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Diffusion processes play an important role in describing systems in many fields of science, as in physics, biology, finance and social science. One of the most famous examples of the diffusion process is the Brownian motion.    At mesoscopic scale, the Brownian theory describes the very irregular and animated motion of a particle suspended in a fluid. In this thesis, the dynamics and thermodynamics of diffusion processes driven out of equilibrium, at mesoscopic scale, are investigated.    For dynamics, the theory of Brownian motion for a particle which is able to rotate and translate in three dimensions is presented.  Moreover, it is presented how to treat diffusion process on n-dimensional Riemann manifolds defining the Kolmogorov forward equation on such manifold.   For thermodynamics, this thesis describes how to define thermodynamics quantities at mesoscopic scale using the tools of Brownian theory. The theory of stochastic energetics and how to compute entropy production along a trajectory are presented introducing the new field of stochastic thermodynamics. Moreover, the "anomalous entropy production" is introduced. This anomaly in the entropy production arises when diffusion processes are driven out of equilibrium by space dependent temperature field. The presence of this term expresses the fallacy of the overdamped approximation in computing thermodynamic quantities.    In the first part of the thesis the translational and rotational motion of an ellipsoidal particle in a heterogeneous thermal environment, with a space-dependent temperature field, is analyzed from the point of view of stochastic thermodynamics.    In the final part of the thesis, the motion of a Brownian rigid body three-dimensional space in a homogeneous thermal environment under the presence of an external force field is analyzed, using multiscale method and homogenization. / <p>QC 20170515</p>

A new numerical analysis method, dispersion vs. absorption plots (DISPA), has been developed for ESR. This method may be used for semi-quantitative line-shape studies and is useful both as a diagnostic and analytical tool. In addition it provides a method of automatic phasing for magnetic resonance spectra. Numerous examples of its applications, both simulated and experimental are presented, with emphasis on spin-probe studies. The digital acquisition and processing methods used for these studies are also briefly discussed. ESR and NMR relaxation time studies of the bis(dialkyl-N-carbodithioate) metal(II) class of spin-probes have been performed. The T₁,'s of ¹³C and ²H enriched nickel complex were measured by NMR. The line-widths of ⁶³Cu complex were measured by ESR and analysed by Redfield theory. The two sets of results were combined to give the principal elements of the rotational diffusion tensor for the pyrollidine derivative in toluene. This is the first time that ESR and NMR studies have been combined to measure a diffusion tensor. A general strategy for this approach is presented. ESR data from previous work has been re-analysed in the light of the new results. The analysis shows that the commonly used assumption of isotropic diffusion is extremely misleading. / Science, Faculty of / Chemistry, Department of / Graduate

Copper is an integral component of key metabolic enzymes. Numerous physiological processes depend on a fine balance between the biosynthetic incorporation of copper into proteins and the export of excess copper from the cell. The homeostatic control of copper requires the activity of the copper transporting ATPases (Cu-ATPases). In Wilson disease the disruption in the function of the Cu-ATPase ATP7B results in the accumulation of excess copper and a marked deficiency of copper-dependent enzymes. In this work, the structure of ATP7B has been modeled by homology using the Ca-ATPase X-ray structure, enabling a mechanism of copper transport by ATP7B to be proposed. The fourth transmembrane helix (TM4) of Ca-ATPase contains conserved residues critical to cation binding and is predicted to correspond to TM6 of the ATP7B homology model, containing the highly conserved CXXCPC motif. The interaction with Cu(I) and the importance of the 3 cysteines in TM6 of ATP7B has been shown using model peptides. ATP7B has a large cytoplasmic N-terminus comprised of six copper-binding domains (WCBD1-6), each capable of binding one Cu(I). Protein-protein interactions between WCBDs and the copper chaperone Atox1 has been shown, contrary to previous reports, to occur even in the absence of copper. 15N relaxation measurements on the apo and Cu(I)-bound WCBD4-6 show that there is minimal change in the dynamic properties and the relative orientation of the domains in the two states. The domain 4-5 linker remains flexible, and domain 5-6 is not a rigid dimer, with flexibility between the domains. Copper transfer to and between WCBD1-6 likely occurs via protein interactions facilitated by the flexibility of the domains with respect to each other. The flexible linkers connecting the domains are important in giving the domains motional freedom to interact with Atox1, to transfer copper to other domains, and finally to transfer copper to the transmembrane site for transport across the membrane.

Copper is an integral component of key metabolic enzymes. Numerous physiological processes depend on a fine balance between the biosynthetic incorporation of copper into proteins and the export of excess copper from the cell. The homeostatic control of copper requires the activity of the copper transporting ATPases (Cu-ATPases). In Wilson disease the disruption in the function of the Cu-ATPase ATP7B results in the accumulation of excess copper and a marked deficiency of copper-dependent enzymes. In this work, the structure of ATP7B has been modeled by homology using the Ca-ATPase X-ray structure, enabling a mechanism of copper transport by ATP7B to be proposed. The fourth transmembrane helix (TM4) of Ca-ATPase contains conserved residues critical to cation binding and is predicted to correspond to TM6 of the ATP7B homology model, containing the highly conserved CXXCPC motif. The interaction with Cu(I) and the importance of the 3 cysteines in TM6 of ATP7B has been shown using model peptides. ATP7B has a large cytoplasmic N-terminus comprised of six copper-binding domains (WCBD1-6), each capable of binding one Cu(I). Protein-protein interactions between WCBDs and the copper chaperone Atox1 has been shown, contrary to previous reports, to occur even in the absence of copper. 15N relaxation measurements on the apo and Cu(I)-bound WCBD4-6 show that there is minimal change in the dynamic properties and the relative orientation of the domains in the two states. The domain 4-5 linker remains flexible, and domain 5-6 is not a rigid dimer, with flexibility between the domains. Copper transfer to and between WCBD1-6 likely occurs via protein interactions facilitated by the flexibility of the domains with respect to each other. The flexible linkers connecting the domains are important in giving the domains motional freedom to interact with Atox1, to transfer copper to other domains, and finally to transfer copper to the transmembrane site for transport across the membrane.

<p>We have studied fundamental aspects of time-resolved two-photon excited fluorescence depolarisation. The thesis presents experimental as well as theoretical progress. We show that a multi-photon induced instrumental response function obtained from a suspension of gold nanoparticles is appropriate for the analysis of two-photon excited fluorescence decays obtained using time-correlated single photon counting detection. Theoretical expressions have been derived for the fluorescence anisotropy decay obtained upon two-photon excitation of various molecular systems in liquid solutions: a) an anisotropic rigid rotor that undergoes rotational diffusion in the presence of ultrafast unresolved restricted reorientations, e.g. librations. b) a molecular group covalently attached to a stationary macromolecule, and undergoing local reorientation in a uniaxial ordering potential. A new approach to the analysis of two-photon excited fluorescence depolarisation experiments was developed, which combines data obtained by using linearly and circularly polarised excitation light, in a global manner. In the analysis, knowledge about unresolved reorientations was obtained from one-photon excitation studies of the corresponding systems. By means of this procedure it has been possible to obtain quantitative information about the molecular two-photon absorption tensor for perylene and two of its derivatives. Thereby the symmetry of the final excited and intermediate vibronic states could be assigned. The analysis reveals that the two-photon transition studied with the 800 nm laser exhibits mixed character. An important finding from the experiments was that the two-photon absorption tensor appears to be solvent dependent. Furthermore, the thesis presents the first theoretical treatment of two-photon excited donor-donor energy migration in the presence of molecular reorientation and which applies the extended Förster theory. Explicit expressions for molecules that belong to the point groups D_2h, D₂ and C_2v are given. Preliminary experiments are finally also reported on a two-photon excited donor-donor energy migration system consisting of a bisanthryl-bisteroid. </p>

We have studied fundamental aspects of time-resolved two-photon excited fluorescence depolarisation. The thesis presents experimental as well as theoretical progress. We show that a multi-photon induced instrumental response function obtained from a suspension of gold nanoparticles is appropriate for the analysis of two-photon excited fluorescence decays obtained using time-correlated single photon counting detection. Theoretical expressions have been derived for the fluorescence anisotropy decay obtained upon two-photon excitation of various molecular systems in liquid solutions: a) an anisotropic rigid rotor that undergoes rotational diffusion in the presence of ultrafast unresolved restricted reorientations, e.g. librations. b) a molecular group covalently attached to a stationary macromolecule, and undergoing local reorientation in a uniaxial ordering potential. A new approach to the analysis of two-photon excited fluorescence depolarisation experiments was developed, which combines data obtained by using linearly and circularly polarised excitation light, in a global manner. In the analysis, knowledge about unresolved reorientations was obtained from one-photon excitation studies of the corresponding systems. By means of this procedure it has been possible to obtain quantitative information about the molecular two-photon absorption tensor for perylene and two of its derivatives. Thereby the symmetry of the final excited and intermediate vibronic states could be assigned. The analysis reveals that the two-photon transition studied with the 800 nm laser exhibits mixed character. An important finding from the experiments was that the two-photon absorption tensor appears to be solvent dependent. Furthermore, the thesis presents the first theoretical treatment of two-photon excited donor-donor energy migration in the presence of molecular reorientation and which applies the extended Förster theory. Explicit expressions for molecules that belong to the point groups D2h, D2 and C2v are given. Preliminary experiments are finally also reported on a two-photon excited donor-donor energy migration system consisting of a bisanthryl-bisteroid.

Diffusion plays an important role in a number of processes like heterogeneous catalysis, corrosion, separation and purification of chemicals of industrial importance, steel hardening, fuel cells, and solid electrolytes for batteries. It also plays a vital role in several biological processes like transport across biomembranes, nerve impulse, flow of blood and permeation of ingested drug. The elementary process of diffusion in solids is quite different from those in liquids. Similarly, the mode of diffusion in porous solid where different regimes such Knudsen regime exists bears little similarity to those in a dense close-packed crystalline solid. Chapter 1 provides a brief introduction to basics of diffusion in different phases of condensed matter. Among the various phases discussed are liquids, close-packed crystalline solids (e.g., body-centered cubic solids), amorphous solids (e.g. glasses) and microporous crystalline solids (e.g., zeolites). Diffusion in these widely differing phases often bears no resemblance to each other; the rate of diffusion in these phases varies over many orders of magnitude and the elementary step and mechanism in the diffusion process are very different. Brief introduction to theories for diffusion in these phases is provided. Various experimental techniques to measure diffusivities are discussed. Different microscopic models to explain the Quasi Elastic Neutron Scattering (QENS) spectra of these phases yield an insight into the elementary step of the diffusion process. Notwithstanding the fact that completely different models are invoked to explain diffusion in different phases, there are certain underlying generic behaviour across these widely differing phases as the recent work on size dependence of diffusion in these phases demonstrate. Diffusion of a molecule or species (in the context of diffusion within condensed phases) without loss of generality may be said to occur in a medium. A universal behaviour observed is that self diffusivity exhibits a maximum as a function of the size of the diffusant when the diffusant is confined to a medium, as a result of what is known as the Levitation Effect. Such a maximum in self diffusivity has been seen in widely differing medium: microporous solids, dense liquids, ions in polar solvents, etc. The aim of the thesis is to investigate and further explore such universal behaviour and demonstrate for the first time the existence of common trends across different condensed phases in spite of difference in the detail at the microscopic level. In Chapter 2, we report a molecular dynamics study of diffusion of diatomic species AB within zeolite Y. The bond length of A-B as well as the interaction of A and B with the host zeolite atoms are varied. The results demonstrate that for the symmetric case (when A=B or AA), there exists a preferred bond length (determined by the bottleneck or window diameter) when the diffusivity is maximum. This is in agreement with previous results on monatomic species which also exhibit a similar diffusivity maximum. More importantly, no such maximum is seen when the interaction asymmetric is introduced in AB. Slight asymmetry in the interaction gives rise to a weak maximum while large asymmetry in interaction obliterates the diffusivity maximum. These results suggest that the importance of interaction between the diffusant and the medium in Levitation Effect or size-dependent diffusivity maximum. Further, it also demonstrates for the first time the close association between an inversion centre (in a statistical sense and not in the crystallographic sense) and the Levitation Effect. In Chapter 3, a study of size dependence of solutes in a Lennard-Jones liquid is reported. Einstein and others derived the reciprocal dependence of the self-diffusivity D on the solute radius ru for large solutes based on kinetic theory. We examine here (a) the range of ru over which Stokes-Einstein (SE) dependence is valid and (b) the precise dependence for small solutes outside of the SE regime. We show through molecular dynamics simulations that there are two distinct regimes for smaller solutes: (i) the interaction or Levitation Effect (LE) regime for solutes of intermediate sizes and (ii) the D 1/ru2 for still smaller solutes. We show that as the solute-solvent size ratio decreases, the breakdown in the Stokes-Einstein relationship leading to the LE regime has its origin in dispersion interaction between the solute and the solvent. These results explain reports of enhanced solute diffusion in solvents existing in the literature seen for small solutes for which no explanation exists. Several properties have been computed to understand the nature of solute motion in different regimes. We investigate in Chapter 4, the dependence of self diffusivity on the size of the diffusant in a disordered medium with the objective of understanding the experimentally observed correlation between self diffusivity and activation energy seen in a wide variety of glasses. Typically, it is found in many ionic glasses that a higher conductivity is associated with lower activation energy and vice versa. Our understanding of transport in glasses as provided by existing theories does not offer an explanation of this correlation. We have carried out molecular dynamics simulation as a function of the size of the impurity atom or diffusant (both neutral and charged) in a model host amorphous matrix. We find that there is a maximum in self diffusivity as a function of the size of the impurity atom suggesting that there is an appropriate size for which the diffusivity is maximum. The activation energy is found to be the lowest for this size of the impurity. A similar maximum has previously been found in other condensed phases such as confined fluids and dense liquids and has its origin in the Levitation Effect. The implications of this result for understanding ionic conductivity in glasses are discussed. Our result suggests that there is a relation between microscopic structure of the amorphous solid, diffusivity or conductivity and activation energy. The nature of this relationship is discussed in terms of the Levitation parameter showing that diffusivity is maximum when the size of the neck or doorway radius is comparable with the size of the diffusant. Our computational results here are in excellent agreement with independent experimental results which show that structural features of the glass are important in determining the ionic conductivity. In Chapter 5, we report results of molecular dynamics investigations into neutral impurity diffusing within an amorphous solid as a function of the size of the diffusant and density of the host amorphous matrix. We find that self diffusivity exhibits an anomalous maximum as a function of the size of the impurity species. An analysis of properties of the impurity atom with maximum diffusivity shows that it is associated with lower mean square force, reduced backscattering of velocity autocorrelation function, near-exponential decay of the intermediate scattering function (as compared to stretched-exponential decay for other sizes of the impurity species) and lower activation energy. These results demonstrate the existence of well known size-dependent diffusivity maximum in disordered solids. Further, we show that the diffusivity maximum is observed at lower impurity diameters with increase in density. This is explained in terms of the levitation parameter and the void structure of the amorphous solid. We demonstrate that these results imply contrasting dependence of self diffusivity (D) on the density of the amorphous matrix, . D increases with  for small sizes of the impurity but shows an increase followed by a decrease for intermediate sizes of the impurity atom. For large sizes of the impurity atom, D decreases with increase in . These contrasting dependence arises naturally from the existence of Levitation Effect. In Chapter 6, we discuss size dependence of impurity diffusion in an ordered system. We report molecular dynamics simulation studies to understand the role of impurity size and impurity-host interaction on impurity diffusivity in a body centered cubic solid. The simulation studies have been performed for a set of impurity-host interaction parameter ih (i=impurity, h=host atom) for a range of impurity sizes in rigid and flexible bcc solids. A double maximum is seen corresponding to two different sizes of the impurity species. Anomalous maximum is seen for a larger size of the impurity species in the case of the rigid host as compared to flexible host. The second anomalous diffusivity disappears with decrease in ih in flexible bcc solid. For one of the ih where double diffusivity maxima are observed, various properties are analysed to understand the anomalous diffusion behaviour. The impurity with anomalous diffusion has lower activation energy as compared to other impurities. Among the two anomalous impurities, the impurity with higher diffusivity has lower activation energy. The anomalous regime impurities as associated with velocity autocorrelation function with little or no backscattering, minimum average mean square force due to host atoms, lower activation energy. The self intermediate scattering function shows faster decay and a single relaxation time for anomalous regime impurity and two relaxation times for other impurity sizes. The wavenumber dependence of diffusivity of impurities shows oscillatory behaviour except for the anomalous regime impurities which show monotonic dependence on wavenumber. Chapter 7 discusses the influence of temperature induced solid-liquid phase transition on the size-dependent diffusivity. We report results for two distinct cases: (a) when the phase change is associated with corresponding changes in density and (b) when the phase change occurs at constant density. The latter is carried out so as to obtain the influence of disorder on the size-dependent diffusion or Levitation Effect. Studies with variable density are useful to understand the effect of disorder as well as change in density on size-dependent diffusivity. Two diffusivity maxima in the solid face-centred cubic phase is seen when the impurity-medium interaction is sufficiently large. One of these diffusivity maximum disappears with decrease in h. The impurities near the diffusivity maximum show velocity autocorrelation function with little backscattering, minimum in the average mean square force, lower activation energy, faster decay of self intermediate scattering function with a single relaxation time and a monotonic decay in wavevector dependence of diffusivity. Chapter 8 reports molecular dynamics simulations of a model guest tetrahedral molecule AX4 with differing bond lengths lAX have been carried out in a sphere with different surface roughness. The rotational-diffusion coefficient Dr shows a maximum for a particular value of lAX. This corresponds to the distance at which the interaction of the guest with the atoms of the host is most favourable. Although, the intensity of the maximum decreases with increase in the roughness of the confining surface, it is seen that the maximum exists even for a reasonably high degree of roughness. The observed maximum arises from the minimum in the torque on the tetrahedral molecule from its interaction with the confining medium due to mutual cancellation of forces. Activation energy for rotation is seen to be also a minimum for the bond length for which Dr is a maximum. These results suggest that there is a maximum in the rotational-diffusion coefficient when the rotating molecule is confined to a sphere of comparable size similar to the maximum in translational diffusion coefficient seen in porous solids and known as the Levitation Effect. On increase in the roughness of the sphere surface, the value of lAX at which the maximum in Dr is seen decreases. This is similar to the shift seen in the size of the diffusant corresponding to maximum diffusivity in the case of translational diffusivity. In Chapter 9 possible extensions to the work reported in the previous chapters and the directions to take are discussed. Symmetry plays an important role in size dependent diffusivity maximum in microporous crystalline solids; it would be interesting to investigate if similar role of symmetry exists in case of liquids and other disordered solids. Previous work from this laboratory on ions in water has shown the importance of electrostatic interactions. In the light of this, it would be interesting to see the influence of long-range interactions in breakdown of Stokes-Einstein relationship in liquids. Effect of density of the medium on impurity diffusion can be studied over a wide range of densities in case of supercritical fluids such as ions in water (where electrostatic interactions are present) and these can provide greater insight into dependence of diffusion on density. The origin of two diffusivity maxima in case of body-centered and face-centred cubic solids needs a detailed investigation to understand its origin. Quantification of disorder and its effect on size dependence of diffusion would be of interest. A detailed comparison with experimental data of matrix isolated molecules to understand and verify the dependence of rotational diffusivity on the size of the molecule as well as the cavity radius would be instructive.

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Knowledge of the properties of single molecule diffusion is important for controlling dynamic self-assembly of molecular structures. A powerful experimental technique for determining diffusion coefficients is the recording of diffusion-induced signal fluctuations by a locally fixed point-like probe. Here, the signal becomes modified, whenever a molecule enters a certain detection area on the surface under the probe. The technique is minimal invasive and has a very good time resolution, enabling the investigation of highly mobile molecules. Theories are necessary for the analysis of the fluctuations and the extraction of diffusion properties. In this thesis, three methods are presented, which are based on the autocorrelation function, the distribution of peak widths and the distribution of interpeak intervals. Analytical expressions are derived for the distributions and the autocorrelation function in case of molecules, which can be described by circular or rectangular shapes. For rectangular shaped molecules, rotational diffusion can influence the recorded fluctuations. To allow for a simultaneous determination of rotational and translational diffusion coefficients the analytical treatment is extended. Furthermore, new methods are developed to determine the diffusion tensor for anisotropic stochastic molecular motion, using either one linearly extended probe or two individual probes. Coarse-graining the signal recorded by a point-like probe, which repeatedly moves on a line or a circle, is suggested for experimental implementation. All facets of the evaluation methods are verified against kinetic Monte Carlo simulations. Applications to experimental data, recorded by a locally fixed scanning tunneling microscope tip, are demonstrated for copperphthalocyanine and PTCDA molecules diffusing on Ag(100).