Phase visual representations

We employ the frequency space to visualize resonances, and besides we use phase space representation to get features of the frequency space. 

It is possible to use the Nernst equation, which relates reversible potential to pH and cation concentration, to generate phase stability plots in potential/pH space. Such diagrams are called Pourbaix diagrams, after Marcel Pourbaix who pioneered their development. Pourbaix diagrams are visual representations of the equilibrium conditions in potential/pH space. They are based on thermodynamics, and indicate the stable phase for given conditions, but say nothing about rates of reactions from one phase to another. 

The primary limitation for Pourbaix diagrams is that they are constructed purely from thermodynamic data. They are simply visual representations of the thermodynamic data. As such, they provide no information about corrosion rates. It is possible for substances that are thermodynamically unstable to be metastable, and exert a strong influence on the corrosion kinetics. For instance, Ni is quite resistant to acids, even though thermodynamics predicts that Ni + should be the stable phase at potentials above the Ni/Ni + reversible potential because of the metastabihty of NiO. On the other hand, some thermodynamically stable phases provide little protection. 

The PUs microstructure can be also investigated by means of atomic force microscopy (AFM). Phase images obtained via AFM, enable visual representation of the PUs microphase separated morphology. AFM records the surface topography of materials by measuring attractive or repulsive forces between the probe and the sample. Vertical deflections caused by surface variations are monitored as a raster scan drugs a fine tip over the sample. A detailed description of different modes in AFM technology has been described in [195]. 

A T-Q diagram is a visual representation of the energy balance equation for each streara Because, for sin e-phase streams with constant Cp and no pressure effect on the enthalpy,... 

Fig. 2.6.9 Visualization of gas flow through a cylindrical surface represents the rock, (b) Only porous sandstone rock. A 3D phase encoding a slice through the center of the rock is sequence with a hard encoding pulse was used, displayed, showing the origin of the gas that is (a) 3D representation of an isochronal surface flowing through the detector at different times at different times after the encoding step. The after the tagging [figure taken from 43].

Phase diagrams are often constructed to provide a visual picture of the existence and extent of the presence of solid and liquid phases in binary, ternary and other mixtures of substances. Phase diagrams are normally two-component (binary) representations but multicomponent phase diagrams can also be constructed. Interactions between active substances and excipients can often be evaluated using phase diagrams. 

A remarkable improvement in the visualization of the signal can be obtained by displaying the 33S power spectrum,22 although this is feasible only when a single signal is present (information about the phases and intensities of peaks are lost in the power spectrum representation). 

We introduce a simple model to investigate and calculate a diffusion coefficient as a basic quantity describing transport in Section II, and then we visualize resonances to detect the structure of the Arnold web and overlapped resonances in Section III. With the aid of this representation, to clarify the relevance of Arnold diffusion and diffusion induced by resonance overlap to global transport in the phase space, we compute transition diagrams in the frequency space in Section IV. In Section V, we extend the resonance overlap criterion to multidimensional systems to identify the pathway for fast transport, and in Section VI we revisit the diffusion coefficient to ensure fast transport affecting the global diffusion. A brief summary is given in Section VII. 

The crystallization behaviour of the glasses containing MgO is more difficult to describe. Unlike ternary systems, four component phase diagrams are properly represented by a regular tetrahedron with each apex representing 100% of each component. The crystallization path is therefore in three dimensions which is sometimes difficult to visualize. The method of representation chosen to describe the observed behaviour is as planes of constant MgO content in the regions of interest to this study. Figures 2, 3, and 4 are planes of the quaternary system at 5, 10 and 15 wt% levels respectively and have been compiled from various sources... 

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