Spheres phase

Experimentally, tire hard-sphere phase transition was observed using non-aqueous polymer lattices [79, 80]. Samples are prepared, brought into the fluid state by tumbling and tlien left to stand. Depending on particle size and concentration, colloidal crystals tlien fonn on a time scale from minutes to days. Experimentally, tliere is always some uncertainty in the actual volume fraction. Often tire concentrations are tlierefore rescaled so freezing occurs at ( )p = 0.49. The widtli of tire coexistence region agrees well witli simulations [Jd, 80]. 

Interactions of such glassy polymeric particles should resemble the collisions of hard spheres. Phase diagrams of the type shown in Fig. 36 have been obtained for various polymer-organic solvent mixtures [85,94,345-353]. 

Podolsky method, Renner-Teller effect, triatomic molecules, Hamiltonian equations, 612—615 Poincare sphere, phase properties, 206 Point group symmetry ... 

In essence, the model divides a reactive polymer solution into a dispersed polymer-rich phase (phase 1), within which the concentration of functional groups is defined by the polymer morphology and structure, and a solvent-rich phase which contains no functional groups (phase 0). The individual polymer molecules are modeled as spheres of polymer-rich phase stuck at points of an imaginary lattice in solution. If the polymer concentration is sufficiently high, another phase enters the calculations which consists of overlapping polymer-rich spheres (phase 2). 

PE-PEP diblock were similar to each other at high PE content (50-90%). This was because the mechanical properties were determined predominantly by the behaviour of the more continuous PE phase. For lower PE contents (7-29%) there were major differences in the mechanical properties of polymers with different architectures, all of which formed a cubic-packed sphere phase. PE-PEP-PE triblocks were found to be thermoplastic elastomers, whereas PEP-PE-PEP triblocks behaved like particulate filled rubber.The difference was proposed to result from bridging of PE domains across spheres in PE-PEP-PE triblocks, which acted as physical cross-links due to anchorage of the PE blocks in the semicrystalline domains. No such arrangement is possible for the PEP-PE-PEP or PE-PEP copolymers (Mohajer et al. 1982). 

Fig. 11 Schematic representation of all the phases considered. Dark a, white b, gray e. (a) Lamellar phase, (b) Coaxed cylinder phase, (c) Lamella-cylinder phase, (d) Lamella-sphere phase, (e) Cylinder-ring phase, (f) Cylindrical domains in a square lattice structure, (g) Spherical domains in the CsCI type structure, (h) Lamella-cylinder-II. (i) Lamella-sphere-II. (j) Cylinder-sphere. (k) Concentric spherical domain in the bcc structure. Reprinted with permission from Zheng et el. [104]. Copyright 1995 American Chemical Society...

Figure 2.1 The hard-sphere phase diagram. Below volume fraction < (f>] = 0.494, the suspension is a disordered fluid. Between <) >i = 0.494 and 02 = 0.545, there is coexistence of this disordered phase with a colloidal crystalline phase with FCC (or HCP) order the colloidal crystalline phase is the equilibrium one up to the maximum close-packing limit of 0cp = 0.74. Nonequilibrium colloidal glassy behavior can also occur between

Figure 12.2 Regions of stability of spheres, cylinders, and lamellae in oil predicted by a mean-field film theory using Eq. (12-2). The hatched region has coexisting spheres and cylinders. Above the region of spheres, there is a region of emulsification failure, where the sphere phase coexists with an excess water phase. In the ordinate, 8 is the thickness of the surfactant film, which is roughly the surfactant molecular length, 4>a and arc the volume fractions of surfactant (or amphiphile) and water, and / opt = (1 + 0.5k/k)/Ho. (From Safran 1994, with permission from Addison-Wesley Publishing Company, Copyright 1994.)...

Iversen et al. [6] found that for a polymer strucmre similar to the interstices between closely packed spheres (phase inversion membrane), Equation 38.4 is able to well describe the tortuosity-porosity relationship whereas for a polymer structure similar to random spheres or clusters (stretched membrane), Equation 38.5 has to be used. 

The hard-sphere phase transition was only the first of many important discoveries that have emerged from these techniques. As has been described frequently in the past, these simulations can serve us in three ways. First, they provide essentially exact results for a given molecular model, which can be used to test the accuracy of approximate theories. Second, through comparison with experiment they can be used to assess the quantitative accuracy of a given molecular model. Third, they can be used for exploring the phenomenological behavior of molecular models in a parallel manner to that in which an experimentalist studies the behavior of a real system. We do not review the techniques themselves because there are now excellent textbooks on the subject [32-34]. We focus instead on the specific aspects of these methods associated with the calculation of SFE. 

Physical Insights on the Hard-Sphere Phase Diagram from a DFT Perspective... 

Baus [10] has presented an interesting argument about how the hard-sphere phase transition can be understood from the perspective of DFT. The argument is based on an analysis of the difference between the Helmholtz energies of the solid and fluid phases as a function of the density, p. This can be written exactly within the context of DFT as... 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

An outstanding problem concerns itself with the structure of a hard sphere phase. This is a special instance of the more. general difficulty of the specification of the structure of infinitely extended random media. These questions will perhaps be the subject of a future mathematical discipline-stochastic geometry. The pair correlation function g(r), even if it is known, hardly suffices to specify uniqudy the stochastic metric properties of a random structure. For a finite N and V finite) system in equilibrium in thermal contact with a heat reservoir at temperature T, the density in the configuration space of the N particles [Eq. (2)]... 

However, this maximum is for a solid phase, wherein spheres are so closely packed that long-range order is preserved and there is little, if any, net diffusion of spheres. For the pure hard-sphere Mid, the upper boxmd on T) is even less the fluid-solid phase transition occurs at ii = 2ii a /3 = 0.494 [12]. For it < 0.494 the substance is fluid and long-range order is disrupted by molecular motions. Without attractive forces between spheres, no vapor-liquid phase transition occurs and we refer to the material at Ti < 0.494 as merely "fluid." The hard-sphere phase diagram is shown in Figure 4.8. 

These relations determine the spin / of the compound state, the hard sphere phase shift cpi and the incident orbital momentum forming the compound state. For neutrons the phase shifts neutron energy increases towards the resonance, Eq. (17.4) shows that a dip precedes the peak. 

For large q q> 0.6), the triple point can be approximated easily from (3.46) and (3.47). It can be observed that the fluid-solid coexistence of the triple point occurs at nearly similar colloid concentrations as the pure hard sphere phase transition. For large q values, (3.46) and (3.47) can be written as Jif = Jij = and Pf = = Pj, because g(< ) and h (j)) vanish for large q. In the coexisting... 

Figure 3. POM images under crossed polarizers taken at 112.7°C (parts a and c) for hex-cylinder phase and at 118.6X1 (part bjfor bcc-sphere phase.

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