Dimensional phase transitions

Fig. 12.19 Examples of dimensional phase transitions in Jourjine s model [jour85] (a) onedimensional phases inside of a two-dimensional phase, (b) a bubble of a two-dimensional phase inside of a one-dimensional phase.

Pd(lll). However, Pd(lll) shows little or no evidence for the stoichiometric 2Bi + L + 3L process. This could be due to the presence of longer range order on the single crystal than on the Pd particles, leading to processes more akin to two dimensional phase transitions on the Pd(lll) crystal surface, rather than a more local species conversion on the small metal crystallites. 

Innocenti et al. have studied the kinetics [101] of two-dimensional phase transitions of sulfide and halide ions, as well as electrosorption valency [102] of these ions adsorbed on Ag(lll). The electrode potential was stepped up from the value negative enough to exclude anionic adsorption to the potential range providing stability of either the first or the second, more compressed, ordered overlayer of the anions. The kinetic behavior was interpreted in terms of a model that accounts for diffusion-controlled random adsorption of the anions, followed by the progressive polynucleation and growth. 

Studies with monolayers and bilayers as models show that modifications of chain conformation can alter the effective thickness of the hydrocarbon portion of a bilayer this, in turn, can have profound effects on its permeability. Suppose, for example, that a two-dimensional phase transition involved cooperatively inducing just one kink per molecule in otherwise fully extended chains. Such a kink shortens the extension of the chain by 0.127 nm, or thins a bilayer by 0.25 nm. We saw in Example 8.2 that this distance is about the length projected by two carbon-carbon bonds in the direction of the chain. Therefore, cooperative induction of such kinks has the same effect on the thickness of the hydrocarbon part of the bilayer as removing two carbon atoms from the tail groups. 

N 001 Theory of the One-Dimensional Phase Transition in Polypeptide Chains ... 

The above described order-disorder transitions are all three-dimensional phase transitions and occur with essentially infinite sharpness unless the condensed phase exists in a colloidally dispersed state, Recently, it has... 

This canonical ensemble partition function predicts a first-order two dimensional phase transition as shown by Hill (8). 

The exchange of NaBr and KBr with HC1 reaches an equilibrium in which, near to room temperature, a mixed Cl /Br surface is at equilibrium with a gaseous HCl/HBr mixture whose composition depends on the surface anion ratio, but which is typically about 1 mole % HBr. This equilibrium was discovered in the laboratory of one of us (LGH) about eighteen years ago [96], but the topic proved difficult to pursue because of the difficulty of quantitative analysis of the gas mixture. Eventually, a temperamental but workable gas chromatographic technique [97] having provided the key to this analysis, we were able to show that the equilibrium is highly non-ideal [98]. The bromide surface apparently undergoes a two-dimensional phase transition, induced by HC1 or HBr adsorption, close to room temperature, in which the halide ions become mobile,... 

V. B. Fainerman and D. Vollhardt, Equations of State for Langmuir Monolayers with Two-dimensional Phase Transition, J. Phys. Chem. B 103, 145-150 (1999). 

The interaction of CO with the solid surface produces several physical and chemical effects on the vibrational properties of the adsorbed species. The adsorption of CO can be envisaged as a two-dimensional condensation, leading to lateral coupling between adsorbed molecules. The vibrational properties of adsorbed CO can thus be used to monitor the effects of other interface properties, such as surface defects, two-dimensional phase transitions [45] and co-adsorption. Finally, CO is formed as an intermediate or poison during the oxidation of several organic molecules at electrodes, thus constituting one of the subjects of interest in electrocatalysis. 

The theoretical model contains four parameters, (3, F K, and K2, whose values are to be obtained from the best fit of the experimental data. Note that all 11 curves in Figure 5.2 are fitted simultaneously." In other words, the parameters (3, F K, and K2 are the same for all curves. The value of F, obtained from the best fit of the data in Figure 5.2, corresponds to 1/ F = 31 A. The respective value of is 82.2 mVmol, which in view of Equation 5.49 gives a standard free energy of surfactant adsorption = 12.3 kT per DS- ion, that is, 30.0 kJ/mol. The determined value of K2 is 8.8 X 10 mVmol, which after substitution in Equation 5.49 yields a standard free energy of counterion binding = 1.9 kT per Na" " ion, that is, 4.7 kJ/mol. The value of the parameter P is positive, 2p TJkT = h-2.89, which indicates attraction between the hydrocarbon tails of the adsorbed surfactant molecules. However, this attraction is too weak to cause two-dimensional phase transition. The van der Waals isotherm predicts such transition for l TJkT> 6.75. 

J. Suzanne, J.P. Coulomb, and M. Bienfait. Two-Dimensional Phase Transition in Xenon Submonolayer Films Adsorbed on (0001) Graphite. Surf. Set. 47 204 (1976). 

A. Thorny, X. Duval, and J. Regnier. Two-Dimensional Phase Transitions as Displayed by Adsorption Isotherms on Graphite and Other Lamellar Solids. Surf. Sci. Rep. 1 1 (1981). 

In conclusion, the analysis of the two-dimensional phase transitions of the supramolecular thin films can be an important means to understand the structural properties of polymolecular assemblies as well as of the components themselves.The AFM image of 1 monolayer of CH20H-derivatized grid G5 reveals rod-like faults. Their typical width is 30 nm. The structure of the rods varies significantly with surface pressure and film thickness. One very important aim now is the preparation and visualization of superstructures of grids by the LB method. This would lead to ordered superlattices suitable for electronic functionalization (Hassmann, J. Hallschmid, N. Hahn, C. Y. Muller, P. Schubert, U. S. Lehn, J.-M., work in progress.). 

Two-dimensional phase transitions occur at a clean or adsorbate-covered surface because a system in thermodynamic equilibrium seeks to minimize its free energy, F [1, 31-33]... 

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