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This thesis discusses the development of software architecture to support the computational investigation of random polygons in 3 space. The random polygons themselves are a simple model of long polymer chains. (A DNA molecule is one example of a polymer.) This software architecture includes "building blocks" which specify the actual manipulations and computations to be performed, and a structural framework which allows the user to specify which manipulations/computations to perform, in which order and with how many repetitions. The overall framework is designed in such a way that new building blocks can easily be added in the future. The development of three different building blocks to be used in this architecture which are entitled: Reducer, Lengthener and OutsideInLengthener are also discussed in this thesis. These building blocks manipulate the existing polygons - increasing or decreasing their size.

Miniaturization of electronic devices for enhancing their performance is associated with higher heat fluxes and cooling requirements. Surface modifi cation by texturing or coating is the most cost-effective approach to enhance the cooling of electronic devices. Experiments on carbon nanotube coated heater surfaces have shown heat transfer enhancement of 60 percent. In addition, silicon nanotubes etched on the silicon substrates have shown heat flux enhancement by as much as 120 percent. The heat flux augmentation is attributed to the combined effects of increase in the surface area due to the protruding nanotubes (nano- n eff ect), disruption of vapor lms and modi fication of the thermal/mass di ffusion boundary layers. Since the e ffects of disruption of vapor lms and modifi cation of the thermal/mass di ffusion boundary layers are similar in the above experiments, the difference in enhancement in heat transfer is the consequence of dissimilar nano- n eff ect. The thermal conductivity of carbon nanotubes is of the order of 6000 W/mK while that of silicon is 150 W/mK. However, in the experiments, carbon nanotubes have shown poor performance compared to silicon. This is the consequence of interfacial thermal resistance between the carbon nanotubes and the surrounding fluid since earlier studies have shown that there is comparatively smaller interface resistance to the heat flow from the silicon surface to the surrounding liquids. At the molecular level, atomic interactions of the coolant molecules with the solid substrate as well as their thermal-physical-chemical properties can play a vital role in the heat transfer from the nanotubes. Characterization of the e ffect of the molecular scale chemistry and structure can help to simulate the performance of a nano fin in diff erent kinds of coolants. So in this work to elucidate the eff ect of the molecular composition and structures on the interfacial thermal resistance, water, ethyl alcohol, 1-hexene, n-heptane and its isomers and chains are considered. Non equilibrium molecular dynamic simulations have been performed to compute the interfacial thermal resistance between the carbon nanotube and different coolants as well as to study the diff erent modes of heat transfer. The approach used in these simulations is based on the lumped capacitance method. This method is applicable due to the very high thermal conductivity of the carbon nanotubes, leading to orders of magnitude smaller temperature gradients within the nanotube than between the nanotube and the coolants. To perform the simulations, a single wall carbon nanotube (nano-fin) is placed at the center of the simulation domain surrounded by fluid molecules. The system is minimized and equilibrated to a certain reference temperature. Subsequently, the temperature of the nanotube is raised and the system is allowed to relax under constant energy. The heat transfer from the nano- fin to the surrounding fluid molecules is calculated as a function of time. The temperature decay rate of the nanotube is used to estimate the relaxation time constant and hence the e ffective thermal interfacial resistance between the nano-fi n and the fluid molecules. From the results it can be concluded that the interfacial thermal resistance depends upon the chemical composition, molecular structure, size of the polymer chains and the composition of their mixtures. By calculating the vibration spectra of the molecules of the fluids, it was observed that the heat transfer from the nanotube to the surrounding fluid occurs mutually via the coupling of the low frequency vibration modes.

Ο περιορισμός αλυσιδωτών μακρομορίων σε νανο-πόρους με διαστάσεις συγκρίσιμες με τις χαρακτηριστικές διαστάσεις της μοριακής αλυσίδας όπως και η δυνατότητα αποδέσμευσης (απελευθέρωσης) της αλυσίδας από το νανο-πόρο παίζουν σημαντικό ρόλο σε πολλές βιολογικές διαδικασίες και βιοτεχνολογικές εφαρμογές. Η διαδικασία της αποδέσμευσης μιας αρχικά εγκλωβισμένης πολυμερικής αλυσίδας εξαρτάται από μια σειρά παράγοντες όπως: το μοριακό βάρος της αλυσίδας, το μέγεθος του νανο-δοχείου σε σχέση με το μήκος της αλυσίδας, το σχήμα του νανο-δοχείου, το μέγεθος του πόρου διαφυγής/εισόδου της αλυσίδας στο νανοδοχείο, η ποιότητα του διαλύτη εντός και εκτός του νανοδοχείου κ.α. Το φαινόμενο και ο ρόλος των παραμέτρων που το επηρεάζουν θα μελετηθούν με αναλυτικές μεθόδους περιγράφοντας το μακρομόριο με μοντέλα ιδανικών αλυσίδων και το νανοδοχείο ως ένα αδιαπέραστο περιορισμό (σφαιρικού σχήματος) με ένα πόρο διαφυγής για την αλυσίδα. Η μελέτη αυτή αποτελεί το πρώτο βήμα για την εφαρμογή ρεαλιστικών μοντέλων για την περιγραφή των διαδικασιών δέσμευσης/αποδέσμευσης γραμμικών πολυμερών σε σφαιρικές κοιλότητες νανοσκοπικών μεγεθών. / The confinement of the macromolecules into nanopores with dimensions translocation of the chain through the nanopore has a key role in many biological processes and biotechnological applications. The process of packing/releasing of the polymer chain depends on many factors like molecular weight of the chain, the size of the nano-container in comparison with the length of the chain, the shape of the nano-container, the size of the nanopore, the quality of the solvent into and out of the nano-container etc. In the present work we will study the translocation of an ideal chain, confined initially into an impenetrable spherical cavity, through a tiny pore on the surface of the spherical cavity. Using analytical tools and Monte Carlo simulations we examine the effects of the molecular weight and the size of the spherical cavity in the translocation process. This study is a first step towards more realistic descriptions of the confinement/translocation problem of linear polymers from nanopores.

Σκοπός της παρούσας διπλωματικής εργασίας είναι η μελέτη της ροής υγρών μέσω νανοπορώδων μεμβρανών που φέρουν προσροφημένα συσταδικά συμπολυμερή διαφόρων μοριακών βαρών υπό συνθήκες ροής με καλό διαλύτη (τολουένιο) καθώς και με μίξη διαλυτών καλού και κακού όπως είναι το τολουένιο και το κυκλοεξάνιο αντιστοίχως. Μελετήθηκαν δύο κατηγορίες συμπολυμερών δισυσταδικά και τα τρισυσταδικά συμπολυμερή. Η πρώτη κατηγορία αναφέρεται στα polystyrene-b-polyethylene oxide (PS-PEO) δισυσταδικά συμπολυμερή διαφόρων μοριακών βαρών. Τα μοριακά βάρη που μελετήσαμε ήταν το 32.000, 70.500, 80.000, 147.000, 182.700, 184.000 , 239.000 και 322.000 g/mol. Το πρωτόκολλο που ακολουθήθηκε περιλαμβάνει τα παρακάτω βήματα: αρχικά μέτρηση της γυμνής μεμβράνης στο φασματοφωτόμετρο FTIR και μετά έλεγχος σε ροή με διαλύτη τολουένιο. Στη συνέχεια έγινε η τοποθέτηση της μεμβράνης σε διάλυμα του εκάστοτε πολυμερούς στο τολουένιο, Το τουλουένιο επιλέχτηκε διότι είναι καλός διαλύτης προς την συστάδα PS. Η μεμβράνη παρέμεινε στο διάλυμα για κάποιο χρονικό διάστημα της τάξεως ημερών έτσι ώστε να επιτευχθεί η διαδικασία της φυσιοπροσρόφησης. Λόγω του ότι το τουλουένιο αποτελεί κακό διαλύτη για το PEO έχουμε αγκύστρωση των πολυμερικών βουρτσών PEO στα κυλινδρικά τοιχώματα της νανοπορώδους μεμβράνης. Αφού πραγματοποιηθεί μέτρηση του υπέρυθρου φάσματος της μεμβράνης για να τεκμηριωθεί η προσρόφηση του πολυμερούς, ακολούθως γινόταν προσαρμογή της προσροφημένης μεμβράνης στην διάταξη και ροή με διαλύτη τολουένιο. Με χρήση του νόμου του Poiseuille βρίσκεται το πάχος των πολυμερικών βουρτσών, υπολογίζοντας έτσι κατά πόσο οι πολυμερικές βούρτσες με την εισχώρηση του διαλύτη από τους πόρους εκτείνονται κάθετα της επιφάνειας που έχουν προσροφηθεί. Τέλος με χρήση του φασματοφωτόμετρου FTIR και πάλι γινόταν έλεγχος εάν έχει γίνει αποκόλληση των πολυμερικών βουρτσών από τα τοιχώματα της μεμβράνης λόγω της ροής. Επίσης πραγματοποιήθηκε η μελέτη της συμπεριφοράς κάποιων από αυτά τα συμπολυμερή σε μίγμα διαλυτών τολουολίου και κυκλοεξανίου. Ακολουθώντας την ίδια διαδικασία όπως προηγουμένως αλλάζοντας μόνο το στάδιο της ροής, ο έλεγχος σε αυτή την περίπτωση ξεκινούσε από 100% τολουένιο, 70% τολουένιο-30% κυκλοεξάνιο, 50%τολουένιο – 50%κυκλοεξάνιο ,30% τολουένιο – 70 % κυκλοεξάνιο και καταλήγαμε με 100% κυκλοεξάνιο. Ελεγχόταν έτσι η απόκριση των πολυμερικών βουρτσών κατά την εναλλαγή του διαλύτη. Με χρήση του νόμου του Poiseuille μπορούσε να υπολογισθεί το πάχος των πολυμερικών βουρτσών και να προσδιορισθεί κατά πόσο μετά το τέντωμα τους από την επαφή τους με τον καλό διαλύτη το τολουένιο, συρρικνώνονται όταν έρχονται σε επαφή με τον κακό διαλύτη το κυκλοεξάνιο. Στην δεύτερη κατηγορία ανήκουν τα τρισυσταδικά συμπολυμερή . Τα τρισυσταδικά πολυμερή που μελετήσαμε ήταν το polyvinyl-2-pyridine-b-polystyrene-b- polyvinyl-2-pyridine (PV2P-PS-PV2P) 30K-180K-30K, το PEO-PS-PEO 18.6K-45.2K-18.6K και το PEO-PS-PEO 2.3K-16.7K-2.3K. Ακολουθώντας το ίδιο πρωτόκολλο πειραματικής διαδικασίας όπως στα δισυσταδικά συμπολυμερή, έγινε η μελέτη της συμπεριφοράς των τρισυσταδικών συμπολυμερών σε ροή με τουλουένιο, το οποίο αποτελεί καλό διαλύτη προς την μια συστάδα (PS) του συμπολυμερούς. Τα τρισυσταδικά συμπολυμερή κατά την διαδικασία της φυσιοπροσρόφησης, δηλαδή της αγκίστρωσης των πολυμερικών βουρτσών μέσω των συστάδων PEO και PV2P στα κυλινδρικά τοιχώματα της νανοπορώδους μεμβράνης παρουσιάζουν τον σχηματισμό βρόχου (loop) . Στην συνέχεια γινόταν τοποθέτηση της μεμβράνης στην διάταξη και ο έλεγχος σε ροή με διαλύτη τολουένιο όπως και στην περίπτωση των δισυσταδικών συμπολυμερών. Τέλος με την χρήση του φασματοφωτόμετρου FTIR γινόταν έλεγχος εάν έχει γίνει αποκόλληση των πολυμερικών βουρτσών από τα τοιχώματα της μεμβράνης. Το πάχος της προσροφημένης βούρτσας υπολογιζόταν με χρήση του νόμου του Poiseuille όπως και με τα δισυσταδικά συμπολυμερή. / -

Traditional topics such as physics, chemistry and mathematics have immensely changed the world in the twentieth century, but the twenty-first century seems to be that of soft condensed matter physics, which has already shown its tremendous possibilities to influence the everyday human life through its technological manifestations such as biotechnology and soft nano technology. Unlike the traditional topics, soft condensed matter physics has an interdisciplinary nature. It studies systems that usually come under chemistry or biology, using the methods of physics and mathematics and hence, transcends the frontiers between the subjects. Soft matter may be classified into three main classes; colloidal dispersions, polymers and polymer melts and liquid crystals. Study of single polymer chains is a fascinating topic that provides insights to understand many processes occurring in biological systems. Here, we present analytical studies of a few such processes, involving single polymer chains. In fact, there are a number of biological processes, which involve the dynamics of a single polymer chain. Due to the importance of Brownian motion at the mesoscopic level, soft matter systems are always studied using the analytical as well as computational methods of statistical mechanics. The statistical mechanics of polymers has been developed into a fascinating topic due to the contributions from the theory of random walks and path integrals. The dynamical behavior of many-particle systems has been described traditionally by the so-called rate theories. Here, we use these classical approaches to study a few biological processes that involve single polymer chains. The kind of processes that we have investigated may be categorized into two, namely the processes that leads to conformational changes in a chain molecule and the processes involving spatial translocation of a polymer. In the first category, we have considered the dynamics of semiflexible polymer loops. Loop formation of chain molecules has a key role in biological processes like DNA replication, gene regulation and protein folding. Hence, the dynamics of a polymer closing to form a loop as well as opening of the loop are topics of considerable theoretical/experimental interest. For closing, results are available in the completely flexible limit. Wilemski and Fixman, (J. Chem. Phys.60, 878 (1974)) have studied the closing and opening reactions in a single flexible polymer chain and using their approach Doi found the closing time to vary with the length of the chain as L2 . Szabo, Schul-ten, and Schulten, (J. Chem. Phys. 72, 4350 (1980)) have used mean first passage time approach and they find that the closure time vary as L3/2. Both approaches have been compared with simulations (Pastor et.al, J. Chem. Phys., 105, 3878 (1996), Srinivas et. al,116,7276 (2002)). In the case of semiflexible chains, studies are fewer in comparison. However, real polymers such as DNA, RNA or proteins are not flexible and therefore, it is important to incorporate the intrinsic stiffness of the chain into account. In the worm-like chain model, the chain is described as a continuous, inextensible and differentiable space curve represented by the position vector r(s), where s is the arc length. Inextensibility of the chain requires that the tangent vector, u(s) = ∂r(s)/∂s, at any point on the curve must have unit magnitude, i.e, |u(s)| = 1. But incorporating this constraint has been a difficult problem in dealing with semiflexible polymers. Yamakawa and Stockmayer (J. Chem. Phys., 57, 2843 (1972)) and Shimada and Yamakawa (Macromolecules, 17, 689 (1984)) have calculated ring closure probabilities for worm-like chains and helical worm-like chains. Cherayil and Dua (J. Chem. Phys., 116, 399 (2002)) have calculated closure time for a semiflexible chain using the approximate model for semiflexible chains by Har-nau, Winkler and Reineker (J. Chem. Phys., 101, 8119 (1994)) and find that the closure time ~ Lν where ν is in the range 2.2 to 2.4. Physically, one expects that the closing time should decrease exponentially with length in the very short chain limit and then increase with length for longer chains. Hence, the closing time has a minimum at an intermediate length. The reason for this behavior is that, for short chains, the bending energy contributes significantly to the activation energy for the process. The activation energy ~ const./L and hence, the closing time τ ~ exp(const./L). For longer chains, the free energy barrier for closing is due to the configurational entropy and hence, τ obeys a power law. Recently, Jun et. al (Europhys. Lett., 64, 420 (2003)) have followed an approximate one dimensional Kramers approach to reproduce this behavior and obtained the minimum at a length Lmin = 3.4lp, where lp is the persistence length of the chain. Monte Carlo simulations by Chen et.al (Europhys. Lett., 65, 407 (2004)) lead Lmin = 2.85lp. We investigate (K. P. Santo and K. L. Sebastian, Phys. Rev. E, 73, 031923, (2006)) in detail the problem of loop opening for semiflexible polymers. The inextensibility constraint is incorporated rigorously by setting u(s) to be a unit vector in the angular direction (θ, φ) and the conformations of the polymer are then represented by Brownian motion over a unit sphere in the tangent vector space. We use the worm-like chain model, which takes into account the bending rigidity of the polymer. The bending energy can then be given in terms of the angle coordinates θ and φ. For the dynamics, we make use of a semiclassical approach, which is based on expanding the bending energy about a minimum energy path. For the sake of simplicity, we take the great circle on the unit sphere to be the minimum energy configuration of the loop and expand the bending energy up to second order in terms of fluctuations about this configuration. We find that, this is a very good approximation in the large stiffness limit, as this approach leads to a minimum energy value, which is very close to the exact calculations. The loop is unstable, unless the ends are bound to each other with a potential. Once the two ends have been brought together, they can separate from each other in any of the three directions in space. Considering the ring to be in the XY plane with the ends meeting in the Y-axis, we find that the separation in the X and Z directions are unstable as motion in these directions lead to decrease in bending energy. But the motion in the other direction, that is, the Y direction leads to increase in energy and is stable. Therefore, we choose the potential to be of Morse type in the X-direction and stable harmonic ones in the other two directions. With this, the potential energy surface for opening can be found and the rate of opening can be calculated using classical Transition State Theory. The effects of friction on the rate can also be incorporated using the standard coupling to a bath of harmonic oscillators . We find that for short chains, the rate is strongly length dependent and is well-described by the equation Aexp(B/x)/xν, with A and B constants, x = L/lp, L the length of the chain, lp the persistence length and ν ~ 1.2. However, for long chains, the rate is found to obey a power law. But in view of the fact that our approximations, while sensible for short semiflexible chains, are not expected to be valid for long flexible chains and therefore, this result is not expected to be correct. We also present results for the seemingly more biologically important reverse process, the closing of a semiflexible polymer, thus presenting a rather complete theory of dynamics of semiflexible polymer loops. In this work, we give a detailed multidimensional analysis of the closing dynamics of semiflexible chains by making use of the approximation scheme developed in the previous study of loop opening. We use the formalism of Wilemski and Fixman for the diffusion-controlled intra-chain reactions of polymers and their "closure" approximation for an arbitrary sink function. In this procedure, the closing time is expressed in terms of a sink-sink correlation function. We calculate this sink-sink correlation function through a normal mode analysis on the chain. The closing times, τclose for different lengths of the chain are then obtained. We find that τclose(L) ~ L4.5W(L), where W(L) was found to be described by B'exp(A'/L) with A' and B' constants. τclose(L) is found to have a minimum at Lmin = 2.4lp, which is to be compared with the values obtained through a one dimensional analysis (Europhys. Lett., 64, 420 (2003)) and simulations (Europhys. Lett., 65, 407 (2004)). We thus present a multidimensional analysis that give results that are physically expected. There does not seem to be any previous analysis which leads to these results shown through one-dimensional studies and simulations. In the category of translocation problems, we consider DNA packaging in viruses. DNA Packaging into the viral capsid is an essential step in any kind of viral infection. The mechanism of packaging in bacteriophage φ — 29 has recently been studied (Simpson et. al, Nature (London), 408, 745 (2000)). The study revealed the structure of the molecular motor that packages the DNA. In another experimental study, Smith et. Al (Nature (London), 408, 745 (2001)) have investigated the effect of applied external force on the packaging. Motivated by this study, we suggest (K. P. Santo and K. L. Sebastian, Phys. Rev. E, 65, 052902 (2002)) a simple model to explain the kinetics of packaging of DNA the external force, which tries to prevent it. The model suggests a Butler-Volmer kind of dependence of the rate of packaging on the pulling force. We find that our model explains the experimental data very well. Another very interesting situation that arises in biological contexts is the translocation of a polymer across a membrane through a pore. The uptake of DNA into the cell nucleus and the translocation of cytosolic protein into endoplasmic reticulum are examples. There have been two main classes of polymer translocation problems; translocation in presence of a field or driven by a molecular motor and the translocation assisted by the adsorption of molecules onto the chain in the region into which it is translocated. While the first class of problems is reasonably well understood, for the second class of problems a complete understanding still does not exist in the literature. The existing understanding of this kind of polymer translocation is mainly due to Simon, Peskin and Oster (Proc. Natl. Accad. Sci. USA, 89, 3770 (1992)), who describe the translocation as kind of biased Brownian motion, which is known as the Brownian Ratchet. But Brownian Ratchet is an idealization and can only be realized in certain limits and therefore, it does not account for the detailed dynamics of polymer and the binding particles. We present a simple statistical description of the problem. We find that in the regime where number of binding particles are larger than the number of adsorption sites on the chain, the translocation proceeds as if it is driven by a constant force and hence, seems to be governed by a mechanism similar to the kink mechanism (K. L. Sebastian and Alok. K. R. Paul, Phys. Rev. E, 62, 927 (2000), K. L. Sebastian, 61, 3245 (2000)) that has been suggested in the case translocation in presence of an external field. In the other regime, where the number of binding particles are less than the number of binding sites on the chain, the translocation was found to be predominantly diffusive.

This thesis presents our recent study of few interesting problems involving activated processes. This chapter gives an overview of the thesis. It is now possible to do single molecule experiments involving enzyme molecules. The kinetics of such reactions exhibits dynamic disorder associated with conformational changes of the enzyme-substrate complex. The static disorder and dynamic disorder of reaction rates, which are essentially indistinguishable in ensemble-averaged experiments, can be determined separately by the real-time single-molecule approach. In our present work we have given a theoretical description of how rate of reactions involving dynamic disorder is studied using path integral approach. It is possible to write the survival probability and the rate of the process as path integrals and then use variational approaches to get bounds for both. Though the method is of general validity, we illustrate it in the case of electronic relaxation in stochastic environment modeled by a particle experiencing diffusive motion in harmonic potential in presence of delta function sink. The exact solution of corresponding Smoluchowski equation was found earlier[1] analytically in Laplace domain with sink having arbitrary strength and position. Exact evaluation of path integral calculation to survival probability is not possible analytically. Wolynes et al.[2] have done an approximate calculation to get bounds to the survival probability in the Laplace domain. A bound in the Laplace domain is not as useful as a bound in the time domain and hence we use the direct approximate variational path integral technique to calculate both lower and upper bound of survival probability in time domain. We mimic the delta function sink by quadratic sink for which the path integral can be solved exactly. The strength of the quadratic sink is treated as variational parameter and using the optimized value for it, one can estimate the optimized lower as well as upper bound of survival probability. We have also calculated a lower bound to the rate. The variational results are compared with the exact ones, and it is found that the results for the two parameter case are better than those of one parameter case. To understand how good our approximation is, we calculate the bounds in survival time and found them to be in good agreement with exact results. Our approach is valid for any arbitrary initial distribution that one may start with. We consider the Kramers problem for a long chain polymer trapped in a biased double well potential. Initially the polymer is in the less stable well and it can escape from this well to the other well by the motion of its N beads across the barrier to attain the configuration having lower free energy. In one dimension we simulate the crossing and show that the results are in agreement with the kink mechanism suggested earlier. In three dimensions, it has not been possible to get analytical “kink solution”for an arbitrary po-tential; however, one can assume the form of the solution of the non-linear equation as a kink solution and then find a double well potential in three dimensions. To verify the kink mechanism, simulations of the dynamics of a discrete Rouse polymer model in a double well in three dimensions were done. We find that the time of crossing is proportional to the chain length which is in agreement with the results of kink mechanism. The shape of the kink solution is also in agreement with the analytical solution in both one and three dimensions. We then consider the dynamics of a short chain polymer crossing over a free energy barrier in space. Adopting the continuum version of the Rouse model, we find exact expressions for the activation energy and the rate of crossing. For this model, the analysis of barrier crossing is analogous to semiclassical treatment of quantum tunneling. Finding the saddle point for the process requires solving a Newton-like equation of motion for a fictitious particle. The analysis shows that short chains would cross the barrier as a globule. The activation free energy for this would increase linearly with the number of units N in the polymer. The saddle point for longer chains is an extended conformation, in which the chain is stretched out. The stretching out lowers the energy and hence the activation free energy is no longer linear in N . The rates in both the cases are calculated using a multidimensional approach and analytical expressions are derived using a new formula for evaluating the infinite products. However, due to the harmonic approximation made in the derivation, the rates are found to diverge at the point where the saddle point changes over from the globule to the stretched out conformation. The reason for this is identified to be the bifurcation of the saddle to give two new saddles. A correction formula is derived for the rate in the vicinity of this point. Numerical results using the formulae are presented. It is possible for the rate to have a minimum as a function of N . This is due to the confinement effects in the initial state. We analyze the dynamics of a star polymer of F arms confined to a double well potential. Initially the molecule is confined to one of the minima and can cross over the barrier to the other side. We use the continuum version of Rouse-Ham model. The rate of crossing is calculated using the multidimensional approach due to Langer[3].Finding the transition state for the process is shown to be equivalent to the solution of Newton’s equations for F independent particles, moving in an inverted potential. For each star polymer, there is a critical total length N Tc below which the polymer crosses over as a globule. The value of NTc depends on the curvature at the top of the barrier as well as the individual arm lengths. So we keep the lengths of (F -1) arms fixed and increase the length of the F th arm to get the minimum total length NTc. Below NTc the activation energy is proportional to the total arm length of the star. Above N Tc the star crosses the barrier in a stretched state. Thus, there is a multifurcation of the transition state at NTc. Above NTc, the activation energy at first increases and then decreases with increasing arm length. This particular variation of activation energy results from the fact that in the stretched state, only one arm of the polymer is stretched across the top of the barrier, while others need not to be. We calculate the rate by expanding the energy around the saddle upto second order in the fluctuations. As we use the continuum model, there are infinite modes for the polymer and consequently, the prefactor has infinite products. We show that these infinite products can be reduced to a simple expression, and evaluated easily. However, the rate diverges near N Tc due to the multifurcation, which results in more than one unstable mode. The cure for this divergence is to keep terms upto fourth order in the expansion of energy for these modes. Performing this, we have calculated the rate as a function of the length of the star. It is found that the rate has a nonmonotonic dependence on the length, suggesting that longer stars may actually cross over faster.

Recent research involving magnetic polymers have given very promising results. It is observed in these polymers, the ability to remain magnetized even in the absence of magnetic field which is of great importance to developing new technologies. In order to deepen the knowledge these polymers, in special, its magnetic properties, many researchers have developed theories and methods. In this paper, we use the model developed by Hubbard in 1963. It is a relatively simple model, with important approaches that do not compromise on fundamental analysis, such polymers. With the Hubbard model as a basis, we take a model for the polymer chain in the case of a chain of sub-lattices which one contains the term and the other metal would be the term ligand of the chain. The chain is described as AXBX+1, where A and B are sub-lattices and X gives the size of the chain. We developed this model observing their magnetic properties according to the interaction applied and the number of particles per site building the magnetic phase diagram for different networks AXBX+1. / Recentemente as pesquisas envolvendo polímeros magnéticos têm dado resultados muito promissores. Observa-se nestes polímeros, a capacidade de manteremse magnetizados mesmo na ausência de campo magnético o que é de grande importância para desenvolvimento de novas tecnologias. A fim de aprofundar o conhecimento sobre estes polímeros, em especial, suas propriedades magnéticas, muitos pesquisadores desenvolveram teorias e métodos. Neste trabalho, utilizaremos o modelo desenvolvido por Hubbard em 1963. Trata-se de um modelo relativamente simples, com aproximações importantes que não comprometem análises fundamentais sobre estes polímeros. Com o modelo de Hubbard como base, tomamos um modelo para a cadeia polimérica tratando-se de uma cadeia com sub-redes as quais uma delas contém o termo metálico e a outra teria o termo ligante da cadeia. A cadeia é descrita como AXBX+1, onde A e B são as sub-redes e X dá a dimensão da cadeia. Desenvolvemos este modelo observando suas propriedades magnéticas de acordo com a interação aplicada e o número de partículas por sítio da rede construindo o diagrama de fases magnéticas para diversas redes AXBX+1.

This thesis is mainly concerned with some important properties of complex fluids, and how these properties are influenced by structures in the nano/mesoscopic scale. Short-range assembly of the constituent molecules results in an amazing variety of phase behavior in these systems. Liquid-liquid phase transitions, or transitions from a homogeneous(mixed) phase to an immiscible phase (two-phase coexistence), are the outcome of a competition between entropy and short-ranged attractive forces, and form an important part of this thesis. A rich phase behavior is uncovered by a detailed study of liquid-liquid phase transitions in a mixture of ethanol(E) and water(W), induced by the addition of ammonium sulfate(AS) ions (E and W are otherwise completely soluble in each other). This is the main motivation for choosing this system. Furthermore, experimental evidence of the presence of supramolecular association in alcohol-water mixtures [J.-H. Guo et al., Phys. Rev Lett, 91, 15401(2003)] enhances our interest to study the phase behavior in more detail. The presence of a critical point, at which there is a second order phase transition, is quite common in complex fluids. An issue which has been the subject of extensive scientific research in recent years is the influence of nano/mesoscopic structure on the critical behavior of these fluids corresponds to the Ising universality class. However, the approach to the asymptotic regime is governed by a competition between the correlation length of critical concentration fluctuations and the additional length scale arising due to structuring., which results in a crossover from the universal Ising behavior to the mean-field behavior, sometimes within the critical domain. This phenomenon of crossover criticality is presently explored in the E + W + AS system. A significant portion of the thesis presents explorations on the critical behavior in the vicinity of special critical points (SCP), which are formed by the coalescence of two or more critical points. Recentrant liquid-liquid phase transitions observed in the E + W + AS system, furnishes an unique opportunity for the realization of three SCPs – the double critical point(DCP) and the critical double point(CDP) formed by the merger of two critical points , and a critical inflection point(CIP), formed by the merger of three critical points. A CIP had not been experimentally realized prior to the studies presented in this thesis. Apart from the above studies investigations are also carried out on the conformational changes of a technologically important conducting polymer, polyethylene dioxythiophene doped with polystyrene suflonate (PEDOT-PSS), in various solvents. The electrical and optical properties of the polymer films get enhanced when solution processed with specific solvents. The experiments presented in this thesis are directed at unraveling the role of conformational modifications in the electrical and optical properties of these systems. The experimental techniques that were employed in the present studies are: Laser light scattering, small-angle X-ray scattering(SAXS) measurements and visual observations. The eoexistence surface of the system E + W + AS was determined by visual observations. Laser light scattering measurements were conducted to study the critical behavior of osmotic susceptibility (xr) of E + W + As, whereas SAXS studies were conducted to ascertain the existence, and quantify the spatial extent of the additional length scale in the two systems investigated. The main objectives of this research were: (i) to study the phase behavior of the ternary mixture E + W + AS at atmospheric pressure; (ii) to check the existence of crossover from 3-D Ising to mean-field critical behavior while moving away from Tc in this system; (iii) to determine the nature (monotonic or nonmonotonic) of crossover; (iv) to provide some insight into the origin of this crossover behavior in terms of an additional length scale characteristic of the system; (v) to understand the evolution of the critical behavior in the proximity of CDP, and DCP; (vi) to experimentally realize the CIP; and (vii) to investigate the presence of solvent-induced conformational changes in conducting polymer.

Mit dieser Arbeit ist weltweit zum ersten Mal die molekulare Dynamik von vereinzelten,d.h. einander nicht berührenden Polymerketten experimentell bestimmt worden. Die Grundlagen dafür sind einerseits die breitbandige dielektrische Spektroskopie mit ihrer außerordentlich hohen experimentellen Empfindlichkeit und andererseits die Weiterentwicklung einer speziellen Probenanordnung, bei der hochleitfähige Silizium-Elektroden durch elektrisch isolierende Siliziumdioxid-Nanostrukturen in einem vordefinierten Abstand gehalten werden und so den Probenkondensator bilden. Im Rahmen dieser Arbeit wurde die Höhe der Nanostrukturen (und damit des Elektrodenabstands) auf nur 35 nm reduziert. Damit gelang der Nachweis, dass selbst vereinzelte kondensierte Polymer-Knäuel - im Rahmen der Messgenauigkeit - dieselbe Segmentdynamik (bzw. denselben dynamischen Glasübergang), gemessen in ihrer mittleren Relaxationsrate, wie die makroskopische Schmelze („bulk“) aufweisen. Nur ein kleiner Anteil der Segmente zeigt eine langsamere Dynamik, was auf attraktive Wechselwirkungen mit dem Substrat zurückzuführen ist, wie komplementäre Untersuchungen mittels Infrarot-Spektroskopie zeigen. Zudem bieten diese Experimente die Möglichkeit, nach der dielektrischen Messung die mit Nanostrukturen versehene obere Elektrode zu entfernen und die Verteilung der vereinzelten Polymerketten, deren Oberflächenprofile und Volumen mit dem Rasterkraftmikroskop zu bestimmen. Erst damit gelingt der Nachweis, dass die Polymer-Knäuel im Mittel aus einer einzelnen Kette bestehen. Die Kombination dieser drei unabhängigen Messmethoden liefert ein schlüssiges und detailliertes Bild, gekennzeichnet dadurch, dass attraktive Oberflächenwechselwirkungen die Glasdynamik nur über ca. 0,5nm direkt beeinflussen. In einem zweiten Teil trägt die Arbeit mit der Untersuchung dünner Polymerschichten im Nanometer-Bereich zu einer international geführten, kontroversen Diskussion um die Frage, ob sich im Falle solcher räumlichen Begrenzungen der dynamische und kalorimetrische Glasübergang ändern, bei. Dabei zeigt mit den präsentierten dielektrischen und ellipsometrischen Messungen eine Kombination aus einer Methode, die im Gleichgewichtszustand misst und einer, die den Übergang in den Nichtgleichgewichtszustand bestimmt, dass sich sowohl Polystyrol-Schichten verschiedener Molekulargewichte bis zu einer Dicke von nur 5 nm als auch Polymethylmethacrylat-Schichten auf unterschiedlichen (hydrophilen und hydrophoben) Substraten bis zu einer Dicke von 10 nm weder in ihrem dynamischen noch ihrem kalorimetrischen Glasübergang von der makroskopischen Schmelze unterscheiden.