fs(k,t) is well fitted by the Kohlrausch-Williams-Watts (KWW) stretched exponential form : Largest Smallest 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 Spatially heterogeneous dynamics in highly supercooled liquids Recent time domain experiments, het  2-3 nm Near Tg , dynamics differ by 1-5 orders of magnitude between the fastest and slowest regions Why do these heterogeneities arise ?

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

45. Rotational Dynamics in Relaxing Inhomogeneous Domains RI=2.5 VFT form Orientational correlation function Av. Rotational correlation time 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  1013 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. Frequency-Dependent Friction from Mode-Coupling Theory *=0.85 T*=0.85 Enskog friction 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

48. Frequency Dependent Viscosity *=0.85 T*=0.73 Maxwell relation Maxwell relation MCT Viscoelastic relaxation time 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

49. Barrier Crossing Rate *=0.6-1.05, T*=0.85 Transmission coefficient  strongly depends on b b  21013 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