Mass density wave

Note that due to the constraint pe, + pi, + pc + pa — 1/4, there is no fourth independent component. The order parameter components defined in eq. (11) are not orthogonal with each other, and do not bring out the symmetry properties of the structure in a natural way thus in practice one proceeds differently, by considering the expansion of the ordering in terms of mass density waves, as will be discussed below. 

Prost, and later Prost and Barois, developed a phenomenological theory of the A phases in which the free energy density is expressed in terms of two coupled order parameters, p r) describing the mass density wave, and 4 r), the dipolar density wave arising from the antiparallel associations of neighbouring molecules. The total free energy is written as... 

Of these, the most important (and which, for simplicity, we shall consider exclusively) is the chiral smectic C, denoted smectic C. In this phase, like all smectics, partial translational ordering of the molecular centers of gravity is superimposed on the orientational ordering. In the case of smectic C, this translational order can be thought of as a mass density wave as follows ... 

The smectic A is an untilted phase in which the mass density wave is parallel to the director. The cost in free energy of buckling the layers into saddle-shaped deformations is low, with the result that it is relatively easy to construct devices that show bistability between a scattering focal conic director configuration in which the layers are buckled and a clear homeotropic configuration in which the director is perpendicular to the cell walls and the layers parallel to the walls. Transitions between these two textures have been exploited in laser-written projection displays and in both thermo-optic and electrooptic matrix displays. The various mechanisms employed are summarized in Fig. 12. 

Magnetic fields introduce hydromagnetic waves, which are transverse modes of ion motion and wave propagation that do not exist in the absence of an apphed B field. The first of these are Alfven, A, waves and their frequency depends on B and p, the mass density. Such waves move parallel to the apphed field having the following velocity ... 

Acoustic Wave Sensors. Another emerging physical transduction technique involves the use of acoustic waves to detect the accumulation of species in or on a chemically sensitive film. This technique originated with the use of quartz resonators excited into thickness-shear resonance to monitor vacuum deposition of metals (11). The device is operated in an oscillator configuration. Changes in resonant frequency are simply related to the areal mass density accumulated on the crystal face. These sensors, often referred to as quartz crystal microbalances (QCMs), have been coated with chemically sensitive films to produce gas and vapor detectors (12), and have been operated in solution as Hquid-phase microbalances (13). A dual QCM that has one smooth surface and one textured surface can be used to measure both the density and viscosity of many Hquids in real time (14). 

Ceroplastol synthesis, 1, 428 Cetyl alcohol synthesis, 1, 478 Chaetoglobasins structures, 4, 376 Chalcone, o -azido-2 -oxy-synthesis, 3, 823 Chalcone, 2-hydroxy-reduction, 3, 751 Chalcone, 2 -hydroxy-mass spectra, 3, 618 Chalcone dibromides flavone synthesis from, 3, 823 Chalcones polymers, 1, 304 Chanoclavine synthesis, 6, 423 Charge density waves in stacks of ions, 1, 351-352 Chartreusin... 

Figure 7. Countour map of the thermodynamic potential in the dynamical mass (M) - wave number (q) plane. The absolute minimum is denoted by the cross for given density. We have the first order phase transitions in this calculation.

In Eq. 6.1, p = b dJM, where is the scattering length of carbon, its mass density, and the molecular weight, q is the magnitude of the transfer wave vector, g=(47t/)i)sin( /2). An important difference between SAXS and SANS is the contribution in the latter case from incoherent scattering, I This originates... 

Quartz crystal microbalance — The quartz crystal microbalance (QCM) or nanobalance (QCN) is a thickness-shear-mode acoustic wave mass-sensitive detector based on the effect of an attached foreign mass on the resonant frequency of an oscillating quartz crystal. The QCM responds to any interfacial mass change. The response of a QCM is also extremely sensitive to the mass (density) and viscoelastic changes at the solid-solution interface [i-vi]. 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

For longitudinal plane waves propagating in the x direction through a homogeneous medium of mass density p, the wave equation is... 

Example 2.9 If a plane wave propagates in a medium in which the mass density changes, how is wave velocity effected ... 

Thus, from Equation 2.48, the fractional chan in wave velocity is minus the fractional change in mass density of the medium ... 

Acoustic wave (AW) devices are ideally suited to thin film characterization due to their extreme sensitivity to thin film properties [10]. The sensitivity of AW devices to a variety of film properties (see Chapter 3), such as mass density, viscoelasticity and conductivity, makes them versatile characterization tools. The ability to rapidly monitor changes in device responses resulting from changes in thin film properties permits their use for monitoring dynamic processes such as film deposition, chemical modification (e.g., photo-polymerization, corrosion), and diffusion of species into and out of films. 

The results presented here demonstrate that thin films can be characterized based on acoustical monitoring of changes in film mass density, conductivity, and viscoelasticity. Additional sensing mechanisms are available to probe film properties. Some examples are thin-film dielectric constant, stress, and structure (e.g., roughness). Some of these sensing mechanisms will be hard to quantify since they involve a complex interaction (e.g., wave attenuation based on wave scattering due to film roughness) however, they may still be useful to provide a qualitative monitor based on empirical data. 

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