Confinement by Curved Interfaces

Now let us consider curved interfaces. As mentioned in Section 9.1, to calculate the shift of the triple point in a system containing phases confined by curved interfaces, compared to the triple point for the bulk phases, on the basis of Equation 9.4 one needs to establish a relation between the pressure of vapors in equilibrium with bulk condensed phases and the corresponding phases confined by curved interfaces. To determine the vapor pressure in equilibrium with the condensed phase under the curved interface, it is necessary to take into account two physical reasons for the deviation of this pressure from the pressure above a plane interface (p°). These are the increasing (decreasing) pressure inside a condensed phase under a convex (concave) meniscus and the surface forces-induced non-uniformity of structure of the condensed media near the interface. We will analyze the influence of above factors separately and show that they additively contribute to the shift of the triple point of boundary layers. 

As a first step we consider the part of the condensed media confined by the curved interface that is far from the interface and can be considered as uniform. 

Since at the triple point we can always consider the boundary between the vapor phase and each of the condensed phases (solid and liquid), we will use the Laplace equation to relate the pressure in vapor phase Py and the pressure in liquid phase Pj/ 

for the pressure of vapor in equilibrium with a condensed phase confined by a curved interface the Kelvin equation is applicable  

if at any temperature Tc the solid phase confined by the interface with the effective curvature Ks is in the equilibrium with the liquid phase confined by the interface with the curvature Kl, the equality should be held. By equating 

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In this review, we introduce another approach to study the multiscale structures of polymer materials based on a lattice model. We first show the development of a Helmholtz energy model of mixing for polymers based on close-packed lattice model by combining molecular simulation with statistical mechanics. Then, holes are introduced to account for the effect of pressure. Combined with WDA, this model of Helmholtz energy is further applied to develop a new lattice DFT to calculate the adsorption of polymers at solid-liquid interface. Finally, we develop a framework based on the strong segregation limit (SSL) theory to predict the morphologies of micro-phase separation of diblock copolymers confined in curved surfaces. 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

One of the most important points to be discussed in this section is the mutual influence of the bulk and the boundary part of the medium confined by the curved interface on their melting behavior. From Equation 9.38 it follows that the shift of the triple point temperature for the sublayers of a curved boundary phase will differ from that for the case of plane interface (as described by Equation 9.22), first fall due to the effect of curvature described by the terms in the second set of brackets (within []). This shift, in turn, affects the values of y l and nl (because of their temperature dependence), which should be substituted into the first set of brackets and the magnitude of the effective latent heat of fusion (see the discussion after Equation 9.36). As a mle, the terms in the first and in the second sets of brackets act in the same direction (while it is not necessary for the general case). This synergetic action makes the sequential phase transitions in boundary sublayers more pronounced and more separated on the temperature scale compared to the case of plane interfaces. Therefore, the phase transitions in deeper boundary sublayers become experimentally detectable. This effect is revealed in most experimental methods as an apparent increase in the thickness of a skin layer (melted layer in the case of premelting) for the curved interfaces [11, 60]. 

Another aspect of the mutual influence of the bulk and the boundary part of the medium confined by the curved interface on their melting behavior consists in the effect of the existence of premelted (prefreezed) sublayers on the shift of the triple point for the rest of the particle in comparison to T°. Equations 9.37 and 9.39 already account for this effect if one correctly uses the value of Ysv for solid particle having melted sublayers in the case of premelting or Ylv for liquid particle having... 

It is not obvious why (13.1.31) is called an electrocapillary equation. The name is a historic artifact derived from the early application of this equation to the interpretation of measurements of surface tension at mercury-electrolyte interfaces (1-4, 6-8). The earliest measurements of this sort were carried out by Lippmann, who invented a device called a capillary electrometer for the purpose (9). Its principle involves null balance. The downward pressure created by a mercury column is controlled so that the mercury-solution interface, which is confined to a capillary, does not move. In this balanced condition, the upward force exerted by the surface tension exactly equals the downward mechanical force. Because the method relies on null detection, it is capable of great precision. Elaborated approaches are still used. These instruments yield electrocapillary curves, which are simply plots of surface tension versus potential. 

While the barrier confines the amphiphiles to a smaller area, the force exerted by the monolayer is continuously measured and a surface pressure (I7)-area (A) isotherm can be drawn at a constant temperature. This curve plots the surface pressure (force per unit length) versus the mean molecular area occupied by the amphiphiles at the air/water interface. Usually, a U-A isotherm shows four interesting regions [21]. An initial horizontal region where the mean molecular area is large and the interaction between molecules is small so the surface pressure is approximately constant. The first linear region deviates from the... 

Figure 18 (a) Binodals of a symmetric, binary polymer blend confined into a film of thickness D= 2.6/ e as obtained by self-consistent field calculations. The strength of preference at one surface is kept constant. The surface interactions at the opposite surface vary, and the ratio of the surface interactions is indicated in the key. +1.0 corresponds to a strictly symmetric film, and -1.0 marks the interface localization-delocalization transition that occurs in an antisymmetric film. The dashed curve shows the location of the critical points. Filled circles mark critical points and open circles/dashed horizontal lines denote the three-phase coexistence (triple point) for - 0.735 and -1.0. The inset presents part of the phase boundary for antisymmetric boundaries, (b) Schematic temperature dependence for antisymmetric boundaries. The three profiles correspond to the situations (u), (m), and (I) in the inset of (a), (c) Coexistence curves in the// /-A/y plane. The ratio of surface interactions varies according to the key. The analogs of the prewetting lines for A//pw< 0 and ratios of the surface interactions, -0.735 and -1.0, are indistinguishable, because they are associated with the prewetting behavior of the surface with interaction, which attracts the A-component. Reproduced from Muller, M. Binder, K. Albano, E. V. Europhys. Lett. 2000, 50, 724-730, with authorization of http //epljournal.edpsciences.org/... 

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