Theoretical background

Theoretical treatments of unimolecular ionic reactions consider the ideal situation in which the ions, once formed, suffer no interaction with any other ion, molecule, electron or surface and neither absorb nor emit radiation. This being the case, the specific amounts of internal energy, E, and angular momentum with which the ions are formed must be conserved. 

Unimolecular reaction theory with specific reference to ions has been reviewed fairly recently [548, 883 see also 759, 837]. 

The theoretical data essential for understanding liuninescence phenomenon may be found in many books, but we believe that for the specific field of mineral liuninescence the fundamental books of Marfunin (1979a, 1979b) are the best. Below we tried to present very shortly only the most essential data, especially data connected with kinetic considerations, which are the basis of the time-resolved technique. 

The theoretical treatment of the dynamics of a polymer chain in solution usually starts by writing down Kirkwood s diffusion equation (Smoluchowski equation)  

Here the F) denote the positions of the monomers, so that r) specifies the conformation of the polymer chain. / ( /v, t F, 0) is the conditional probability density for a transition from r at time 0 to F at time t. The forces Fj are defined thermodynamically via the configurational equilibrium distribution function p( F ) at absolute temperature T (ks is Boltzmann s constant)  

1) describes the diffusive behavior of a chain (i.e., the movement of the center of mass) as well as its conformational rearrangements as a function of time. The equation is stochastic because the chain performs Brownian motion, and it has many different conformations which all have the same probability. The monomer-monomer interactions are described by the Fj. We will assume here that there are no long-range interactions present (in marked contrast to the case of polyelectrolytes ) and that hence the chain s structure is a random or self-avoiding walk. Motion in three-dimensional space is assumed throughout. The diffusion tensor Dy specifies the dynamics. Mathematical consistency of eq. (3.1) requires that Dy is symmetric and positive-definite for all possible golymer conformations (no other property is necessary). In the Rouse case, Dy is simply diagonal. 

For the more microscopic approach of an MD simulation of a chain in solvent particles, it is useful to also look at the theory from a more microscopic point of view, in particular in order to assess its limitations. The derivation of equations of motion of the Smoluchowski type and the discussion of the involved errors is a standard problem in modem transport theory. In the present case, the form of the hydrodynamic interaction tensor has to be derived from the microscopies, However, analytical 

A treatment that studies the case of unphysical dynamics of the solvent particles has been carried out in Ref. 31, also using a microscopic approach. This was done in order to discuss if it is justified to replace an MD algorithm in which the solvent particles obey strictly Newton s equations of motion by a Langevin simulation in which every solvent particle is artificially coupled to a weak friction and a weak heat bath (this latter method has some 

Although the detailed theoretical interpretation of primary isotope effects is a problem of great complexity and considerable uncertainty [1-3], many of the important aspects of the problem may be treated in a simplified and approximate manner with a precision which is adequate for most experimental work and which provides at least a qualitative insight into the mechanism of the observed effects. 

The transition-state theory relates the rate constant (k,) for a reaction with the equihbrium constant Kf) for the equihbrium between reactants and the transition states (Eqn. 1) [5], 

The K is a transmission coefficient which expresses the fraction of transition-state species going on to products relative to those returning to reactant and is usually considered, for no very good reason, to be insensitive to isotopic substitution. Boltzmann s constant, absolute temperature and Plancks constant are k, T and h respectively. The isotope effect is simply the ratio of the two rate constants (Eqn. 3). 

Michael I. Page (Ed), The Chemistry of Enzyme Action 1984 Elsevier Science Publishers B. V. 

The species A is not a molecule and is not a normal equilibrium constant. The transition state has one vibration less than a normal molecule (3 — 7 instead of 3n — 6) the missing vibration mode is motion along the reaction coordinate. 

The following description provides a theoretical framework for understanding 

Since electrons in filled orbitals are normally paired (i.e., with opposite spins), the ground electronic state is also referred to as a singlet state. The absorption of chemical energy can produce electronically excited singlet states whose wave function (x ) can then be represented as 

In the case of fiuorescence, the transition between states x and x is produced by light itself, i.e., by electromagnetic radiation, and the energy difference A between ground and excited singlet states is given by 

If light emission does occur, the energy difference between the lowest vibrational level of the first excited singlet or triplet state and the ground state 5g determines the frequency (v) of that emission, by Eq. (4). Further, since 

Further, since 1 mole of photons (defined by modem convention as 1 Einstein) contains N photons (where N is Avogadro s number, 6 x 10 ), then 

This section introduces some of the main theoretical concepts necessary to describe the dynamics of making and breaking bonds at surfaces. Several simple models that often capture the essence of the dynamics are also introduced. For more details, a recent book that emphasizes many aspects of theoretical dynamics is Ref. [2]. 

In the following, a theoretical background is given with a view to providing a Arm qualitative understanding of the factors which affect the hydrodynamically modulated signal. Reference should be made to the original literature cited for a more extensive treatment. 

In this section, we provide a short introduction to the topic of friction. Without such a background, it is difficult to ask meaningful questions and interpret the outcome of simulations. After all, our goals extend beyond 

The discussion begins below with an overview of proposed energy dissipation mechanisms that lead to friction. This is followed by brief discussions of phenomenological friction laws that describe the dependence of friction upon normal load and sliding velocity. The dependence of friction on the symmetry of the surfaces that are in contact is discussed later. 

As already mentioned, one of the merits of the virial equation is that it has a firm foundation in statistical thermodynamics and molecular theory. The theoretical derivation of the series has been described in numerous texts and will not be discussed in detail here. The most complete derivation for a mixture containing an arbitrary number of components is made by means of an expansion of the grand partition function. This leads to expressions for the virial coelficients in terms of cluster integrals involving two molecules for B, three molecules for C etc. These expressions are completely general and involve no restrictive assumptions about the nature of molecular interactions. Nevertheless, to simplify the expressions for the virial coefficients, a number of assumptions are often made as follows  

The first of these assumption, often implicitly made, affects the virial coefficients of all orders, as do the second and third assumptions. The fourth assumption affects the third and high-order virial coefficients. 

It should be emphasised that none of the assumptions outlined above is either necessary or even correct, but taken together they lead to simple and informative results. In particular, under these simplifications, the second virial coefficient is given by 

1 Virial Coefficients of Hard-Core-Square-Well Molecules 

5 and 3.7 are easily evaluated by numerical quadrature for any assumed intermolecular potential-energy function uy. In a few simple cases, analytical results may be obtained and we consider here the case of the hard-core-square-well potential defined by 

The fundamentals of Spectro-Spatial Interferometry (Mariotti and Ridgway 1988), also called Double Fourier Modulation (DFM) or Multi-Fourier Transform Spectroscopy (Ohta et al. 2007, 2006), are presented here and provide the background for the understanding of the following chapters. 

The goal of Double Fourier Modulation is to measure the spectral and spatial characteristics of an object simultaneously and it can be understood as the combination of two well known techniques Fourier transform spectroscopy (FTS) and Stellar Interferometry. The literature regarding both FTS and Stellar Interferometry is extensive, and the concepts presented here are the ones related to the work of this Thesis. 

In Sect. 2.2 the fundamentals of Stellar Interferometry are shown. Starting with the Young s double slit experiment the interferometric observables are explained, this is, the complex visibility function. The data synthesis relevant to the work of this thesis is then presented. Finally, in Sect. 2.3 the concept of Multi-Fourier Transform Interferometry is developed. 

The surface free energy of a solid can be described as the sum of the dispersive and specific contributions. Dispersive (apolar) interactions, also known as Lifshitz-van der Waals interactions, consist of London interactions which originate from electron density changes but may include both Keesom and Debye interactions [6, 7]. Other forces influencing the magnitude of surface energy are Lewis acid-base interactions which are generated between an electron acceptor (acid) and an electron donor (base). Details of the widely accepted theoretical 

The dispersive component of the surface energy 7 can be calculated from the retention time obtained from iGC measurements of a series of n-alkane probes injected at infinite dilution (concentration within the Henry s portion of the adsorption isotherm) [8]. Two approaches are used for the evaluation of these dependencies, the first one according to Schultz and co-workers [9] Equation 4.1 and the second one according to Dorris and Gray [10] Equation 4.2  

7 Dispersive component of surface free energy of the liquid probe 7 Dispersive component of the surface free energy of the solid Vj Retention volume and C A constant. 

The shear stress (x) generated along a defined plane depends on the normal stress (a) exerted on this plane. If a material is subjected to a shearing action, a characteristic relation is obtained between normal and shear stresses for each material. This relationship is graphically shown in o-x coordinates (Mohr diagrams) and the straight line obtained finally is the yield locus for a bulk material [11]. 

Although the static and multiple headspace modes use similar equipment, the two rely on rather different principles. On the other hand, purge and trap, and dynamic headspace, possess the same foundation, the only difference between them being the location of the tubing used to transfer the carrier gas to the sample container. 

The vapour-liquid distribution (partition) coefficient, K, is defined as the ratio of the equilibrium concentration of the solute in the sample (CJ to that in the gas phase (C )  

The mass balance for a volatile analyte between two phases can be expressed with reference to one of the states as follows [56]  

If the phase ratio, fS, is defined as then eq. (4.2) can be transformed into 

The peak area A obtained after chromatographic separation and detection is proportional to the analyte concentration in the gas phase  

The electron, discovered by J. J. Thomson in 1895, was first considered as a corpuscule, a piece of matter with a mass and a charge. Nowadays things are viewed differently. We rather speak of a wave-particle duality whereby electrons exhibit a wavelike behavior. But, in Levine s own words [45], quanmm mechanics does not say that an electron is distributed over a large region of space as a wave is distributed it is the probability patterns (wavefunctions) used to describe the electron s motion that behave like waves and satisfy a wave equation. 

Here we benefit from the notion of stationary electron density. The particles are not at rest, but the probability density does not change with time. 

and PM3 are based on the same semiempirical model [12, 13], and differ only in minor details of the implementation of the core-core repulsions. Their parameterization has focused mainly on heats of formation and geometries, with the use of ionization potentials and dipole moments as additional reference data. Given the larger number of adjustable parameters and the greater effort spent on their development, AMI and PM3 may be regarded as methods which attempt to explore the limits of the MNDO model through careful and extensive parameterization. 

and PM3 employ an sp basis without d orbitals [13, 19, 20]. Hence, they cannot be applied to most transition metal compounds, and difficulties are expected for hypervalent compounds of main-group elements where the importance of d orbitals for quantitative accuracy is well documented at the ab initio level [34], To overcome these limitations, the MNDO formalism has been extended to d orbitals. The resulting MNDO/d approach [15-18] retains all the essential features of the MNDO model. 

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31], 

Usually, nuclear relaxation data for the study of reorientational motions of molecules and molecular segments are obtained for non-viscous liquids in the extreme narrowing region where the product of the resonance frequency and the reorientational correlation time is much less than unity [1, 3, 5]. The dipolar spin-lattice relaxation rate of nucleus i is then directly proportional to the reorientational correlation time p 

Ionic liquids, however, are often quite viscous, and the measurements are thus beyond the extreme narrowing region. The relaxation rates hence become frequency-dependent. Under these conditions, the equation for the spin-lattice relaxation rate becomes more complex  

the J terms are the spectral densities with the resonance frequencies co of the and nuclei, respectively. It is now necessary to find an appropriate spectral density to describe the reorientational motions properly (cf [6, 7]). The simplest spectral density commonly used for interpretation of NMR relaxation data is the one introduced by Bloembergen, Purcell, and Pound [8]. 

Cole and Davidson s continuous distribution of correlation times [9] has found broad application in the interpretation of relaxation data of viscous liquids and glassy solids. The corresponding spectral density is  

Another way to describe deviations from the simple BPP spectral density is the so-called model-free approach of Lipari and Szabo [10]. This takes account of the reduction of the spectral density usually observed in NMR relaxation experiments. Although the model-free approach was first applied mainly to the interpretation of relaxation data of macromolecules, it is now also used for fast internal dynamics of small and middle-sized molecules. For very fast internal motions the spectral density is given by  

For states distributed in energy or for N discrete states very close to each other, equation (2.3) transforms [8]  

The po and Pi ratio in equation (2.3) determines which of two factors—namely, equilibrium or nonequilibrium (due to emission from traps) carriers—dominate in the relaxation process. That is, the depolarization current contains two maximum one is related to release of carriers from trap the origin of the other lies in the change of conductivity with temperature [14-18]. Although only one of the peaks mentioned contains information about trap parameters, it is possible to discriminate between simultaneously occurring processes, e.g., thermally stimulated depolarization and thermally stimulated dielectric relaxation. 

The expressions (2.1)-(2.6) are similar to those describing TSC processes obeying first-order kinetics and represent an asymmetrical glow curve the amplitude of which is a function of heating rate. 

Detailed solutions have not been discussed in the literature so far. The time dependence of the rate equations (2.4) and (2.5) is replaced by the temperature dependence via the heating program, which is taken to be linear T(f) = Tq + v t, where Vt = dr/dt is the heating rate. Retrapping can be neglected in a sample at high electric fields. From equations (2.5) and (2.6), we have 

Equation (2.2) is solved numerically with parameters AE, C, p, E, and given. It was found to be a good approximation for a wide range of physically reasonable trapping and other parameters, typical for a wide range of amorphous semiconductors  

With knowledge of the full OPDF in Eq. (6), the averages in Eq. (8) can be evaluated by integration with respect to the angles f and , which is straightforward due to the orthogonality properties of the spherical harmonics.46 

As can be seen, only the 2nd rank part of the OPDF, described by the five coefficients c2q, contributes to the observation of RDCs. These coefficients 

It is common to work with an expression for RDCs written in the PAS of alignment since this leads to simplification of Eq. (8), with the remaining nonzero elements (Y29( (f) (0)) related to the diagonalized Saupe order tensor elements (Szz and (Sxx—Syy)) which describe the magnitudes of alignment. When written in its PAS, Eq. (8) becomes, 

The explicit expansion of the spherical harmonic functions describing the orientation of the z/th internuclear vector leads to the following expression, 

From the expression of RDCs in the PAS of alignment (whether Eq. (12) or Eq. (13)), it is clear that a given RDC measurement, Djjes, does not correspond to a unique orientation of the internuclear vector. In fact the 

In optical experiments the absorption of radiation is governed by the Beer-Lambert law. If I0(E) is the measured intensity of a beam of electromagnetic radiation of energy E and 1(E) that following absorption by a gas of thickness L containing n molecules per unit volume, then 21 

An analogous expression holds for electron scattering, except that in such an experiment we usually desire to measure that proportion of the incident beam (of impact energy E0) that has lost energy E. In general, the intensity of such an inelastically scattered beam will depend on the polar angles 8s, j s (with respect to the main beam) at which it is measured. If the incident electron beam has an intensity I0, the inelastically scattered beam will have an intensity 

The cross section a is a fundamental property of the molecule and as such is related to the molecular wave functions for the two states between which a transition is induced. Hence it is desirable to separate the contributions to a that arise from purely kinematic quantities such as the impact energy of the electron beam from those that depend solely on the properties of the molecule. To this end, a dimensionless quantity, the oscillator strength, is introduced in optical absorption spectroscopy, defined by the relation22 

An analogous quantity, the generalized oscillator strength, is found to be useful in electron-scattering theory. It is a function of the momentum K transferred from the incident electron to the molecule and has the form5 

The cross sections j,( ) and oel(0s, / s, , o) are related5-21 to those two oscillator strengths through the equations 

The general theory of the quantum mechanical treatment of magnetic properties is far beyond the scope of this book. For details of the fundamental theory as well as on many technical aspects regarding the calculation of NMR parameters in the context of various quantum chemical techniques we refer the interested reader to the clear and competent discussion in the recent review by Helgaker, Jaszunski, and Ruud, 1999. These authors focus mainly on the Hartree-Fock and related correlated methods but briefly touch also on density functional theory. A more introductory exposition of the general aspects can be found in standard text books such as McWeeny, 1992, or Atkins and Friedman, 1997. As mentioned above we will in the following provide just a very general overview of this 

47 For example, despite the high symmetry of C60, no calculation of its NMR properties at the MP2 level has been reported so far. DFT-based methods, on the other hand, can readily be applied to this and to larger fullerenes, see Buhl et al., 1999. 

Fortunately, as shown by Lee, Handy, and Colwell, 1995, it seems that the consequences of this approximation with regard to the accuracy of the computed chemical shifts are rather modest and are of less significance than the general shortcomings inherent in the functionals used. Hence, from an application-oriented, pragmatic point of view one does not need to worry too much about using functionals which are formally inadequate because they neglect the required dependence on j(r). 

In Chapter 2 it was shown that the flux of a component i through a pervaporation membrane can be expressed in terms of the partial vapor pressures on either side of the membrane, pio and pie, by the equation 

Equation (9.1) is the preferred method of describing membrane performance because it separates the two contributions to the membrane flux the membrane contribution, P /C and the driving force contribution, (pio — p,r). Normalizing membrane performance to a membrane permeability allows results obtained under different operating conditions to be compared with the effect of the operating condition removed. To calculate the membrane permeabilities using Equation (9.1), it is necessary to know the partial vapor pressure of the components on both sides of the membrane. The partial pressures on the permeate side of the membrane, p,e and pje, are easily obtained from the total permeate pressure and the permeate composition. However, the partial vapor pressures of components i and j in the feed liquid are less accessible. In the past, such data for common, simple mixtures would have to be found in published tables or calculated from an appropriate equation of state. Now, commercial computer process simulation programs calculate partial pressures automatically for even complex mixtures with reasonable reliability. This makes determination of the feed liquid partial pressures a trivial exercise. 

Having said this, the bulk of the pervaporation literature continues to report membrane performance in terms of the total flux through the membrane and a separation factor, /3pervap, defined for a two-component fluid as the ratio of the two components on the permeate side of the membrane divided by the ratio of the two components on the feed side of the membrane. The term /3pervap can be written in several ways. 

The second step in the process is permeation of components i and j through the membrane this step is equivalent to conventional gas separation. The driving force for permeation is the difference in the vapor pressures of the components in the feed and permeate vapors. The separation achieved in this step, /fmem, can be defined as the ratio of the components in the permeate vapor to the ratio of the components in the feed vapor 

Equation (9.4) shows that the separation achieved in pervaporation is equal to the product of the separation achieved by evaporation of the liquid and the separation achieved by selective permeation through the membrane.1 

The effective spin Hamiltonian (SH) that includes Zeeman and hyperfine interaction for a single unpaired electron and N nuclei is  

If two immiscible phases are placed in contact, one containing a solute soluble to some extent in both phases, the solute will distribute itself so that when equilibrium is attained no further net transfer of solute takes place, as then the chemical potential of the solute in one phase is equal to its chemical potential in the other phase. If we think of an aqueous (w) and an organic (o) phase, we can write, according to equations (3.49) and (3.52), 

The term on the left hand side of equation (5.28) is constant at a given temperature and pressure, so it follows that aja = constant and, of course aja = constant. These constants are the partition coefficients or distribution coefficients, P. If the solute forms an ideal solution in both solvents, activities can be replaced by concentration, so that 

P is therefore a measure of the relative affinities of the solute for an aqueous and a non-aqueous or lipid phase. Unless otherwise stated, P is calculated according to the convention in equation (5.29), where the concentration in the nonaqueous (oily) phase is divided by the concentration in the aqueous phase. The greater the value of P, the higher the lipid solubility of the solute. 

It has been shown for several systems that the partition coefficient can be approximated by the solubility of the agent in the organic phase divided by its solubility in the aqueous phase, a useful starting point for estimating relative affinities. 

In many systems the ionisation of the solute in one or both phases or the association of the solute in one of the solvents complicates the calculation of partition coefficient. As early as 1891, Nemst stressed the fact that the partition coefficient as a function of concentration would be constant only if a single molecular species was involved. If the solute forms aggregates or otherwise self-associates, the following equilibrium between the two phases 1 and 2 occurs when dimerisation occurs in phase 2  

It is necessary to establish an appropriate analytical framework for the calculations, and through this a clear definition of the variables mapping all information about molecular structure and dynamics. Over the past decades, there has been many excellent texts on the statistical (i.e., density matrix) quantum theory providing the basis for analytical 

The state of an NMR-relevant physical system changes over time as described by the Schrodinger equation which within a statistical density operator formalism may be recast in form of the so-called Liouville-von Neuman equation 

The dynamics of the spin system is governed by the Hamiltonian which in addition to a term reflecting external radio-frequency (rf) manipulations displays dependencies on chemical shifts, scalar (electron-mediated) J coupling, dipole-dipole coupling, and quadrupolar coupling, i.e., 

For simplicity, the direct Zeeman interaction with the external magnetic field is parametrized out by representing the system in the standard rotating frame. Within the high-field approximation, the Hamiltonian for the external rf irradiation may be expressed as 

The Wigner rotations describe the coordinate transformations from the principal axis frame (P ) in which the tensor describing the interaction X is diagonal, via a molecule-fixed frame (C) and the rotor-fixed frame (R) to the laboratory frame (L) as illustrated in an ORTEP representation in Fig. 1. 

Density is defined as the mass per unit volume at constant temperature and pressure 

The unit of density is kg m . In practice, smaller units, e.g. (g cm ), are frequently used. The reciprocal value of density is the specific volume. 

The density of melts depends on the temperature and pressure. The dependence on temperature is expressed by the coefficient of thermal expansion 

For simple melts in which the composition does not change with temperature, density is a linear function of temperature. 

The molar volume can be calculated from density according to the relation 

A monochromatic X-ray beam ol cncrg . I. is attenuated by ilia passage through a muienal ol Ihickness il according to equation Id. I 

This line structure has been known for a long time—II. Fricke 2 and G. Ilert/. 3 discovered il in 1920—but the effect could not he explained satisfactorily by theory at the time. R. de L. Kronig 14,5] already had the correct fundamental ideas ia the 1930s, but the interpretation remained confusing until the 1970s when D.E. Saveis. I .A. Stem and I. W. Lytle [6,7] formulated a theory that has remained generally accepted until today. This theory will he briefly outlined below. 

The probability for absorption of an X-ray photon by an electron of an inner shell is dcperuleni on the initial and final states of tire excited electron. The initial stale is dial of the core electron, while the final Male is more difficult to describe. The absorption of X-ray radiation generates an electron 

According to the above considerations, precise measurements of the energy dependence ol the X ray ahsotption coeflicicnt make il possible 

The next step is to convert the function ( ) n the function x(A). where k is the magnitude of the photoeleciron wave vector, k can be calculated according to equation (10.3) from the energy of the incident X-ra photon and the position of the absorption edge . 

The identification of the stacking mode in an MDO polytype is based on two orthogonal projections, which are sufficient to characterize reliably any structure. For 

Owing to the efficiency of atomic scattering factors as a function of sind/X, the diffractions close to the origin of the reciprocal lattice are best suited for identification purposes. Moreover, any family structure in micas is trigonal or hexagonal and from Friedel s law it follows that the reciprocal lattice rows 20/, 13/, 13/, 20/, 13/ and 13/ 

The recognition of the significance of the YZ projections (and thus also the five MDO groups given in Table 7), which can be derived also directly from the full polytype symbols (Durovic et al. 1984), is very important also for the interpretation of HRTEM images (Kogure, this volume). 

The identification of the stacking mode of an MDO polytype in the homo-octahedral approximation is straightforward. It can be performed by visual inspection of the intensity distribution along two rows (one D and one X), and from visual inspection of the geometry of the diffraction pattern. 

Reciprocal lattice rows parallel to c in the hOl r.p. have 1 (subfamily A) or 2 (subfamily B) reflections in the c i repeat  

Another way to describe deviations from the simple BPP spectral density is the so-called model-free approach by Lipari and Szabo [9]. This takes into account the reduction of the spectral density usually observed in NMR relaxation experiments. 

The specific impulse. Is, is often expressed in terms of the absolute temperature in the combustion chamber 1, and the number of moles o( gaseous products produced per unit weight of propellant N by the simplified relationship given as Eq. (1) [1]. 

This proportionality can be rationalized in the following manner based on kinetic theory. 

Is is directly related to the thrust that is developed by a propellant when it undergoes combustion thrust is the recoil force that is produced by the formation and discharge of the gaseous products. 

From kinetic theory, the kinetic energy of one mole of a gas having molecular weight M and at an absolute temperature T is 

Considering only the contribution to the kinetic energy coming from the velocity component in the x direction 

Analysis of the dynamic magnetic resonance experiments is achieved by using the density operator formalism, outlined elsewhere [10, 35, 49]. Here we summarize important features of this treatment and introduce the simulation paranKters. The spin Hamiltonion representing Zeeman, quadrupole or hyperfine interactions 

In order to describe the time evolution of the density matrix Q(t) during some arbitrary pulse sequence, we divide the sequence into regions, where a pulse is present and regions where there is no pulse. The action of the different non-selective puls (including a single 90° pulse for the FID which after FT yields the CW frequency spectrum) is considered by unitary transformations employing Wigner rotation matrices [10, 49]. After the pulse the density matrix is assumed to obey the stochastic Liouville equation [85, 86] 

Notatit for coordinate systems and Eulw trantfcmnations used in the NMTt and ESR relaxation model 

Markov operator for the various rotational processes and Qeq is the equilibrium density matrix. 

In the finite grid point method [87, 88], the Markov operator is represented by a matrix W( l, ft) whose elements give the transition rates between discrete sites of ft. The values of the transition rates depend upon the model used to describe the motion. For the intramolecular dynamics such as tram-gauche isoTnenz tion or ring flips (see Fig. 4) a random jump process is assumed. Consequently [90] 

Both algorithms are based on the least squares method. The difference between the original curve F(v,) and the calculated curve is kept as small as possi- 

In the frame of the Levenberg-Marquardt algorithm the sum of the quadratic deviations between measured and calculated data points is given by 

The Local Least Squares algorithm performs an independent fit for each individual peak. The calculation is thereby restricted to the range around the band maximum. This drastically reduces the amount of data required for the calculation, enhancing the speed compared to the Levenberg-Marquardt method. Some loss of precision versus the Levenberg-Marquardt method occurs. The Local Least Squares algorithm has some conditions  

The given model can be considered as an area in an n-dimensional space with n being the total number of band parameters. In most cases this area has one absolute minimum and several local minima. The quality of the calculation depends on the quality of the selected model, i.e. does the calculation find the absolute minimum or does it orbit in the vicinity of the local minima. The latter case can be detected by a relatively large error and a visually obvious poor fit. In the case of a poor fit, start again with a new optimized model. 

Casagrande (1936) performed a series of drain, strain-controlled triaxial tests on initially loose and initially dense sand specimens. Results showed that all specimens tested at the same effective confining pressure approached the same density or void ratio when sheared 

Permeability of Sediments from Abyssal Plains and Selected Regions 

Location Sediment Type Coefficient of Permeability (cm/s) Reference 

Madeira Abyssal Plain Fine-grained turbidites 6x10-5 Schuttenhelm et al. (1985) 

MAP Great Meteor East Nanno-fossil turbidites (top) 1x10-5 Schultheiss and Gunn (1985) 

Any chromatographic process relates to the selective distribution of an analyte between the mobile and the stationary phase of a given chromatographic system. In liquid chromatography the solvent, with a volume V in the interparticle space, moving along the column at a certain velocity, is the mobile phase and the porous adsorbent, having a pore volume Vp, is the stationary phase. The distribution coefficient equals the ratio of the concentrations of the analyte in the stationary and the mobile phases. In classical SEC a distribution of the analyte between the interparticle volume and the accessible pore volume takes place and the retention volume Vr is determined by 

Kd is related to the change in Gibbs free energy AG at the point where the analyte molecules pass from the mobile into the stationary phase  

The change in Gibbs free energy may be due to different effects  

inside the pore, which is limited in dimensions, the macromolecule can not occupy all possible conformations and, therefore, the conformational entropy AS decreases  

when penetrating the pores, the macromolecule may interact with the pore walls resulting in a change in enthalpy AH. 

Every photon of light carries energy Huj and momentum hk so, if light is absorbed by an object, the momentum transferred from a light beam of power, P, leads to a reaction force, F, on the object, given by 

Embedded iead users can use knowiedge derived from their own use experience as well as their interactions with other users to devise new solutions to user problems and to iimovate. To understand to what extent embedded lead users effectively innovate on behalf of their company, 1 staufy their iimovative work behavior (IWB). 

Embedded iead users can also deploy their unique knowledge of the firm as well as the user environment for internal boundary spatming behavior (BSBINT) in the iimovation process. They can interact with R D as well as marketing and sales, and thus foster information dissemination and exchange inside the boundaries of the firm. 

Finally, embedded lead users can leverage their embeddedness in the use enviromnent to communicate with other users and foster customer satisfaction. Even if customer orientation behavior is especially relevant for marketing, it may also contribute to innovation, because customer orientation helps to extract irmovation-related user knowledge from other customers. I capture these behaviors by a well-established construct called customer orientation behavior [COB] and seek to understand whether embedded lead userness predicts higher levels of customer orientation behavior. 

Next to these behaviors, which clearly relate to the boundary role of embedded lead users, I will Investigate the influence of lead userness on organizational citizenship behavior [OCB], Organizational citizenship behavior is related to an employee s tendency to reciprocate to the organization beyond what is explicitly captured by formal incentive systems. As the firm environment is expected to be veiy accommodating for embedded lead users, they are likely to be good citizens of the Arm. 

By this approach, I study organizational behavior as determined by a hitherto ne ected variable, employees lead userness. 1 use constructs well established in the innovation, organizational behavior, and marketing literatures and combine them to build new theory. In the next two sections, I provide a detailed deduction of my research hypotheses on the antecedents and consequences of embedded lead userness. 

In this section, we give a short overview of the models and methods used to compute the capillary force exerted by a liquid meniscus bridging two solid surfaces. We describe the main practical cases of 

The chemical potential i (i.e. the partial molar free energy) is the most fundamental physical quantity to describe the thermodynamic properties of polymer solutions. If p is represented as functions of absolute temperature (7), pressure (7 ) and the composition (for example, the polymer concentration for a binary mixture), one can determine not only the molecular weight, M, of the solute (the polymer), but also many other thermodynamic quantities, such as the partial molar entropy, the partial molar enthalpy and the partial molar volume of each component. 

The difference between the chemical potential of the solvent in the solution Po and that of pure solvent pS, chosen as the standard state, is usually denoted by Apo, which is simply referred to as the chemical potential of the solvent, relating directly to the activity ao through the equation  

It has been widely confirmed with numerous experiments that polymer solutions reveal remarkable deviation from the behaviour expected for an ideal solution [5]. According to the Flory-Huggins lattice theory of polymer solutions [6,7], Apo is given by  

When the polymer solution is placed under an atmosphere of saturated vapour phase and both phases are in equilibrium, the vapour pressure of the solvent component in the solution Pq is smaller than the vapour pressure of the pure solvent Then their ratio Po/Po is equal to Oq and eqn (5.12) holds  

Accurate determination of Pq and pS for solutions with various values of c allows one to evaluate M,j by extrapolating the plot of In (Po/Po) versus c to c = 0. Unfortunately, the difference between Po and Pq is extremely small in dilute solution, and the maximum molecular weight (M ) determinable by the vapour-pressure depression is never beyond 1000. As will become clear, this limit is overcome by an indirect measurement technique. 

The retention of the test solutes is expressed as the corrected retention time 

For physico-chemical calculations, the retention data are often presented as the specific retention volume Vg (cm g )  

It is worth noting two points with regards to the background of IGC  

Nastasovic and Onija [19] presented the advantages and drawbacks of the IGC over traditionally used methods when determining the glass transition temperature (Tg). The precision and accuracy of Tg determination is influenced by (i) the inert support type (ii) the coating thickness (iii) the sorbate type and (iv) the flow rate. The report of Nastasovic and Onija is, and the references cited therein, should be taken into account when examining polymer properties by means of IGC below and above Tg. 

Solute equilibrium conditions between the mobile and stationary phases are achieved at zero surface coverage of the surface. The chromatogram must be symmetric and the maximum of the chromatographic peak should be independent of the amount of retained adsorbate [20]. The concentration of adsorbate in the gas phase is minimal, and the sorption process is derived from real adsorbate-adsorbent interactions. The adsorbate might be considered as an ideal gas both in the gas phase and in the adsorbed state. At infinite dilution, the net retention volume (Vj,f) is related to the concentration of adsorbate in the gas phase c, as follows  

The method sketched so far is based on the assumption that there exists a smooth solution path x(s) without bifurcation and turning points. Before describing continuation methods in more algorithmic details we look for criteria for an existence of such a solution path. 

For this end we differentiate H x s) s) = 0 with respect to the parameter s and obtain 

If Hx x, s) is regular we get to the so-called Davidenko differential equation [Dav53]  

The existence of a solution path x s) at least in a neighborhood of (x(0),0) can be ensured by standard existence theorems for ordinary differential equations as long as Hx has a bounded inverse in that neighborhood. For the global homotopy (3.8.1) this requirement is met if F satisfies the conditions of the Inverse Function Theorem 3.2.1. 

For the global homotopy the Davidenko differential equation reads 

Whenever addressing the structure-property relationships of a polymer, its rheological behavior must be one of the first considerations, more specifically, its intrinsic viscosity [ ], also called the Staudinger index, because it is often used in industry as a 

Studies on Perfect Hyperbranched Chains Free in Solution and Confined in a Cylindrical Pore, Springer Theses, DOI 10.1007/978-3-319-06097-2 5, 

5 Solution Properties of Perfect Hyperbranched Homopolymers and Copolymers 

As mentioned in Chap. 1, randomly hyperbranched chains are even more complicated than dendrimers. It has not been completely clear whether they are fractal objects [11, 12] and whether those previously reported M-dependent intrinsic viscosities from an on-line combination of the size exclusion chromatograph (SEC) with viscosity and multi-angle laser light scattering (MALLS) detectors actually captured its structure-property relationship [11, 13]. In the next section, we ll discuss our experimental results in detail. 

If the protein has anomalous scatters in its molecule, the difference in intensity between the Bijvoet pairs, Fm/( + ) and Fa /( —), can be used for the phase angle determination. In the MAD method the wavelength 

In infinitely dilute solutions, the polyelectrolyte chains are essentially isolated. In Muthukumar s treatment [21] of this limit, the degrees of freedom of counterions 

Both parameters a and w are known in our simulations, a is already discussed above. For the repulsive LJ potential used in our simulations, w is given by the binary duster integral between beads, 

2 Dependence of Radius of Gyration on Sah with Monovalent Counterions 

Cp/g = 8 X 10 ). Open symbols are simulation data and the curve is ssM theory without any adjustable parameters, (b) is the double logarithmic plot of (a). 

To a large extent, the formalism of crossover functions has been developed by Sengers and coworkers [10] in recent years. It has been shown that the crossover from mean field to Ising behavior could be non-universal and rather complex. In the asymptotic critical Ising type regime the correla- 

Ising model. The ratios of the Ising and mean field critical amplitudes are given as 

In the present case of a zero microscopic length (A cx)) the crossover between the two universality classes is determined by a single parameter, namely uAf which is proportional to the Ginzburg number Gi and thereby to the crossover temperature Tx (see Fig. 1). In conformity with our earlier 

The expressions for the critical amplitudes are given for the mean field case in Eq. 11. For the Ising case they are expressed in terms of the molecular volumes for symmetric blends according to C+ oc and oc y l-v) 

The above expressions for Gi are valid for UCST systems showing phase decomposition at low temperatures. These systems are characterized by (Th and Ter) 0 and Fa Fc if F 0 as otherwise no miscibility is achieved. There are several blends, showing phase decomposition at high temperatures, and which are so-called LCST systems. The EH parameter of such blends have to fulflll the following conditions, namely (Fa and Fa) 0 and IFcrl / c = 2/y, in order to show phase transitions at a finite critical temperature. The corresponding critical amplitude for the mean field susceptibility becomes Cj[ p = 2 [- (2/V -F Ter)] because of the necessary redefinition 

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