Isotropic fluids

For an isotropic fluid, the one-particle correlation fimction is independent of tire position and... 

Consider an isotropic fluid in which viscous phenomena are neglected. Concentrations and temperature are non-unifonn in this system. The rate of entropy production may be written... 

Fig. 2. Schematic representation of the orientational distribution function f 6) for three classes of condensed media that are composed of elongated molecules A, soHd phase, where /(0) is highly peaked about an angle (here, 0 = 0°) which is restricted by the lattice B, isotropic fluid, where aU. orientations are equally probable and C, Hquid crystal, where orientational order of the soHd has not melted completely.

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Fig. 12. A, Schematic representation of parallel arrays of polynuclear aromatic hydrocarbon molecules in a mesophase sphere. B, a) isolated mesophasc spheres in an isotropic fluid pitch matrix b) coalescence of mesophase c) structure of semi-coke after phase inversion and solidification.

Fig. 12 Pn-4 (a, b) and Pn-8 (c, d) P-sheet variation measure by IR circles) and NMR triangles). 10 mg mL peptide solutions prepared in (a, c) D2O and (b, d) 130 mM NaCl in D2O. For Pii-4, I nematic gel, II flocculate, III nematic fluid, IV isotropic fluid. For Pn-8, I isotropic fluid, II biphasic solution. III nematic gel. Adapted from Carrick et al. [23]. Copyright 2007, with permission from Elsevier...

Referring to the plot of G vs. T for the isotropic and nematic fluids, a uni-molecular isomerization process would show the same characteristics if one of the isomers had a higher entropy of formation than the other. In the molecular case, however, the equilibrium constant at a given temperature would derive from the free energies on a per molecule basis, while for the phases this free energy is per collective volume of molecules. The similarities to a molecular isomerization, however, are more important than the differences for the purposes of this discussion. The transition from isotropic fluid to nematic LC can be considered a temperature-driven, or thermotropic, isomerization. 

The starting system is achiral (plates at 90° with isotropic fluid between), but leads to the formation of a chiral TN structure when the fluid becomes nematic. In this case, enantiomeric domains must be formed with equal likelihood and this is precisely what happens. The size of these domains is determined by the geometry and physics of the system, but they are macroscopic. Though the output polarization is identical for a pair of heterochiral domains, domain walls between them can be easily observed by polarized light microscopy. This system represents a type of spontaneous reflection symmetry breaking, leading to formation of a conglomerate of chiral domains. 

Interestingly, it has been suggested that this preference for the layers to form perpendicular to the free surface in smectics forming from the isotropic fluid... 

We present here some very general exact results, which hold for arbitrary reorientation mechanisms of any molecule in an equilibrium isotropic fluid (but not a liquid crystal). A coordinate frame (R) is rigidly attached to the molecule of interest. Its orientation in the laboratory frame (L) is defined by the Euler rotation = (affy) that carries a coordinate frame from coincidence with the laboratory frame L to coincidence with the molecular frame R/ The conditional probability per unit Euler volume [( (0r at time t must depend only on the Euler rotation A = 1 (i.e., rotate first by < 0 then... 

Evidently, correlation functions for different spherical harmonic functions of two different vectors in the same molecule are also orthogonal under equilibrium averaging for an isotropic fluid. Thus, if the excitation process photoselects particular Im components of the (solid) angular distribution of absorption dipoles, then only those same Im components of the (solid) angular distribution of emission dipoles will contribute to observed signal, regardless of the other Im components that may in principle be detected, and vice versa. The result in this case is likewise independent of the index n = N. Equation (4.7) is just the special case of Eq. (4.9) when the two dipoles coincide. 

As shown by the data in Figure 2, annealing the biphasic composition (i.e. phase A, coexisting with the isotropic fluid) at 40 for 10 hr leads to enhanced values of Rhv(q) Rw(q) over the... 

Note In the case of an isolated spherical particle in a viscous isotropic fluid, /is a scalar anAF =fu. 

The void is considered to be in an infinite isotropic fluid medium. 

Clearing point the temperature at which the mesophase transforms into an isotropic fluid. 

Various other instances of hydrodynamic and electrohydrodynamic instabilities in nematic and, to a lesser extent, smectic liquid crystals have been investigated. No attempt is made here to review this work. For the present discussion, it is sufficient to note that (a) most of the work has dealt with oriented layers having anisotropic properties, and (b) some interesting instabilities arise in oriented layers which do not occur for isotropic materials. An example of the latter is cellular convection in a fluid layer confined between horizontal plates maintained at different temperatures. With an isotropic fluid, convection can arise only if the lower plate is hotter than the upper plate. Then, fluid near the lower plate is less dense and tends to rise while fluid near the upper plate is denser and tends to sink. With an oriented layer, however, convection can arise even when the upper plate is hotter if the anisotropy of thermal conduction properties is of a particular type (8). 

The quantities h, (p0 and /, v(/o constitute (to within a constant multiplier) two pairs of the canonically conjugate arbitrary constants. Therefore we may choose them as the phase variables while averaging over T in (27). We note that these quantities refer to a local phase space corresponding to any chosen direction of the symmetry axis. Hence, integration performed in the overall phase space dTls corresponding to an isotropic fluid should additionally include averaging over all possible inclinations 0 of the symmetry axis C to the a.c. field vector E. Thus,... 

Qij = jS (rnrij - j5y)] and the layer displacement u and the modulus in the smectic A case [cp = S(s> exp ir/o(z - n) ]. Here, as in the rest of the chapter, we refer to the system of coordinates defined in Sect. 2.1. We note that u is only a good variable if we consider small deformations of the layers. For large layer deformations the phase

further discussion, we will concentrate on the parts due to symmetry variables and the order parameters, while for terms already present in the isotropic fluid see, e.g., [30, 31]. 

Scons> which is identical to the isotropic fluid, is discussed elsewhere [30, 31]. The symmetry part reads... 

In the synthesis of 45 [73], a solution of CuCl2 in ethanol is added to a solution of bis(4-decylbenzoyl)methane in THF at 40 °C. The mixture is stirred at this temperature for 30 min, then slowly mixed with 20% ammonia solution and stirred for a further 2h. The resulting precipitate is isolated by filtration and recrystallized from 2-propanol to yield greenish bronze crystals. They transform into liquid crystals at 87 °C and enter the isotropic fluid state at 130 °C. 

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