Shape dependence, first-order transition

Another unique feature of ionized NIPA gel has been found recently both of the equilibrium swelling ratio a and the first-order transition temperature T0 depend strongly on the shape of samples [31]. The measurement of equilibrium a has been made on ionized NIPA gel rods of various diameters, and also on plates and cubes. The gel contained 680 mM NIPA, 20 mM acrylic acid (AA), and 8.6 mM BIS. All samples were prepared from the same pregel solution at the same time so as to guarantee that the composition and the structure of all samples were the same. 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

Our review draws a number of important physical properties for the MTH curves regarding the sign of the quadrupolar interactions (K0) and the values of sing-ion anisotropy (D0) within the NPs. (1) During heating-cooling processes, we observe TH behaviours for the particles magnetization, which is indicative of a first-order transition when the quadrupolar interaction is repulsive. The positions and shapes of the loops and the loop area depend on the values of crystal-field parameter as well as the size of the particle. (2) We also show that the thermal... 

The solution x = corresponding to the isotropic distribution is trivially satisfied at all densities and in any approximation. Below a certain critical density, (2.2.7) has only the isotropic root. Above this density two new roots appear the one which corresponds to low x gives t and the others are discarded. Using (2.2.5), the pressure can then be evaluated as a function of density or volume. The transition predicted by Onsager s theory in the second virial approximation is confirmed by this model even when all virial coefficients up to the seventh are included but the properties of the anisotropic phase depend rather sensitively on the order of the approximation. Fig. 2.2.2 gives the isotherm obtained in the sixth approximation, and it can be seen that the shape of the curve is characteristic of a first order transition. 

This function has been introduced to account for the first order like transition in the process of the compression of the film. The function F(jc) may be thus represented as an "S"-shape function (Figure 8) [30,31]. In analogy with the section 2, the time dependent changes of concentrations, [S], [DiIlt] and [Dsllb] are calculated from the above equations and the rectangular cell model based on division of the air/water interface into twenty cells. In the present work, we take the approximation that the dynamic surface pressure is directly proportional to [S] and [Dint] [44,45]. 

Line shape analysis of the proton transfer in THF gave A// =4.5 kcalmol 1 and AS = -21 eu. Actually, the proton transfer rate depends critically on solvent. It is slow in diethyl ether-<7 0. Adding up to 2 mol% THF-d8 to the ether solution, the rate is first order in THF-r/g- However, above the latter I I IL -z/g concentration the kinetic order in THF- 8 increases. Faster transfer in the presence of superior ligands for lithium in conjunction with the large negative entropy of activation implies that development of the transition state involves increased solvation around Li+. In principle, that should facilitate transfer of lithium between two ends of the bridge. 

The frequency shift can cause the non-central transitions (i.e. m 2) to be shifted sufficiently far from the Larmor frequency that these transitions become difficult to observe with conventional pulse techniques. Equation 2.116 also shows that there is an orientational dependence of the frequency so that very broad lines occur in a powder. This is important for spin-1 nuclei as there is no central transition and all transitions are broadened to first-order. For the satellite transitions the shape extent will depend on Xq and the shape will depend on -q (Figure 2.9A). It should be noted that for m — m transitions there is no first-order effect. 

The shape of adsorption isotherms measured on lamellar halides [102], on graphite [103] and on NaCl [101] strongly depends on strrface homogeneity. Poor substrates exhibit broad distributions of adsorption energies whereas uniform strrfaces allow the observation of step wise isotherms. The steps are clear signatures of first order two-dimensional phase transitions [6,8,14]. 

With continuous Him compression, a first-order phase transition from the gaseous phase to a two-dimensional expanded liquid (LE) can be observed [5]. In this region, the n-A dependency is of hyperbolic shape for most compounds, and the monolayer compressibility is considerably lower than in the gas phase, because cohesion forces act between hydrocarbon chains. Harkins and Jura [6] applied the relationship between film compressibility and molecular area as a possible interpretation of the liquid expanded state compressibility, Cs,was defined as G = —j j. ... 

The Mossbauer spectra of NpSe below at 38 K exhibit a fairly broad distribution of hyperfine fields pointing towards an incommensurate spin structure. The field distribution clearly changes shape with temperature meaning that details of the spin modulation are also temperature dependent. The spectral shape near indicates a first-order phase transition. A positive determination of spin structure by neutron diffraction is still outstanding (Blaise et al. 1992). At low temperatures type-II multi-fe order seems to be indicated, but this does not explain the distribution in hyperfine fields which still prevails at 4.2 K in the Mossbauer spectrum. 

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