An increase in the co-operativity in the motion of particles and a growth of a suitably defined dynamical correlation length seem to be generic features exhibited by all liquids upon supercooling. These features have been observed both in experiments and in numerical simulations of glass-forming liquids. Specially designed NMR experiments have estimated that the rough magnitude of this correlation length is of the order of a few nanometers near the glass transition. Simulations also predict that there are regions in the system which are more liquid-like than other regions. A complete theoretical understanding of this behaviour is not available at present. In recent calculations, Berthier, Biroli and coworkers [1, 2] extended the simple mode coupling theory (MCT) to incorporate the effects of dynamic heterogeneity and predicted the existence of a growing dynamical correlation length associated with the cooperativity of the dynamics. MCT also predicts a power law divergence of different dynamical quantities at the mode coupling temperature and at temperatures somewhat higher than the mode coupling temperature, these predictions are found to be consistent with experimental and simulation results. The system size dependence of these quantities should exhibit finite size scaling (FSS) similar to that observed near a continuous phase transition in the temperature range where they show power law growth. Hence we have used the method of finite size scaling in the context of the dynamics of supercooled liquids. In chapter 2, we present the results of extensive molecular dynamics simulations of a model glass forming liquid and extract a dynamical correlation length ξ associated with dynamic heterogeneity by performing a detailed finite size scaling analysis of a four-point dynamic susceptibility χ4(t)  and the associated Binder cumulant. We find that although these quantities show the “normal” finite size scaling behaviour expected for a system with a growing correlation length, the relaxation time τ does not. Thus glassy dynamics can not be fully understood in terms of “standard” critical phenomena. Inspired by the success of the empirical Adam-Gibbs relation  which relates dynamics with the configurational entropy, we have calculated the configurational entropy for different system sizes and temperatures to explain the nontrivial scaling behaviour of the relaxation time. We find that the behaviour of the relaxation time τ can be explained in terms of the Adam-Gibbs relation  for all temperatures and system sizes. This observation raises serious questions about the validity of the mode coupling theory which does not include the effects of the potential energy (or free energy) landscape on the dynamics. On the other hand, in the “random first order transition” theory (RFOT), introduced by Wolynes and coworkers , the configurational entropy plays a central role in determining the dynamics. So we also tried to explain our simulation results in terms of RFOT. However, this interpretation has the drawback that the value of one of the exponents of this theory extracted from our numerical results does not satisfy an expected physical bound, and there is no clear explanation for the obtained values of other exponents. Thus we find puzzling values for the exponents relevant to the applicability of RFOT, which are in need of explanation. This can be due to the fact that RFOT focuses only near the glass transition, while all our simulation results are for temperatures far above the glass transition temperature (actually, above the mode coupling temperature). Interestingly, results similar to ours were obtained in a recent analysis  of experimental data near the laboratory glass transition, on a large class of glass-forming materials. Thus right now we do not have any theory which can explain our simulation data consistently from all perspectives. There have been some attempts to extend the RFOT analysis to temperatures above the mode coupling temperature [7, 8] and to estimate a length scale associated with the configurational entropy at such temperatures. We compare our results with the predictions arising from these analyses. In chapter 3, we present simulation results that suggest that finite size scaling analysis is probably the only feasible method for obtaining reliable estimates of the dynamical correlation length for supercooled liquids. As mentioned before, although there exists a growing correlation length, the behaviour of all measured quantities (specifically, the relaxation time) is not in accordance with the behaviour expected in “standard” critical phenomena. So one might suspect the results for the correlation length extracted from the scaling analysis. To find out whether the results obtained by doing finite size scaling are correct, we have done simulations of very large system sizes for the same model glass forming liquid. In earlier studies, the correlation length has been extracted from the wave vector dependence of the dynamic susceptibility in the limit of zero wave vector, but to estimate the correlation length with reasonable accuracy one needs data in the small wave vector range. This implies that one needs to simulate very large systems. But as far as we know, in all previous studies typical system sizes of the order of 10, 000 particles have been used to do this analysis. In this chapter we show by comparing results for systems of 28, 000 and 350, 000 particles that these previous estimates are not reliable. We also show that one needs to simulate systems with at least a million particles to estimate the correlation length correctly near the mode coupling temperature and this size increases with decreasing temperature. We compare the correlation length obtained by analyzing the wave vector dependence of the dynamic susceptibility for a 350, 000particle system with the results obtained from the finite size scaling analysis. We were only able to compare the results in the high temperature range due to obvious reasons. However the agreement in the high temperature range shows that the finite size scaling analysis is robust and also establishes the fact that finite size scaling is the only practical method to extract reliable correlation lengths in supercooled liquids. In chapter 4, we present a free energy landscape analysis of dynamic heterogeneity for a monodisperse hard sphere system. The importance of the potential energy landscape for particles interacting with soft potentials is well known in the glass community from the work of Sastry et al.  and others, but the hard sphere system which does not have any well defined potential energy landscape also exhibits similar slow dynamics in the high density limit. Thus it is not clear how to treat the hard sphere systems within the same energy landscape formalism. Dasgupta et al. [10, 11, 12, 13, 14, 15] showed that one can explain the slow dynamics of these hard core systems in term of a free energy landscape picture. They and other researchers showed that these system have many aperiodic local minima in its free energy landscape, with free energy lower than that of the liquid. Using the Ramkrishnan-Yussouff free energy functional, we have performed multi parameter variational minimizations to map out the detailed density distribution of glassy free energy minima. We found that the distribution of the widths of local density peaks at glassy minima is spatially heterogeneous. By performing hard sphere event driven molecular dynamics simulation, we show that there exists strong correlation between these density inhomogeneity and the local Debye-Waller factor which provides a measure of the dynamic heterogeneity observed in simulations. This result unifies the system of hard core particles with the other soft core particles in terms of a landscapebased description of dynamic heterogeneity. In chapter 5, we extend the same free energy analysis to a polydisperse system and show that there is a critical polydispersity beyond which the crystal state is not stable and glassy states are thermodynamically stable. We also found a reentrant behaviour in the liquid-solid phase transition within this free-energy based formalism. These results are in qualitative agreement with experimental observations for colloidal systems
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