Black film equilibrium

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. 

Figure 14. Typical FRAP data curves obtained with (a) 2 mM SDS in 2 mM sodium phosphate buffer, pH 7.0 containing 0.1 M NaCl and 14 /xM ODAF (b) FITC-BSA (0.5 mg/ml) in distilled water, pH 8.0, at an equilibrium film thickness of 83 nm (c) FITC-BSA (0.2 mg/ml) in 50 mM Na acetate buffer, pH 5.4 at an equilibrium common black film thickness of 14 nm.

Between the time a liquid film is first formed, until it thins to an equilibrium black film, the thickness can be measured by techniques based on the interference of light. For monochromatic light, the intensity of light, IR, reflected at angle 8 from a fluid film compared to the maximum intensity yielded by constructive interference, I0, enables the determination of the film thickness, t, as ... 

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. 

II.G. Model Calculations. In what follows it will be shown that the general behavior discussed above is consistent with realistic calculations. An important parameter, which is however unknown, is the equilibrium constant of the association—dissociation equilibrium, Kv. The dissociation constant of the ion pair NaSQr was estimated to be Ku = 10 0 7 mol/dm3 = 0.1995 mol/dm3.11 Because of the repulsion among the headgroups on the interface, the dissociation constant is expected to be lower in the present case. In what follows, we will use Ku = 0.050 mol/dm3. For this value, the pressures at which the transitions from the common to the Newton black films occur are in agreement with the experiments of Exerowa et al.2... 

III.B. The Role of Thermal Fluctuations on the Transition from Common Black Films to Newton Black Films. The method described in the previous section will be now applied to thin films with fluctuating interfaces, with the interaction energy calculated as in section II.G. For low values ofthe external pressure, the enthalpy has two metastable minima at Zk and 2c, and a stable one at 2 - 0 (the former two correspond to the Newton and to the common black films, respectively, and the latter implies the rupture of the film), separated by two maxima located at Z and 22 (see Figure 7a). At metastable equilibrium the distances between the surfaces are distributed between 21 and 22 for the Newton black film and between z2 and 2 —°° for the common black film. The stability of the metastable states depends on the chance for a small area S of the interface to reach the... 

In what follows, it will be considered that the lifetime of a metastable state exceeds the duration of experiment if the potential barrier for a small surface of area S, SAH, exceeds 3kT. For the first minimum in Figure 7a, the potential barrier is higher than 3kT for Kc > 20 x 10 19 J hence the Newton black films, once formed, remain stable in cases 4 and 5 during the experiment. In contrast, if the interfaces are more flexible (low Kc, cases 1,2, and 3), the probability for an individual small surface to reach the height which separates two minima is much higher. In this case, the metastable equilibrium of the Newton black film has a shorter lifetime while the film can either rupture or have a transition to a common black film, the second process has a higher chance. On the other hand, the common black films are more stable, because, while the minimum of the enthalpy is higher, the interfaces have more room to fluctuate. 

The fluctuations increase the equilibrium thicknesses of the films (see Figures 5, 6a, and 7a), an effect which is particularly important for the common black films (see Figure 7a). This effect was found in the experiments of Exerowa et al.,2 who noted that the DLVO theory (for planar surfaces) predicts a too small repulsion and suggested that the hydration force (which has a shorter range) cannot account for the discrepancy. 

An equilibrium black film of radius rz and contact angle 6z is formed from the initial non-equilibrium thick film having parameters r and d = 0 during its expansion at V = const. The value of 0z can be determined from the experimentally measured values of r(, r2, 6 and R (the tube radius) according to the formula... 

This device was used in the study of the kinetics of common thin film thinning [16,106], in the determination of the critical thickness of rupture of macroscopic films having an area of about 1 cm2 as well as in the measurement of black film thickness [107]. In equilibrium films this technique does not give reliable results, since there are difficulties in the evaluation of the capillary pressure in the menisci. 

As it is well known, the contacts between drops (in emulsions), solid particles (in suspensions) and gas bubbles (in foams) are accomplished by films of different thickness. These films, as already discussed, can thin, reaching very small thickness. Observed under a microscope these films reflect very little light and appear black when their thickness is below 20 nm. Therefore, they can be called nano foam films. IUPAC nomenclature (1994) distinguishes two equilibrium states of black films common black films (CBF) and Newton black films (NBF). It will be shown that there is a pronounced transition between them, i.e. CBFs can transform into NBFs (or the reverse). The latter are bilayer formations without a free aqueous core between the two layers of surfactant molecules. Thus, the contact between droplets, particles and bubbles in disperse systems can be achieved by bilayers from amphiphile molecules. 

This foam film with a smaller equilibrium thickness hi is called Newton black film (NBF). Its point of equilibrium is situated on the rising left hand side of the isotherm and, alike the preceding minimum, is not described by the DLVO-theory. In Section 3.3 it was shown that the departure from the DVLO-theory begins to be expressed in the experimentally obtained fl(/i) isotherms at film thickness below 20 nm [254]. There are many other experimental data on black foam films [e.g. 18,96,201,202,253,254] which also indicate a deviation of the 1T(/i) isotherm from DLVO-theory that cannot be explained even if the various corrections reflecting the theory refinements are accounted for [e.g. 148,166,171,172,221,255-259]. One of the divergences from the DVLO-theory is the discrepancy between experimental and theoretical data about the interaction energy in black films. 

The results of the measurements equilibrium thickness of foam films from lyso PC as a function of NaCl concentration are shown in Fig. 3.49. At low electrolyte concentration thick equilibrium films that gradually decreased in thickness with increase in Cei were formed. When Cei exceed 10 3 mol dm 3, black spot formation occurred and spontaneous transition from silver to 7.6 nm thick black films was observed in some experiments. At 1.3-10 3 mol dm 3 NaCl predominantly black films were formed. 

Quantitative studies performed by Bulgarian and Dutch scientists [e.g. 14,95,159,160] in the period of 1962 - 1964 proved that two different equilibrium states of black films exist which are realised under certain conditions, i.e. capillary pressure, electrolyte and surfactant concentration, film radius, etc. Studies with macroscopic film [e.g. 308] under a variety of conditions confirmed that fact. 

The analysis of the above techniques (Section 3.4.2.2) developed to estimate the conditions under which stable CBF and NBF exist, and reveals the equilibrium character of the transition between them and the particular features of the two types of black films. Furthermore the difference between the techniques of investigation as well as the difference between their intrinsic characteristics proves to be a valuable source of information of these thinnest liquid formations. The transition theory of microscopic films evidences the existence of metastable black films. Due to the deformation of the diffuse electric layer of the CBF, the electrostatic component of disjoining pressure 1 L( appears and when it becomes equal to the capillary pressure plus Ylvw, the film is in equilibrium (in the case of DLVO-forces). As it is shown in Section 3.4.2.3, CBF exhibit several deviations from the DLVO-theory. The experimentally obtained value of ntheoretically calculated. This is valid also for the experimental dependence CeiiCr(r). Systematic divergences from the DLVO-theory are found also for the h(CeiXr) dependence of NaDoS microscopic films at thickness less than 20 nm. 

The linear energy of the contact line in three-phase equilibrium system could have either positive or negative values. This does not violate the mechanical equilibrium stability condition in such systems. This is proved experimentally by determining k in the case of liquid black films in equilibrium with bulk solutions. The absolute values of k obtained are less than about 10 9 J m 1 (10 4 dyn) they are positive at lower and negative at higher electrolyte (NaCl) concentrations. 

The process of expansion of an emulsion film is also quite similar to that of black spots in a foam film at low electrolyte concentrations the spots in the emulsion film expand slowly, at high concentrations the process is very fast (within a second or less) and ends up with the formation of a black film with large contact angle with the bulk phase (meniscus). In the process of transformation of the black spots into a black film, the emulsion film is very sensitive to any external effects (vibrations, temperature variations, etc.) in contrast to the equilibrium black foam film. 

The mechanism of the equilibrium elasticity acts until it is possible to provide a surfactant re-partition between the exterior and interior of the film. In a NBF such a repartition is not possible and this mechanism of elasticity ceases to act. The elasticity properties of bilayer films, in which the hydrodynamic and adsorption processes are characterised with normal time of relaxation, are due to Marangoni effect in the insoluble adsorption layers. That is why stable foams with black films are very sensitive to different local disturbances (heating, vibration, etc.). 

Indeed, a direct relationship between the lifetimes of films and foams and the mechanical properties of the adsorption layers has been proven to exist [e.g. 13,39,61-63], A decrease in stability with the increase in surface viscosity and layer strength has been reported in some earlier works. The structural-mechanical factor in the various systems, for instance, in multilayer stratified films, protein systems, liquid crystals, could act in either directions it might stabilise or destabilise them. Hence, quantitative data about the effect of this factor on the kinetics of thinning, ability (or inability) to form equilibrium films, especially black films, response to the external local disturbances, etc. could be derived only when it is considered along with the other stabilising (kinetic and thermodynamic) factors. Similar quantitative relations have not been established yet. Evidence on this influence can be found in [e.g. 2,13,39,44,63-65]. 

FIGURE 5.36 Main stages of formation and evolution of a thin liquid film between two bubbles or drops (a) mutual approach of slightly deformed surfaces (b) at a given separation, the curvature at the center inverts its sign and a dimple arises (c) the dimple disappears, and eventually an almost plane-parallel film forms (d) due to thermal fluctuations or other disturbances the film either ruptures or transforms into a thinner Newton black film (e), which expands until reaching the final equilibrium state (f). 

After the entire film area is occupied by the Newton black film, the film radius increases until it reaches its equilibrium value, R = (Figure 5.36f). Finally, the equilibrium contact angle is established. For more details about this last stage of film thinning, see part IV.C of Reference 164. 

The equilibrium thickness of foam films drawn from solutions of DMS containing potassium thiocyanate at various concentrations are shown in Figure 2. In this case first black films were formed with a thickness of 975 A at a salt concentration of 4 X 10"4 mole/dm3. The thickness decreased with increasing salt concentration and reached 59 A, those of a second black film, at salt concentrations of the order of 0.5 mole/dm3 potassium thiocyanate. 

At high electrolyte concentrations the films become so thin that they loose ability to reflect light there are the so-called common black films. In addition to that, an increase in electrolyte concentration results in a decrease of the height of potential barrier which preserves the film in the state of this metastable equilibrium, i.e., film stability decreases. Thermal oscillations of interface, i.e., the Mandel shtam waves (See Chapter VI, 1), help the system to overcome a potential barrier. If other stabilizing factors are absent, such (local) overcoming of potential barrier results in film rupture. 

Fig. 2D.3. demonstrates that an equilibrium film, the so-called common black film can reach a critical thickness at which it ruptures due to surface disturbances. Vrij (1966) studied surface fluctuations theoretically on the basis of Mandelstam s theory and computer simulations. Newton black film rupture was studied experimentally and theoretically by Exerowa et al. (1982) and Exerowa Kachiev (1986). They assume the existence of vacancies in the film. The mobility of these vacancies is the mechanism which controls the film stability (Fig. 2D.7),... 

A recently measured disjoining-pressure isotherm of an isolated lamella is shown in Figure 8 for the surfactant sodium dodecyl sulfate (SDS) at 10-3 kmol/m3 in aqueous 0.01 kmol/m3 sodium chloride brine (65). A solid line connects the data points for three independent experimental runs, shown by various symbols. The negative, attractive portion of the isotherm between thicknesses of about 4 and 5 nm is not sketched because equilibrium measurements are not possible there. The measured isotherm indeed obeys the classic S-shape. Film meta-stability demands that the slope of the isotherm be negative (2<5, 72). For positive slopes, even the slightest, infinitesimal disturbance ruptures the film. Thus, the lamella in Figure 8 can exist only along the two repulsive branches near 4 nm and above 7 nm. The thicker branch or common black film arises from electrostatic overlap forces, and the inner branch or Newton black... 

For the static, trapped lamellae, the film reaches an equilibrium thickness set by the local capillary pressure and the film curvature in obedience to equation 1. The capillary pressure, in turn, depends, on the wetting-liquid saturation the film curvature depends on the particular location within the pore structure dictated by an approximately 90° contact angle with the pore wall. As the capillary pressure in the porous medium rises during drainage, the film thickness decreases along the common black branch until nmax is reached, and a Newton black film of 4-nm thickness emerges. 

Black Film Fluid films yield interference colors in reflected white light that are characteristic of their thickness. At a thickness of about 0.1 /xm, the films appear white and are termed silver films. At reduced thicknesses, they first become grey and then black (black films). Among thin equilibrium (black) films, one may distinguish those that correspond to a primary minimum in interaction energy, typically at about 5-nm thickness (Newton black films) from those that correspond to a secondary minimum, typically at about 30-nm thickness (common black films). 

The presence of attractive colloidal forces becomes apparent in the black spots." The black film appears to be in a sort of metastable equilibrium state with a finite thickness in the colloidal size range. What are... 

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