Boundary phase distribution function

Small angle X-ray scattering (SAXS) experiments give information about the radius of a scattering superstructure. The boundary phase distribution function (BPDF), which can be calculated from the scattering curve [57], gives information about the size of the amorphous and crystalline regions. 

Figure 9.16 Boundary phase distribution function (BPDF) [125]. Small angle X-ray diffraction (SAXS) was recorded with a Kratky camera at the same wavelength as the WAXS experiments. For further explanation see text.

Addressing this problem Implies discussing the notion of liquid structure and the influence exerted on It by a nearby, different, phase. The notion of structure of a system In which the molecules are continually changing their positions can only be made rigorously concrete by statistical means, and it is embodied in the notions of radial and angle-dependent distribution functions, g(r) and g[r,B], respectively. Distribution functions have been introduced in secs. I.3.9d and e, the structure of solvents, emphasizing water, in sec. 1.5.3d. Distribution functions are in principle measurable by scattering techniques, see I.App.ll. For liquids near phase boundaries these distribution functions become asymmetrical. However, it is not always possible, and. for that matter, not always necessary to consider the structure in such detail. 

Explain the basis of the penetration theory for mass transfer across a phase boundary. What arc the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed (Do not present a solution of the differential equal ion.) Obtain the age distribution function for the surface ... 

Now we compare the isotropic-liquid crystal phase boundary concentrations for various polymer solution systems with the scaled particle theory for the wormlike spherocylinder. If the equilibrium orientational distribution function f(a) in the coexisting liquid crystal phase is approximated by the Onsager trial... 

Accounting for size differences can also be realized in terms of distribution functions, assuming certain interaction energies. Simply because of size differences between molecules preferential adsorption will take place, i.e. fractionation occurs near a phase boundary. In theories where molecular geometries are not constrained by a lattice, this distribution function is virtually determined by the repulsive part of the interaction. An example of this kind has been provided by Chan et al. who considered binary mixtures of adhesive hard spheres in the Percus-Yevick approximation. The theory incorporates a definition of the Gibbs dividing plane in terms of distribution functions. A more formal thermodynamic description for multicomponent mixtures has been given by Schlby and Ruckenstein ). 

Equation [16] is known as the Liouville equation and is, in fact, a statement of the conservation of the phase space probability density. Indeed, it can be seen that the Liouville equation takes the form of a continuity equation for a flow field on the phase space satisfying the incompressibility condition dIdT F = 0. Thus, given an initial phase space distribution function /(F, 0) and some appropriate boundary conditions on the phase space satisfied by f, Eq. [16] can be used to determine /(F, t) at any time t later. 

Although we have assumed in Eq. [209] that the velocity profile in the confined fluid is linear, it is not immediately obvious that this is technically possible in the absence of moving boundary conditions. A parallel to this situation is the comparison between Nose-Hoover (NH) thermostats and Nose-Hoover chain (NHC) thermostats. Although the Nose-Hoover equations of motion can be shown to generate the canonical phase space distribution function, for a pedagogical problem like the simple harmonic oscillator (SHO), the trajectory obtained from the NH equations of motion has been found not to fill up the phase space, whereas the NHC ones do. The SHO is a stiff system and thus to make it ergodic, one needs additional degrees of freedom in the form of an NHC.2 ... 

It is now possible to consider a normalized distribution function, /(U, t), describing an ensemble of systems evolving according to Hamilton s equations. An equation of motion for the distribution function can be derived by balancing the rate of change of the number of ensemble members inside a phase space volume v by the flux through the boundary surface... 

The calculation of the perturbed radial distribution function g(2) for steady-state viscous flow is carried out using Eq. 34 for continuity in molecular pair phase space. The development is by no means simple but it is carried out without the introduction of additional assumptions beyond the formulation of appropriate boundary conditions. Those used by Kirkwood and co-workers27 38 make use of the following ... 

The optimal control of the two-phase tubular reactors had been formulated by Kassem (1977). A distributed minimum principle was presented and the necessary conditions for optimality were derived. Based on these conditions for optimality a functional gradient aigorichm for synthesizing boundary and distributed controls were deduced. 

As a second example, it is instructive to derive the Kramers stationary flux function which serves as a basis for practical application in the Rayleigh quotient variational method (34,35). In principle there are an infinity of stationary flux functions, as any function in phase space which is constant along a classical trajectory will be stationary. Kramers imposed in addition the boundary condition that the flux is associated with particles that were initiated in the infinite past in the reactant region. Following Pechukas (69), one defines (68) the characteristic function of phase points in phase space Xr, which is unity on all phase space points of a trajectory which was initiated in the infinite past at reactants and is zero otherwise. By definition x, is stationary. The distribution function associated with the characteristic function Xr projected onto the physical phase space is then... 

We can further characterize the structure by the distribution function of u in Fig.7. At low temperature the motion of each particle is nearly harmonic, so that the distribution function is Gaussian. As the temperature increases, the distribution becomes asymmetric, and above it is symmetric again and almost independent of temperature. At 850 K we have found a phase alternation between the aj and a phases, which correspond to the experimental observation that the domain boundary between the two phases is constantly vibrating just below Tc [37]. We cannot tell exactly whether Tc is below or above 850 K in this simulation because the system size is too small. At 900 K the MD-synthesized quartz is clearly in the / ... 

In this paper we consider the problem when evaporation takes place from the condensed phase and investigate the time development of the disturbance, especially the propagation and decay of the discontinuity of the velocity distribution function, and the steady behavior of the evaporation from a plane condensed phase. The relations among the variables at Infinity and on the condensed phase in the steady evaporation serve as the boundary condition for the... 

So far we discussed the problem under the conventional boundary condition on the condensed phase, where the velocity distribution function of the molecules leaving the condensed phase is independent of the velocity distribution of the molecules incident on the condensed phase and its shape is the half of a stationary Maxwellian. Now we Investigate the effect of different boundary conditions at the condensed phase on the steady evaporation. 

The Guinier, Debye-Bueche, Invariant and Porod analyses are all based on the assumption of well defined phases with sharp interfacial boundaries. In addition, the Guinier approach is based on the assumption that the length distribution function (23.15), or probability Poo(r) that a randomly placed rod (length, r) can have both ends in the same scattering particle (phase) is zero beyond a well defined limit. For example, for monodisperse spheres, diameter D, Poo = 0, for r > D. In the Debye-Bueche model, Poo has no cut off and approaches zero via an exponential correlation function only in the limit r oo [45,46]. 

NLP) problem. Near two- and thtee-phase critical points, near phase boundaries, and with chemical reaction, when the phase distribution is uncertain, numerical difficulties may arise. The most obvious singularity appears at a critical point, where two or more phases coalesce. For two- phase critical points, the singularity is expressed explicitly using the tangent plane distance function ... 

For very high densities, however, the boundary between solid and liquid phases is not sharp and a broad metastable region exhibiting liquid-crystal-like behavior is found. This state can be recognized by the comparison of the mean square displacement and radial distribution function for several temperatures with a fixed density. In this case there is no clear-cut change of pattern as the state point moves from solid to liquid (as in Figure 2 and it is seen, instead, a seemingly contradictory liquid-like mean square displacement and solid-like radial distribution function (see Fig 8). 

Fig. 2. Mapping the phase boundary between solid and fluid by the change of profile in the mean square displacement (a) and radial distribution function (b) for rigid dumbbells with interatomic separation A/rr = 0.50 and density 0.22.

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