By classifying particle center positions with a hexagonal grid, evaluation of the two-dimensional rigid sphere partition function is reduced to a special lattice statistics problem, with precisely defined nearest-neighbor effective pair interactions. The hexagonal cell size is chosen to be the maximum consistent with no more than double occupancy. Since the resulting lattice partition function (with three states per site) contains a collectively determined many-cell effective interaction Δ*, as well as nearest-neighbor contributions, it becomes necessary to examine in detail the statistical geometry of available phase space for the original spheres, under varying restraints of nearest-neighbor cell distribution. Accordingly, we obtain for the first time an unambiguous definition of ``random close-packed'' or ``glassy'' arrangements of spheres (which however are not themselves equilibrium states), and to relations between properties of these arrangements, and of Δ*. The key features which subsequently allow description of rigid sphere order—disorder behavior are: (1) the observation that certain nearest-neighbor cell pairs which occur in the glassy state (both unoccupied and both doubly occupied) are geometrically excluded completely in the ordered, close-packed arrangement; (2) Δ* sensitively depends upon these pair distributions. In spite of the fact that Δ* is thereby assigned a specially generalized free-volume form, the theory leads to a proper virial series development at low density. In addition, we report some preliminary results for the effective cell interactions, for glassy state parameters, and some calculations designed to reveal the structure of Δ*. Although this analysis does not yet represent a full quantitative theory of the two-dimensional rigid disk system, it does lead to a novel qualitative explanation of how a fluid—solid transition can occur, and suggestions are given for completing the quantification.
The equatorial:axial ratio and rate of conformational isomerization of cyclohexyl fluoride has been measured by observation of the 19F resonance as a function of temperature (—87.6° to +29.5°) for a 25 vol.% solution in CCl3F. Rate information is readily obtained over a wide temperature range, since the equatorial and axial fluorines differ in chemical shift by 20.5 ppm; the equatorial and axial α protons differ in chemical shift by only 0.46 ppm. Interpretation of the spectra in terms of isomerization rates is complicated by the following: (a) the equatorial and axial conformers are present in unequal proportions; (b) their ratio changes with temperature; and (c) 19F–H coupling produces fine structure and/or peak broadening. The general expression for spectral line shape resulting from exchange between two unequal, chemically shifted spin populations is plotted (using the IBM 7090 computer), chemical shift and conformer ratio data being obtained from the low temperature 19F spectra. The observed spectra fit the calculated spectra very closely, and can be used with their aid to obtain the rates for axial-to-equatorial and equatorial-to-axial isomerization. Excess line broadening resulting from 19F–1H coupling was eliminated by double resonance. Modifications to the rate expressions arising from the differing entropies and enthalpies of the axial and equatorial conformers are presented. Theoretical expectations for the activation entropies and enthalpies are discussed and compared to observed values.
Design of high recording density thinfilm heads for particulate rigid disks
The rate of conformational isomerization of cyclohexane has been measured over the temperature range —75.0 to —47.0° by observation of the NMR spectrum of C6HD11, which is much simpler than that of cyclohexane itself and is amenable to exact kinetic treatment. The line broadening arising from D—H coupling was eliminated by double resonance. It is found that ΔF‡ is equal to ΔH‡ within experimental error for the chair-to-chair reaction and is 10.5 kcal (206.0°K) (10.2 kcal for the chair-to-boat process); ΔS‡ for the chair-to-chair process is thus about zero and for the chair-to-boat process is about +1.4 eu.
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