Hydrophobicity and Composition-Dependent Anomalies in Aqueous Binary Mixtures, along with some Contribution to Diffusion on Rugged Energy Landscape
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I started writing this thesis not only to obtain a doctoral degree, but also to compile in a particular way all the work that I have done during this time. The articles published during these years can only give a short overview of my research task. I decided to give my own perspective of the things I have learned and the results I have obtained. Some sections are directly the published articles, but some other are not and contain a significant amount of unpublished data. Even in some cases the published plots have been modified / altered to provide more insight or to maintain consistency. Historical perspectives often provide a deep understanding of the problems and have been briefly discussed in some chapters. This thesis contains theoretical and computer simulation studies to under-stand effects of spatial correlation on dynamics in several complex systems. Based on the different phenomena studied, the thesis has been divided into three major parts: I. Pair hydrophobicity, composition-dependent anomalies and structural trans-formations in aqueous binary mixtures II. Microscopic analysis of hydrophobic force law in a two dimensional (2D) water-like model system III. Diffusion of a tagged particle on a rugged energy landscape with spatial correlations The three parts have been further divided into ten chapters. In the following we provide part-wise and chapter-wise outline of the thesis. Part I consists of six chapters, where we focus on several important aqueous binary mixtures of amphiphilic molecules. To start with, Chapter 1 provides an introduction to non-ideality often encountered in aqueous binary mixtures. Here we briefly discuss the existing ideas of structural transformations associated with solvation of a foreign molecule in water, with particular emphasis on the classic “iceberg” model. Over the last decade, several investigations, especially neutron scattering and diffraction experiments, have questioned the validity of existing theories and have given rise to an alternate molecular picture involving micro aggregation of amphiphilic co-solvents in their aqueous binary mixtures. Such microheterogeneity was also supported by other experiments and simulations. In Chapter 2, we present our calculation of the separation dependence of potential of mean force (PMF) between two methane molecules in water-dimethyl sulfoxide (DMSO) mixture, using constrained molecular dynamics simulation. It helps us to understand the composition-dependence of pair hydrophobicity in this binary solvent. We find that pair hydrophobicity in the medium is surprisingly enhanced at DMSO mole fraction xDMSO ≈ 0.15, which explains several anomalous properties of this binary mixture – including the age-old mystery of DMSO being a protein stabilizer at lower concentration and protein destabilizer at higher concentration. Chapter 3 starts with discussion of non-monotonic composition dependence of several other properties in water-DMSO binary mixture, like diffusion coefficient, local composition fluctuation and fluctuations in total dipole moment of the system. All these properties exhibit weak to strong anomalies at low solute concentration. We attempt to provide a physical interpretation of such anomalies. Previous analyses often suggested occurrence of a “structural transformation” (or, microheterogeneity) in aqueous binary mixtures of amphiphilic molecules. We show that this structural transformation can be characterized and better understood under the purview of percolation theory. We define the self-aggregates of DMSO as clusters. Analysis of fractal dimension and cluster size distribution with reference to corresponding “universal” scaling exponents, combined with calculation of weight-averaged fraction of largest cluster and cluster size weight average, reveal a percolation transition of the clusters of DMSO in the anomalous concentration range. The percolation threshold appears at xDMSO ≈ 0.15. The molecular picture suggests that DMSO molecules form segregated islands or micro-aggregates at concentrations below the percolation threshold. Close to the critical concentration, DMSO molecules start forming a spanning cluster which gives rise to a bi-continuous phase (of water-rich region and DMSO-rich region) beyond the threshold of xDMSO ≈ 0.15. This percolation transition might be responsible for composition-dependent anomalies of the binary mixture in this low concentration regime. Similar phenomenon is observed for another amphiphilic molecule – ethanol, as discussed in Chapter 4. We again find composition dependent anomalies in several thermophysical properties, such as local composition fluctuation, radial distribution function of ethyl groups and self-diffusion co-efficient of ethanol. Earlier experiments often suggested distinct structural regimes in water-ethanol mixture at different concentrations. Using the statistical mechanical techniques introduced in the previous chapter, we show that ethanol clusters undergo a percolation transition in the anomalous concentration range. Despite the lack of a precise determination of the percolation threshold, estimate lies in the ethanol mole fraction range xEtOH ≈ 0.075 - 0.10. This difficulty is probably due to transient nature of the clusters (as will be discussed in Chapter 6) and finite size of the system. The scaling of ethanol cluster size distribution and the fractal behavior of ethanol clusters, however, conclusively demonstrate their “spanning” nature. To develop a unified understanding, we further study the composition-dependent anomalies and structural transformations in another amphiphilic molecule, tertiary butyl alcohol (TBA) in Chapter 5. Similar to the above-mentioned aqueous binary mixtures of DMSO and ethanol, we demonstrate here that the anomalies occur due to local structural changes involving self-aggregation of TBA molecules and percolation transition of TBA clusters at xTBA ≈ 0.05. At this percolation threshold, we observe a lambda-type divergence in the fluctuation of the size of the largest TBA cluster, reminiscent of a critical point. Interestingly, water molecules themselves exhibit a reverse percolation transition at higher TBA concentration ≈ 0.45, where large spanning water clusters now break-up into small clusters. This is accompanied by significant divergence of the fluctuations in the size of the largest water cluster. This second transition gives rise to another set of anomalies around. We conclude this part of the thesis with Chapter 6, where we introduce a novel method for understanding the stability of fluctuating clusters of DMSO, ethanol and TBA in their respective aqueous binary mixtures. We find that TBA clusters are the most stable, whereas ethanol clusters are the most transient among the three representative amphiphilic co-solvents. This correlates well with the amplitude of anomalies observed in these three binary mixtures. Part II deals with the topic of hydrophobic force law in water. In the introductory Chapter 7 of this part, we briefly discuss the concept of hydrophobicity which is believed to be of importance in understanding / explaining the initial processes involved in protein folding. We also discuss the experimental observations of Israelachvili (on the force between hydrophobic plates) and the empirical hydrophobic force law. We briefly touch upon the theoretical back-ground, including Lum-Chandler-Weeks theory. We conclude this chapter with a brief account of relevant and important in silico studies so far. In Chapter 8, we present our studies on Mercedes-Benz (MB) model – a two dimensional model system where circular disks interact with an anisotropic potential. This model was introduced by Ben-Naim and was later parametrized by Dill and co-workers to reproduce many of the anomalous properties of water. Using molecular dynamics simulation, we show that hydrophobic force law is indeed observed in MB model, with a correlation length of ξ=3.79. The simplicity of the model enables us to unravel the underlying physics that leads to this long range force between hydrophobic plates. In accordance with Lum-Chandler-Weeks theory, density fluctuation of MB particles (leading to cavitation) between the hydrophobic rods is clearly distinguishable – but it is not sufficiently long ranged, with density correlation extending only up to ζ=2.45. We find that relative orientation of MB molecules plays an important role in the origin of the hydrophobic force in long range. We define appropriate order parameters to capture the role of orientation, and briefly discuss a plausible approach of an orientation-dependent theory to explain this phenomenon. Part III consists of two chapters and focuses on the diffusion of a Brownian particle on a Gaussian random energy landscape. We articulate the rich history of the problem in the introductory Chapter 9. Despite broad applicability and historical importance of the problem, we have little knowledge about the effect of ruggedness on diffusion at a quantitative level. Every study seems to use the expression of Zwanzig [Proc. Natl. Acad. U.S.A, 85, 2029 (1988)] who derived the effective diffusion coefficient, Deff =D0 exp (-β2ε2 )for a Gaussian random surface with variance ε, but validity of the same has never been tested rigorously. In Chapter 10, we introduce two models of Gaussian random energy surface – a discrete lattice and a continuous field. Using computer simulation and theoretical analyses, we explore many different aspects of the diffusion process. We show that the elegant expression of Zwanzig can be reproduced ex-actly by Rosenfeld diffusion-entropy scaling relationship. Our simulations show that Zwanzig’s expression overestimates diffusion in the uncorrelated Gaussian random lattice – differing even by more than an order of magnitude at moderately high ruggedness (ε>3.0). The disparity originates from the presence of “three-site traps” (TST) on the landscape – which are formed by deep minima flanked by high barriers on either side. Using mean first passage time (MFPT) formalism, we derive an expression for the effective diffusion coefficient, Deff =D0 exp ( -β2ε2)[1 +erf (βε/2)]−1 in the presence of TSTs. This modified expression reproduces the simulation results accurately. Further, in presence of spatial correlation we derive a general expression, which reduces to Zwanzig’s form in the limit of infinite spatial correlation and to the above-mentioned equation in absence of correlation. The Gaussian random field has an inherent spatial correlation. Diffusion coefficient obtained from the Gaussian field – both by simulations and analytical methods – establish the effect of spatial correlation on random walk. We make special note of the fact that presence of TSTs at large ruggedness gives rise to an apparent breakdown of ergodicity of the type often encountered in glassy liquids. We characterize the same using non-Gaussian order parameter, and show that this “breakdown” scales with ruggedness following an asymptotic power law. We have discussed the scope of future work at the end of each chapter when-ever appropriate.
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