Structural properties static structure factors

The defining property of a structural glass transition is an increase of the structural relaxation time by more than 14 orders in magnitude without the development of any long-range ordered structure.1 Both the static structure and the relaxation behavior of the static structure can be accessed by scattering experiments and they can be calculated from simulations. The collective structure factor of a polymer melt, where one sums over all scattering centers M in the system... 

In Fig. 3.16 dynamic structure factor data from a A =36 kg/mol PE melt are displayed as a function of the Rouse variable VWt (Eq. 3.25) [4]. In Fig. 3.6 the scaled data followed a common master curve but here they spht into different branches which come close together only at small values of the scahng variable. This splitting is a consequence of the existing dynamic length scale, which invalidates the Rouse scaling properties. We note that this length is of purely dynamic character and cannot be observed in static equilibrium experiments. 

For the discussion of the properties of the static structure factors, it is often more convenient to write the scattering functions in terms of a space correlation function y(r)4wr2dr. 

The accuracy of Eq. (1), when dealing with real systems like polystyrene spheres in water, has been usually tested indirectly. Theoretical calculations and/or computer simulated experiments, of static properties such as the static structure factor, are carried out assuming such functional form of the pair... 

Once the multipole analysis of the X-ray data is done, it provides an analytical description of the electron density that can be used to calculate electrostatic properties (static model density, topology of the density, dipole moments, electrostatic potential, net charges, d orbital populations, etc.). It also allows the calculation of accurate structure factors phases which enables the calculation of experimental dynamic deformation density maps [16] ... 

A thorough investigation of simulated structure factor data of ammonia provides us with quantitative figures for the accuracy of BCP properties achievable with the pseudoatom projection [72]. The Pbcp> Pbcp> tBCp(H) indices of the static model density obtained by the multipole refinement of HF/6-31IG static structure factors deviate from the correct values (derived... 

The most obvious experimental manifestations of interionic correlation are found in the Haven ratio (deduced from diffusion and conductivity measurement, see Chapter 4), in the static structure factor S(Q) (deduced from partial occupation factors measured by X-ray or neutron diffraction) and from the dynamical properties (S Q, co), quasi- and inelastic-neutron scattering, frequency dependent conductivity) and e(co) dielectric relaxation. 

In discussing the thermodynamics of the kinetic model description we will concentrate only on those properties which directly affect the behavior of S(k,(o). These are the isothermal compressibility xt, the specific heats Cy and Cp, and the adiabatic sound speed c . The compressibility is the long-wavelength limit of the static structure factor Sik), ... 

Recall the HP relation, which summarizes the effect of grain size on some static mechanical properties. Experimental results indicate that the resistance of materials to fatigue-crack initiation and propagation is significantly influenced by grain size. This applies to ceramics as well. It is widely recognized that, when all the other structural factors are kept approximately fixed, an increase in grain size will... 

In the first two chapters, we learned about thermodynamics (free energy, osmotic pressure, chemical potential, phase diagram) of polymer solutions at equilibrium and static properties (radius of gyration, static structure factor, density correlation function) of dissolved polymer chains. This chapter is about dynamics of polymer solutions. Polymer solutions are not a dead world. Solvent molecules and polymer chains are constantly and vigorously moving to change their positions and shapes. Thermal energy canses these motions in a microscopic world. 

The first static property characterizing the system is the pVT behavior, which was already discussed in the last section. Besides this thermodynamic information, an amorphous polymer melt is mainly characterized through two types of structural information the single chain structure factor and the overall structure factor of the melt. 

Simulations, both MC and MD, have been used to test these scaling predictions and to determine other properties of a star polymer, including the static structure factor in the dilute limit. At present, it is not possible to simulate a melt or even a semi-dilute solution of many-arm star polymers due to the long relaxation times. For few-arm stars f 12) MC methods are clearly most efficient, while for large number of arms, MD methods work very well. For small /, the density of monomers of the star is low almost everywhere and static MC methods in which one generates the chains by constructing walks can be Using this method,... 

This section presents a theoretical study of more concentrated deformable emulsions and microemulsions where higher order interactions become important. The purpose is to relate the microseopic droplet deformability to the structure of such systems and further to their macroscopic (thermodynamic) properties. The radial distribution function and static structure factor are calculated utilising an integral equation approach in an appropriate closure approximation. This method allows us to obtain the virial equation of state as well. A semi-empirical equation of state, based on modifying the Camahan-Starling expression, as well as comparison with Brownian dynamics simulations are also presented. 

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