Equilibrium thickness

There apparently exists a critical amount of liquid phase for the optimization of grain/interface boundary sliding during superplastic deformation. The optimum amount of liquid phase may depend upon the precise material composition and the precise nature of a grain boundary or interface, such as local chemistry (which determines the chemical interactions between atoms in the liquid phase and atoms in its neighboring grains) and misorientation. The existence of an equilibrium thickness of intergranular liquid phase in ceramics has been discussed [14]. This area of detailed study in metal alloys has not been addressed. 

Flow thins protective film to equilibrium thickness which is a function of both mass transfer rate and growth kinetics. Erosion corrosion rate is controlled by the dissolution rate of the protective film. 

The equilibrium state of the layer is found by minimization of F with respect to L, bearing in mind that

varies inversely with L. Both approaches (Equations 2 and 4) yield the correct equilibrium thickness ... 

The shape of a droplet or of the front end of a film can be determined from the surface energies and interaction forces between the interfaces. These also determine the equilibrium thickness of a liquid film that completely wets a surface. The calculation is done by minimization of the free energy of the total system. In a two-dimensional case the free energy of a cylindrical droplet can be expressed as [5] ... 

Let us consider a circular puddle of liquid, L, on solid, S, in the presence of liquid vapor, V. The puddle is of radius Rq and small initial thickness Bq. We assume that holes nucleate spontaneously in the puddle and grow with radius r(t) as time t passes because the equilibrium contact angle. Go, is nonzero. The liquid is unstable as a wetting film. Equilibrium thickness of a film, < is given by [27,28]... 

A vertical belt is moving upward continuously through a liquid bath, at a velocity V. A film of the liquid adheres to the belt, which tends to drain downward due to gravity. The equilibrium thickness of the film is determined by the steady-state condition at which the downward drainage velocity of the surface of the film is exactly equal to the upward velocity of the belt. Derive an equation for the film thickness if the fluid is (a) Newtonian (b) a Bingham plastic. 

Measurements have been carried out on the excess tensions, equilibrium thicknesses, and compositions of aqueous foam films stabilized by either n-decyl methyl sulfoxide or n-decyl trimethyl ammonium-decyl sulfate, and containing inorganic electrolytes. 

Fig.9 Typical F/R vs. D curve between the PMMA brush (L = 87nm, Mn = 121700, Mw/Mn = 1.39) and the sUica probe (attached on an AFM cantilever). The arrowheads indicate critical distances is the equilibrium thickness at which a repulsive force is detectable, and Do is the offset distance beyond which the brush was no more compressible...

The equilibrium thickness of a (meta-)stable soap film will depend on the strength and range of the repulsive forces in the film. Electrostatic forces are long-range in water and hence give rise to thick (0.2 micron) films, which are highly coloured due to the interference of visible light... 

In this situation, the equilibrium thickness at any given height h is determined by the balance between the hydrostatic pressure in the liquid (hpg) and the repulsive pressure in the film, that is n = hpg. Cyril Isenberg gives many beautiful pictures of soap films of different geometries in his book The Science of Soap Films and Soap Bubbles (1992). Sir Isaac Newton published his observations of the colours of soap bubbles in Opticks (1730). This experimental set-up has been used to measure the interaction force between surfactant surfaces, as a function of separation distance or film thickness. These forces are important in stabilizing surfactant lamellar phases and in cell-cell interactions, as well as in colloidal interactions generally. 

The equilibrium thickness of most bubbles is so much less than the radius of curvature of the bubble that the air masses can be regarded as blocks with planar faces like those in our models. In research studies on such systems, the bubbles are allowed to equilibrate on frames that make their compliance with the model even better. 

If a soap film is sufficiently thin, its equilibrium thickness is the result of the double-layer repulsion, given by Equation (82), and van der Waals attraction, given by... 

Fig. 10. Plot of normalized approach to equilibrium thickness against the square root of time for a temperature-sensitive 10x4 PNIPAAm gel sheet swelling and shrinking between 10 and 25 °C. Shown are the curve fits to the kinetic data of theory developed from equations of motion (Tanaka and Fillmore theory) [60]. The equilibrium degree of swelling is 17.0 at 10°Cand 11.1 at 25 °C the diffusion coefficients obtained from the curve fits are 2.3 x 10"7 ctn2/s for swelling and 3.6 x 10 7 cm2/s for shrinking [121]...

Lyklema and Vliet8 determined the equilibrium thickness to of free liquid films stabilized by poly(vinyl alcohol) (PVA) adsorbed at the air-water interface. They estimated to at different applied hydrostatic pressures by measuring the intensities of light reflected from the surface of the film to that of the silvery film. The to values obtained increased with rising hydrostatic pressure and were extrapolated to zero pressure to obtain to for a free film. The extrapolated to should correspond to twice the thickness of the adsorbed PVA layer, but it far exceeded twice the latter determined by ellipsometry. The great difference was interpreted in terms of the presence of long dangling tails which are probably not to be seen by ellipsometry. 

The observed equilibrium thickness represents the film dimensions where the attractive and repulsive forces within the film are balanced. This parameter is very dependent upon the ionic composition of the solution as a major stabilizing force arizes from the ionic double layer interactions between any charged adsorbed layers confining the film. Increasing the ionic strength can reduce the repulsion between layers and at a critical concentration can induce a transition from the primary or common black film to a secondary or Newton black film. These latter films are very thin and contain little or no free interlamellar liquid. Such a transition is observed with SDS films in 0.5 M NaCl and results in a film that is only 5 nm thick. The drainage properties of these films follows that described above but the first black spot spreads instantly and almost explosively to occupy the whole film. This latter process occurs in the millisecond timescale. 

The solution diffusion properties of FITC-labelled BSA were measured by FRAP [12], The results showed that the protein diffused freely in solution with a diffusion coefficient of approximately 3xl0 7 cm2/s. This was in reasonable agreement with previously published values [36]. FRAP measurements were also made on thin films stabilized by FITC-BSA. The films were allowed to drain to equilibrium thickness before measurements were initiated. Thin films covering a range of different thicknesses were studied by careful adjustment of solution conditions. BSA stabilized films that had thicknesses up to 40 nm showed no evidence of surface diffusion as there was no return of fluorescence after the bleach pulse in the recovery part of the FRAP curve (Figure 14(c)). In contrast, experiments performed with thin films that were > 80 nm thick showed partial recovery (55%) of the prebleach level of fluorescence (Figure 14(b)). This suggested the presence of two classes of protein in the film one fraction in an environment where it was unable to diffuse laterally, as seen with the films of thicknesses < 45 nm, and a second fraction that was able to diffuse with a calculated diffusion coefficient of lxlO 7 cm2/s. This latter diffusion coefficient was 3 times slower than that... 

By adding the function u(x) to the phase factor in (4) one can describe departures from the planar (lamellar, one-dimensional) layer arrangement, which is characteristic for the 2D structures. The first term in (3) is the smectic layer compressibility energy. It is zero when layers are of the equilibrium thickness. If cx(T) > 0, the second term in (3) requires the director to be along the smectic layer normal (the smectic-A phase). If c (T) < 0, this term would prefer the director to lie in the smectic plane. So the last term in (3) is needed to stabilize a finite tilt of the director with respect to the smectic layer normal. In addition this term gives the energy penalty for the spatial variation of the smectic layer normal. 

Modern coating technologies require increasingly thinner polymer films. This requirement is opposed by the surface pressure and the chain elasticity. Below a certain equilibrium thickness, the film is either metastable or even unstable and tends to break into droplets regardless of the chemical structure of the substrate [321, 322]. Anomalous wetting behaviour was observed for amphiphilic polymer films whose stability is controlled by the orientation of the surface active moieties [323,324]. All these phenomena belong to the dewetting problem. 

Equilibrium thickness measured by optical methods (interference fringes, and reflectance techniques)... 

Clarke, D.R., Shaw, T.M., Philipse, A.P. and Horn, R.G., Possible electrical doublelayer contribution to the equilibrium thickness of intergranular glass films in polycrystalline ceramics ,/. Am. Ceram. Soc., 1993 76(5) 1201-1204. 

For a suitably high critical value of A, this theoretical model predicts a lower limit on the equilibrium thickness that can be observed. This lower limit on Z, Z n, is defined by the conditions F= 0 and <1F/<1L = 0 since for a stable film F= 0 and dF/dL > 0 (Clarke, 1987). Various solutions to these conditions have been examined by Knowles and Turan (2000). In the absence of capillary pressure and external pressure, Zmin = 2.58. Using reasonable estimates for Knowles and Turan estimate Zmin to be >6.50 A. That in practice the observed intergranular film thicknesses are typically of the order of 1-2 nm in non-oxide engineering ceramics indicates that the relevant Hamaker constants for ceramics are significantly lower than the critical value. 

Experiments on the stability of water/surfactant films at various pressures were performed by Exerowa et al.2,3 For a dilute aqueous solution of a nonionic surfactant,3 tetraoxyethylene decyl ether (D(EO>4,5 x 10-4 mol/dm3) or eicosaoxyethylene nonylphenol ether (NP(EO)2o, 1 x 10-5 mol/dm3), and electrolyte (KC1), thick films (with thicknesses of the order of 100 A) were observed at low electrolyte concentrations. With an increase of the electrolyte concentration, the film thickness first decreased, which suggests that the repulsion was caused by the double layer. This repulsive force was generated because of the different adsorptions of the two species of ions on the water/ surfactant interface. At a critical electrolyte concentration, a black film was formed, and the further addition of electrolyte did not. modify its thickness, which became almost independent of the external pressure, until a critical pressure was reached, at which it ruptured. While for NP(EO)2o only one metastable equilibrium thickness was found at each electrolyte concentration, in the case of D(EO)4 a hysteresis of the film thickness with increasing and decreasing pressure (i.e., two metastable minima) was observed in the range 5 x 10 4 to 3 x 10 mol/dm3 KC1. The maximum pressure used in these experiments was relatively low, 5 x 104 N/m2, and the Newton black films did not rupture in the range of pressures employed. 

The purpose of this article is to present a model and to calculate on its basis the metastable equilibrium thicknesses of the film as a function of the applied pressure. In section II, the interaction energy of the film was calculated, assuming planar interfaces free of thermal fluctuations. The double layer interaction was calculated by accounting for the charge recombination at the surface with increasing electrolyte concentration. An approximate... 

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