Crystallizing volume element

We now relate the rate of liquid accumulation In cavities to the crystallization rate. Within a crystallizing volume element having a saturated amorphous component (Figure 7), the masses of the solvent, M, and polymer, M, are... 

Schematic of a small, crystallizing volume element within a sample undergoing SINC, having a saturated amorphous component, (a) saturated, amorphous phase (c) pure crystalline phase (s) pure liquid phase.

Many-body problems wnth RT potentials are notoriously difficult. It is well known that the Coulomb potential falls off so slowly with distance that mathematical difficulties can arise. The 4-k dependence of the integration volume element, combined with the RT dependence of the potential, produce ill-defined interaction integrals unless attractive and repulsive mteractions are properly combined. The classical or quantum treatment of ionic melts [17], many-body gravitational dynamics [18] and Madelung sums [19] for ionic crystals are all plagued by such difficulties. 

Now, we can calculate a mean volume element that a dye occupies in the zeolite crystal, and if we consider this volume element as a cube, we can calculate the length of the cube. If we use a 1 1 mixture of donor D and acceptor A, this length is approximately the mean distance between D and A ... 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

The lines in the figure divide the crystal into identical unit cells. The array of points at the corners or vertices of unit cells is called the lattice. The unit cell is the smallest and simplest volume element that is completely representative of the whole crystal. If we know the exact contents of the unit cell, we can imagine the whole crystal as an efficiently packed array of many unit cells stacked beside and on top of each other, more or less like identical boxes in a warehouse. 

In Bragg s way of looking at diffraction as reflection from sets of planes in the crystal, each set of parallel planes described here (as well as each additional set of planes interleaved between these sets) is treated as an independent diffractor and produces a single reflection. This model is useful for determining the geometry of data collection. Later, when I discuss structure determination, I will consider another model in which each atom or each small volume element of electron density is treated as an independent diffractor, represented by one term in a Fourier series that describes each reflection. Bragg s model tells us where to look for the data. The Fourier series model tells us what the data has to say about molecular structure. 

As I stated in Chapter 2, computation of the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to convert a Fourier-series description of the reflections to a Fourier-series description of the electron density. A reflection can be described by a structure-factor equation, containing one term for each atom (or each volume element) in the unit cell. In turn, the electron density is described by a Fourier series in which each term is a structure factor. The crystallographer uses the Fourier transform to convert the structure factors to p(.x,y,z), the desired electron density equation. 

With homogeneous strain, the deformation is proportionately identical for each volume element of the body and for the body as a whole. Hence, the principal axes, to which the strain may be referred, remain mutually perpendicular during the deformation. Thus, a unit cube (with its edges parallel to the principal strain directions) in the unstrained body becomes a rectangular parallelepiped, or parallelogram, while a circle becomes an ellipse and a unit sphere becomes a triaxial ellipsoid. Homogeneous strain occurs in crystals subjected to small uniform temperature changes and in crystals subjected to hydrostatic pressure. 

Microindentation hardness is currently measured by static penetration of the specimen with a standard indenter at a known force. After loading with a sharp indenter a residual surface impression is left on the flat test specimen. An adequate measure of the material hardness may be computed by dividing the peak contact load, P, by the projected area of impression (Tabor, 1951). The microhardness, so defined, may be considered as an indicator of the irreversible deformation processes which characterize the material. The strain boundaries for plastic deformation below the indenter are critically dependent, as we shall show in the next chapter, on microstructural factors (crystal size and perfection, degree of crystallinity, etc.). Indentation during a microhardness test permanently deforms only a small volume element of the specimen (V 10 -10 nm ) (non-destructive test). The rest of the specimen acts as a constraint. Thus the contact stress between the indenter and the specimen is much larger than the compressive yield stress of the specimen (about a factor of 3 higher). 

Thus a small volume element of the flowing liquid has the optical symmetry of an orthorhombic crystal, with three different refractive indices corresponding to the three different q values in the indicated directions. However, only the two refractive indices which lie in. the XY plane have ever been experimentally observed. 

We must add Nq electrons to our crystal, in accordance with the Pauli Exclusion Principle. Each level of Equation (5.16) can hold two electrons of opposite spin. The volume element (27r/a) defines a primitive unit cell in k space, each cell contains one energy level. The ground state will fill all levels from A = 0 to a limiting value, k. The Nq electrons will need Nq/2 unit cells, or the number lying in a sphere of radius k-p. 

In the present analyses, prismatic dislocation loops distributed on different slip planes are used as agents for dislocation generation. For copper, sources length of about 0.60 p,m are used. It is worthy to mention that the boundary conditions of the computational cell sides are different in FE and DD parts of the code. In DD, periodic boundary condition for the representative volume element RVE is used to ensure both the continuity of the dislocation curves and the conservation of dislocation flux across the boundaries, by that we take into account the periodicity of single crystals in an infinite media. In FE analysis however, the sides are constrained to move only in the z direction so that a imiaxial strain consistent with the shock experiment is achieved. In order for the boundary conditions in FE and DD to be consistent, periodic FE bormdary condition is implemented as well. The result of this implementation is discussed in the next section. 

Consider a small volume element, dV, of a ciystal (figure 6.2) and let us define the path lengths of the primary (rp) and secondary or reflected (ts) beams as these beams enter and leave the crystal, to and from dV, respectively. The attenuation of the secondary beam originating at dV is given by... 

For the whole crystal this formula is integrated over all volume elements dV in the crystal. 

Figure 6.2 The analytical calculation of a crystal sample correction involves the integration over the whole illuminated volume of the crystal. dV is a volume element, ts and tp are the path lengths through the crystal sample of the secondary (diffracted) and primary (incident) beams. In macromolecular crystallography this calculation is never done because the crystal is bathed in a blob of mother liquor, which may well decrease or swell during the experiment. A better method is to use short wavelengths (e.g. 0.9 A or in future 0.33 A) - see figure 6.3.

Crystal sample volume Volume element of crystal Unit cell volume... 

The process is called mixed suspension-mixedproduct removal crystallization. Because of the above restraints, the nucleation rate, in number of nuclei generated in unit time and unit volume of mother liquor, is constant at all points in the magma the rate of growth, in length per unit time, is constant and independent of crystal size and location all volume elements of mother liquor contain a mixture of particles ranging in size from nuclei to large crystals and the particle-size distribution is independent of location in the crystallizer and is identical to the size distribution in the product. 

This equation gives the change of concentration in a finite volume element with time. In the approach of Barrer and Jost, the diffusivity is assumed to be isotropic throughout the crystal, as Dt is independent of the direction in which the particles diffuse. Assuming spherical particles. Pick s second law can be readily solved in radial coordinates. As a result, all information about the exact shape and connectivity of the pore structure is lost, and only reflected by the value of the diffusion constant. 

Transport in the fluid phase inside the packed bed takes place through convection, axial diffusion and flow to or from the zeoHte crystals. A mass balance for a small volume element of the bed results in the following equation for the concentration Cz in the gas phase... 

To calculate cavitation in a macroscopic specimen, one must determine the concentration and crystallization history of each volume element in the sample during the sorptloii process and apply Equation 6 to each from the time saturation occurs. This has been done for films by solving a non-Ficklan transport equation, an equation for local crystallization and Equation 6 simultaneously. The transport and crystallization equations (without their initial and boundary conditions) are... 

According to equation (4.36), the strain is the derivative of the displacement u of the volume element with respect to the undeformed crystal,... 

Experimentally,/ //f /2 so that AJ Af. Thus the critical elongation of the volume element pervaded by a PS sequence appears to play the major role in governing the crystal growth. Indeed, one may write as a criterion for fractionation (equivalent to the criterion for crystal stability)... 

Figure 6.18. Schematic showing rectangular volume element for polystyrene segments in single crystals of styrene-ethylene oxide block copolymers, the cross-sectional area being given as Z. (Lotz and Kovacs, 1966.)...

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