Computer Simulation Studies Of Phase Transition In Soft-Condensed Matter : Isotropic-Nematic, Gas-Liquid, And Polymer Collapse
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The present thesis reports computer simulation studies of several phase transition related phenomena in a range of soft-condensed matter systems. A coherent unifying theme of the thesis is the understanding of dynamics of phase transitions through free energy calculations using recently developed efficient non-Boltzmann sampling methods. Based on the system/phenomena of interest, the thesis has been classified into four major parts: I. Isotropic-nematic (IN) phase transition in liquid crystals. II. Nucleation phenomena in gas-liquid transition with particular emphasis on the systems close to the spinodal curve. III. Collapse transition in linear hydrocarbon (n-alkane) chains for a varying range of length, solvent and temperature. IV. Crystallization of unbranched polymer chains in dilute solution, with particular emphasis on the temperature dependent crossover between the rod-like crystalline state and spherical molten globule state. The thesis has been further divided into ten chapters running through the four parts mentioned before. In the following we provide a brief chapter-wise outline of the thesis. Part I deals with the power law relaxation and glassy dynamics in thermotropic liquid crystals close to the IN transition and consists of two chapters. To start with, Chapter I.1 provides an introduction to thermotropic liquid crystals. Here we briefly introduce various liquid crystalline phases, the order parameter used to characterize the IN transition, a few well established theoretical models, and we conclude with describing the recent experimental and computer simulation studies that have motivated the work described in the next chapter. In Chapter I.2, we present our molecular dynamics simulation studies on single particle and collective orientational dynamics across the IN transition for Lebwohl Lasher model, which is a well-known lattice model for thermotropic liquid crystals. Even this simplified model without any translational degrees of freedom successfully captures the short-tointermediate time power law decay recently observed in optical heterodyne detected optical Kerr effect (OHDOKE) measurements near the IN transition. The angular velocity time correlation function also exhibits a rather pronounced power law decay near the IN boundary. In the mean squared angular displacement at comparable time scales, we observe the emergence of a sub-diffusive regime which is followed by a super-diffusive regime before the onset of the longtime diffusive behavior. We observe signature of dynamical heterogeneity through pronounced non-Gaussian behavior in the orientational motion particularly at lower temperatures. Interestingly, this behavior closely resembles what is usually observed in supercooled liquids. We obtain the free energy as a function of orientational order parameter by the use of recently developed transition matrix Monte Carlo (TMMC) method. The free energy surface is flat for the system considered here and the barrier between isotropic and nematic phases is vanishingly small for this weakly first-order transition, hence allowing for large scale, collective, and correlated orientational density fluctuations. We attribute this large scale fluctuations as the reason for the observed power law decay of the orientational time correlation functions. Part II consists of three chapters, where we focus on the age old problem of nucleation and growth, both from the perspective of thermodynamics and kinetics. We account for the rich history of the problem in the introductory Chapter II.1. In this chapter we describe various types and examples of the nucleation phenomena, and a brief account of the major theoretical approaches used so far. We begin with the most successful Classical Nucleation Theory (CNT), and then move on to more recent applications of Density Functional Theory (DFT) and other mean-field types of models. We conclude with a comparison between the experiments, theories and computational studies. In the next chapter (Chapter II.2) we attempt to elucidate the mechanism of nucleation near the gas-liquid spinodal from a microscopic point of view. Here we construct a multidimensional free energy surface of nucleation of the liquid phase from the parent supercooled and supersaturated vapor phase near the gas-liquid spinodal. In particular, we remove the Becker-Doring constraint of having only one growing cluster in the system. The free energy, as a function of the size of the largest cluster, develops a pronounced minimum at a subcritical cluster size close to the spinodal. This signifies a two step nature of the process of nucleation, where the rapid formation of subcritical nuclei is followed by further growth by slower density fluctuations on an uphill free energy surface. An alternative free energy pathway involving the participation of many subcritical clusters is envisaged near the spinodal where the growth of the nucleus is found to be promoted by a coalescence mechanism in contrast to the single particle addition assumption within CNT. The growth of the stable phase becomes progressively collective and spatially diffuse, and the significance of a “critical nucleus” is lost for deeper quenches. In this chapter we present our studies both in 3dimensional Lennard-Jones (LJ) system and Ising model (both 2and 3dimensions). Our general findings seem to be independent of the model chosen. While the previous chapter focuses on relatively well-studied 3-dimensional (3D) LJ system, in Chapter II.3 we present our studies on the characteristics of the nucleation phenomena in 2dimensional (2D) Lennard-Jones fluid. To the best of our knowledge this is the first extensive computer simulation study to check the accuracy of CNT in 2D. Using various Monte Carlo methods, we calculate the free energy barrier for nucleation, line tension, and bulk densities of equilibrium liquid and vapor phases, and also investigate the size and shape of the critical nucleus. The study is carried out at an intermediate level of supersaturation (away from the spinoidal limit). In 2D, a surprisingly large cutoff (rc ≥ 7.0σ where σ is the diameter of LJ particles) in the truncation of the LJ potential is required to obtain converged results. A lower cutoff leads to a substantial error in the values of the line tension, nucleation barrier, and characteristics of the critical cluster. Note that typically 2.5σ is sufficient for 3D LJ fluids. We observe that in 2D system CNT fails to provide a reliable estimate of the free energy barrier. While it is known to slightly overestimate the nucleation barrier in 3D, it underestimates the barrier by as much as 50% at the saturation ratio S = 1.1(defined as S = P/Pc, where Pc is the coexistence pressure) and at the reduced temperature T* = 0.427(defined as T* = KBT/ ε, where ε is the depth of the potential well). The reason for the marked inadequacy of the CNT in 2D can be attributed to the non-circular nature of the critical clusters. Although the shape becomes increasingly circular and the clusters become more compact with increase in cutoff radius, an appreciable non-circular nature remains even without any cutoff to make the simple CNT inaccurate. Part III again consists of three chapters and focuses on the conformational equilibria. Collapse transition and self-organized structures of n-alkanes in solution. In Chapter III.1 we carry out a brief survey of the existing theories of polymer in solution, with particular emphasis on the collapse process in poor solvents. We also introduce the concept of “hydrophobicity” and “hydrophobic collapse”, which is now a subject enormous interest, partly because it my help in understanding the initial processes involved in protein folding. We briefly discuss the subject of formation of beautiful self-organized structures by block copolymers, and also simple homopolymers which is essentially the focus of the work embodied in the next two chapters. In Chapter III.2 we demonstrated a chain length dependent crossover in the structural properties of linear hydrocarbon (n-alkane) chains using detailed atomistic simulations in explicit water. We identify a number of exotic structures o the polymer chain through energy minimization of representative snapshots collected from molecular dynamics trajectory. While the collapsed state is ring-like(circular) for small chains(CnH2n+2; n ≤ 20) and spherical for very long ones( n = 100), we find the emergence of ordered helical structures at intermediate lengths (n ~ 40). We find different types of disordered helices and toroid-like structures at n = 60. We also report a sharp transition in the stability of the collapsed state as a function of the chain length through relevant free energy calculations. While the collapsed state is only marginally metastable for C20H42, a clear bistable free energy surface emerges only when the chain is about 30 monomers long. For n = 30, the polymer exhibits an intermittent oscillation(characterized by well-developed 1/f noise, where f is the frequency ) between the collapsed and the coil structures, characteristic of two stable states separated by a small barrier. This appears to support a weakly first order phase transition between the extended and the collapsed states. Chapter III.3 extends the study of previous chapter to much longer chains (n ≥ 100), which irreversibly collapse in water into globular forms. Even though the collapsed form has a nearly spherical shape, close inspection shows a propensity towards local ordering in the alignment of the polymer segments. This tendency to maintain alignment in order to maximize the number of contacts leads to a core-shell like structure, where the shell is often characterized by a bent rod-like shape consisting of two adjacent segments running in parallel. A key event associated with the initial stage of collapse seems to be the formation of a skewed ring (or loop) that serves as a “nucleation center” for rest of the chain to collapse into. Time evolution of the radial distribution function of water surrounding the polymer, shows that the density of neighboring water decreases by only about 15-20% from that of bulk water. Even though interior of the ting-like structures is fully devoid of water, solvent accessible surface representation shows that these regions are geometrically/spatially inaccessible to water molecules. We suggest that the role of water is to stabilize such ring-like structures once formed by natural conformational fluctuations of the polymer chain. This view is confirmed by observation of spontaneous formation and melting away of such ring-like entities in a polar aprotic solvent(DMSO). We also comment on the role of the flexibility of polymer chains in determining the collapse kinetics. The last part(Part IV) of the thesis consists of two chapters that deal with the crystallization of linear polymer chains from dilute solution. The way long chain polymers crystallize is drastically different from their small molecule counterparts due to their topological connectivity. Linear polymers often crystallize from dilute solution in the form of thin lamellae with well-defined crystallographic features. In Chapter IV.1 we briefly survey the current theoretical understanding and confusions associated with the highly debated field of polymer crystallization. While the last few decades have seen the development of many successful phenomenological theories, the molecular mechanism of formation of such self-organized lamellae is extremely complex and very poorly understood. There are clearly two distinct steps in polymer crystallization. Firstly, the individual linear polymers must self-organize into bundles of somewhat regular structures. These structures then further aggregate to lamellar form and crystallize into a lattice. In this respect , it has marked similarity to the problem of protein crystallization. In chapter IV.2 we present Brownian dynamics simulation studies of a single polythelene chain of length 500. Such systems can reasonably mimic the process of crystallization from dilute solutions. Our simulations could successfully reproduce some of the interesting phenomena observed in experiments and very recent computer simulation studies, including multi-center nucleation of rod-like structures within a single polymer chain, an inverse relation between lamellar thickness and temperature etc. But our primary focus has been to understand the nature of the phase transition as one traverses along the melting temperature and the underlying free energy surface. Near the melting temperature we observe a very intriguing fluctuation between the disordered molten globule state and the ordered rod-like crystalline, where these two forms have highly different shape and structure. These fluctuations have strong signature of 1/f noise or intermittency. This clearly indicates the existence of a weakly first order transition, where two widely different states with large difference in values of order parameter are separated by a rather small free energy barrier. This can be related to the experimentally observed density fluctuations that resemble spinodal decomposition. It is important to note that very similar fluctuations have been observed in our previous studies on liquid crystals (Chapter 1.2) and intermediate sized alkalines in water(Chapter III.2) that signifies a universal underlying energy landscape for these systems. We have discussed the scope of future work at the end of each chapter whenever appropriate.