Transition, first-order surface effects

Note that on the mean field level at %c a second-order transition is predicted for f=H2, while taking fluctuations into account renders the transition first order [192,210,211], as also found experimentally [231]. Although the nature of surface effects on this transition is quite different for the first-order case [6] than for the second-order case [12], we discuss mostly the second-order case here. The constant pc in Eq. (43) is the density of the chains (pc=l/N if a Flory-Huggins lattice with lattice spacing unity is invoked), and the constants e0 and u0 can be derived [197] from the random phase approximation as... 

Figure 8. (A) Schematic representation of the shape of the function f(rt). The arrows represent the first order like phase transition effect. The two straight lines are f(tt) = 17.5tt + 20.0 and f(n) = O.Olrc, respectively. (B) Schematic representation of the relationship between the surface pressure (ji) and the effective concentration of surfactant at the air/water interface (T f). The solid and dashed lines represent the expected and ideal relationships, respectively.

It is well know that melting occurs more rapidly if there is a positive pressure at the end of the transition zone. If Film B is considered to be large and if the pressure in the melt Film C is essentially zero at the trailing flight, a first order approximation of this pressure effect can be achieved by adding a pressure dissipation term to the rate of material loss in the y direction of the solid surface adjacent to the Film C ... 

The effect of constraints introduced by confining diblock copolymers between two solid surfaces was examined by Lambooy et al. (1994) and Russell et al. (1995). They studied a symmetric PS-PMMA diblock sandwiched between a silicon substrate, and silicon oxide evaporated onto the top (homopolymer PMMA) surface. Neutron reflectivity showed that lamellae formed parallel to the solid interfaces with PMMA at both surfaces. The period of the confined multilayers deviated from the bulk period in a cyclic manner as a function of the confined film thickness, as illustrated in Fig. 2.60. First-order transitions were observed at t d0 = (n + j)d0, where t is the film thickness and d0 is the bulk lamellar period, between expanded states with n layers and states with (n + 1) layers where d was contracted. Finally, the deviation from the bulk lamellar spacing was found to decrease with increasing film thickness (Lambooy et al. 1994 Russell et al. 1995). These experimental results are complemented by the phenomenologi-... 

In this volume, we will apply the principles developed in Principles and Applications to the description of topics of interest to chemists, such as effects of surfaces and gravitational and centrifugal fields phase equilibria of pure substances (first order and continuous transitions) (vapor + liquid), (liquid 4-liquid), (solid + liquid), and (fluid -f fluid) phase equilibria of mixtures chemical equilibria and properties of both nonelectrolyte and electrolyte mixtures. But do not expect a detailed survey of these topics. This, of course, would require a volume of immense breadth and depth. Instead, representative examples are presented to develop general principles that can then be applied to a wide variety of systems. 

Fig. 12. Schematic variation of the order parameter profile /(z) of a symmetric (f=l/2) diblock copolymer melt as a function of the distance z from a wall situated at z=0. It is assumed that the wall attracts preferentially species A. Case (a) refers to the case % %v where non-linear effects are still negligible, correlation length and wavelength X are then of the same order of magnitude, and it is also assumed that the surface "field" Hj is so weak that at the surface it only induces an order parameter 0.2 n if mb is the order parameter amplitude that appears for %=%t at the first-order transition in the bulk. Case (b) refers to a case where % is only slightly smaller than %t, such that an ordered "wetting layer" of thickness 1 [Eq. (76)] much larger than the interfacial thickness which is of the same order as [Eq. (74)] is stabilized by the wall, while the bulk is still disordered. The envelope (denoted as m(z) in the figure) of the order parameter profile is then essentially identical to an interfacial profile between the coexisting ordered phase at T=Tt for (zl). The quantitative form of this profile [234] is shown in Fig. 13. From Binder [6]...

The classical Clapeyron equation adequately predicts the features of first-order phase transitions, and this has been established for a number of examples of first-order transitions effected by the deliberate variation of temperature or pressure. Second- or higher-order transitions are not readily explained by classical thermodynamics. Unlike the case of first-order transitions, where the free-energy surfaces of the two phases... 

Between temperatures of 28 and 29 K the rms bond length fluctuations of the 13-particle system increase dramatically. Similar results have been obtained for all the other clusters N = 5, 6, and 7) for which S(T) is cal-culated. - The curves of S T) for these systems are similar to those occurring with first-order phase transitions of macroscopic systems.Lindemann s criterion states that melting occurs for such systems when rms fluctuations reach 10%.For the small clusters studied, the rise in this function occurs at values of S slightly below 10%—an effect that can be attributed to the large ratio of surface to core atoms. 

On the other hand, hysteresis of the temperature-induced structural phase transitions in nanostructures with first-order phase transitions reduce useful magnetocaloric effect to transform cycling between martensite (M) and austenite (A) phases under application. In addition, the size, surface and boundary effects on thermal hysteresis loops have been under consideration for the development of research on nanostructured materials. Experimental data indicate that nanostructured materials offer many interesting prospects for the magnetization data and for understanding of temperature-induced martensite/austenite phase transitions. 

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