Equilibrium Periodic Films

In this section, we consider the possibility of existence of equilibrium periodic liquid films that are situated completely in the range of the disjoining pressure action (partial wetting, S-shaped disjoining pressure isotherm). Undersaturatiou is under consideration, that is, 0. 

FIGURE 2.8 Equilibrium periodic film, denotes maximum thickness, and h denotes minimum thickness x is the length of the half-period of the film. 

the profile of the equilibrium periodic film is described by the following 

The following condition 0 (ft) y should be satisfied to have the positive expression under the sqnare root in Equation 2.31. 

As we must have ft (ftj = 0 at the position of the minimum height of the film, it follows from Equation 2.31 that 

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In thin fiat liquid films (oil and aqueous thin films, thin films of aqueous electrolyte and surfactant solutions, and both free films and films on solid substrates), the disjoining pressure acts alone and determines their thickness. However, if the film surface is curved or uneven, both the disjoining and the capillary pressures act together. In the case of partial wetting, their simultaneous action is expected to yield nonfiat equilibrium shapes. For instance, due to the S-shaped disjoining pressure isotherm, microdrops, microdepressions, and equilibrium periodic films could exist on fiat solid substrates. We shall establish a criteria for both existence and stability of such nonfiat equilibrium liquid shapes. On the other hand, we... 

Fig. 5 Schematic cross-sections of thin film morphologies of the topographic pattern grown by a graphoepitaxy method. A micropattern with different lamellar domain orientation is shown, a Surface-parallel lamellae, typical of film thickness t greater than the natural equilibrium period Lq. b Surface-perpendicular lamellae, typical of film thickness t less than L0. (adapted from [41])...

Fig.23. Schematic illustration of wetting geometries expected for ultra-thin films of diblock copolymers a - parallel lamellae, b - surface (pinned) micelles, c - perpendicular lamellae. L corresponds to the equilibrium period of the lamellar morphology...

The situation is more complicated when the aqueous solution contains surfactant micelles, which is a common experimental and practical situation. In such a case the disjoining pressure isotherm H(/j) can exhibit multiple decaying oscillations, whose period is close to the diameter of the micelles (Fig. 8b) (for details see, e.g., Ref 78). The condition for equilibrium liquid film, Eq. (42), can be satisfied at several points, denoted by Hq, h, / 2, and in Fig. 8b the corresponding films contain 0,1,2, and 3 layers of micelles, respectively. The transitions between these multiple equilibrium states represent the phenomenon stratification (see Fig. 9 and Refs 78-91). The presence of dis-... 

Figure 4 (a) The measute period relative to /o as a function of d/Lg. The solid lines are the maximum and minimum deviations of the period from Lo (b) The film thickness divided by the repeat period dlL is shown as a function of the ratio of the film thickness to the equilibrium period d/Lg. (c) Reduced lamellar period, A, as a function of reduced plate separation. d=D/Lg, for6=0.15. Wherever the vertical morphology is favored, the bulk equilibrium lamellar period, Lg, is realized, thus A= 1. Reproduced with permission from Walton, D. G. Kellogg, G. J. Mayes, A. M. et al. Macromolecules 9SA, 27,6225. " ... 

Conditions 2.35 completely determine the equilibrium shape of the periodic film. 

In very thin (nanometer) films, where the potential gradient may exceed 10 V/m, another mechanism of ion migration is observed, which involves periodic jumps of ions between equilibrium positions, hi this case, the rate of migration is not proportional to the potential gradient but obeys the exponential law... 

The structure of a vapor-quenched alloy may be either crystalline, in which the periodicity of the unit cell is repeated within the crystallites, or amorphous, in which there is no translational periodicity even over a distance of several lattice spacings. Mader (64) has given the following criteria for the formation of an amorphous structure the equilibrium diagram must show limited terminal solubilities of the two components, and a size difference of greater than 10% should exist between the component atoms. A ball model simulation experiment has been used to illustrate the effects of size difference and rate of deposition on the structure of quench-cooled alloy films (68). Concentrated alloys of Cu-Ag (35-65%... 

These results for spread film and equilibrium spreading suggest that films of racemic N-(a-methylbenzy 1) stearamide may be resolved by seeding the racemic film with crystals of either pure enantiomer. Indeed, when a monolayer of racemic jV- (a-methylbenzyl) stearamide is compressed to 45 A2/molecule (27 dyn cm-1), deposition of a crystal of either R( +)- or S( — )-enantiomer results in a decay of surface pressure from the initial 28 dyn cm-1 film pressure to 3.0 dyn cm-1, the ESP of the enantiomeric systems on a pure 10n sulfuric acid subphase (Table 1). When the experiment is repeated with racemic crystals, the system reaches an equilibrium surface pressure of 11 dyn cm-1, nearly the ESP of the racemic crystal on the clean acidic interface. In either case, equilibrium pressure is reached within a two hour time period. 

Fig. 7 2D thickness-surface energy gradient library for mapping the effects of these parameters on the self-assembly of PS-b-PMMA block copolymer thin films. See text for a fuU description. Lq is the equilibrium self-assembly period and h is the film thickness. Dashed white lines delineate the neutral surface energy region, which exhibits nanostructures oriented perpendicular to the substrate plane. (Derived from [18] with permission)... 

Another real-time study of the reaction of M-FA films with H2S utilized ellip-sometry to monitor changes in film thickness concurrent with metal sulfide formation (53). The reactions appeared to reach equilibrium within the same period of time (within 2 h), with a change per monolayer of 0.2 nm for CdBe and 0.9 nm for both CuBe and ZnBe. Their ellipsometry results, in agreement with Peng et al. (66), also show a dependence of the reaction rate on the H2S pressure and the surface pressure at which the films were deposited. 

In the last chapter we have used the word order without giving it any precise meaning. Most definitions of order involve thermodynamic concepts. Thus, for example, one might say that the most ordered state of a system is the one to which the system tends as the temperature tends to absolute zero. This definition would, however, be of little service in the present context. Most of the systems which we will discuss are remote from thermodynamic equilibrium. This is true both of the films during their preparation and also of the final prepared films. However, these prepared films are in states of metastable equilibrium which are likely to survive for periods long compared with the time taken to carry out experiments on them and, very often, for periods so long as to be, from a human point of view, infinite. 

The amphiphilic material to be studied is dissolved at a known concentration in a volatile solvent which is not miscible with water and a known quantity is spread at the water surface using a micropipette. In order to study the physical properties of the film thus formed, one needs to be able to confine the film to a definite area and to be able to vary this area at will. It might appear that it would be equally possible to maintain a constant area and vary the amount of material which is spread. For the majority of materials this latter procedure is not satisfactory as equilibrium is not arrived at in a reasonable period of time and this method would not allow one to take the material through successive cycles of compression and expansion. We thus turn to a discussion of the various ways in which a film can be confined and its area varied in a systematic manner. Leaving aside methods which are really only of historical interest, for which reference should be made to the book by... 

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