Disjoining pressure theory

The disjoining pressure theory by Churaev et al. [6] begins with the definition of the disjoining pressure, n. There is a quantity that is a function of the coverage, F, or adsorbed film thickness , t, defined by the equation (for the theory t and F can be used interchangeably)... 

As noted earlier in this chapter, there is definite relationship between the disjoining pressure theory of adsorption and the x theory. In this section, some thermodynamic relationships for the spreading pressure are derived. It is questionable at this point how useful these relationships will be. They may be useful in extending the theory into the solution chemistry since these relationships are important in that area of research. 

Although the parameters are given symbols that would imply some interpretation, one may at this point assign whatever interpretation one wishes to these parameters. In the next section, these parameters will be interpreted in terms of the x theory (or equally so, the disjoining pressure theory). 

Fig. 2. Effective interface potential (left) and corresponding disjoining pressure (right) vs film thickness as predicted by DLVO theory for an aqueous soap film containing 1 mM of 1 1 electrolyte. The local minimum in H(f), marked by °, gives the equiHbrium film thickness in the absence of appHed pressure as 130 nm the disjoining pressure 11 = —(dV/di vanishes at this minimum. The minimum is extremely shallow compared with the stabilizing energy barrier.

Comparison of the proposed dynamic stability theory for the critical capillary pressure shows acceptable agreement to experimental data on 100-/im permeability sandpacks at reservoir rates and with a commercial a-olefin sulfonate surfactant. The importance of the conjoining/disjoining pressure isotherm and its implications on surfactant formulation (i.e., chemical structure, concentration, and physical properties) is discussed in terms of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of classic colloid science. 

Disinfection processes, as advanced wastewater treatment, 25 909-910 Disjoining pressure, 12 6-7 Disk agglomeration, 19 8 Disk atomizer spray dryer, 9 127 Disk bowl centrifuge, theory of performance, 5 518 Disk centrifuges, 18 144... 

Using Poisson-Boltzmann theory we can derive a simple expression for the disjoining pressure. For the linear case (low potentials) and for a monovalent salt, the one-dimensional Poisson-Boltzmann equation (Eq. 4.9) is... 

Figure 10.4 shows the results of some measurements on aqueous sodium oleate films. The sensitivity of the equilibrium film thickness to added electrolyte reflects qualitatively the expected positive contribution of electric double layer repulsion to the disjoining pressure. However, this sensitivity to added electrolyte is much less than that predicted from electric double layer theory and at high electrolyte concentration an equilibrium film thickness of c. 12 nm is attained which is almost independent of the magnitude of the disjoining pressure. To account for this observation, Deryagin and Titijevskaya have postulated the existence of hydration layers... 

The principles of colloid stability, including DLVO theory, disjoining pressure, the Marangoni effect, surface viscosity, and steric stabilization, can be usefully applied to many food systems [291,293], Walstra [291] provides some examples of DLVO calculations, steric stabilization and bridging flocculation for food colloid systems. 

The development of the thermodynamics of thin films is related to the problem of stability of disperse systems. An important contribution to its solving are the works of the Russian scientists Derjaguin and Landau [1] and the Dutch scientists Verwey and Overbeek [2], known today as the DVLO theory. According to their concept the particular state of the thin liquid films is due to the change in the potential energy of molecular interaction in the film and the deformation of the diffuse electric layers. The thermodynamic characteristic of a state of the liquid in the thin film, as shown in Section 3.1, appears to be the dependence of disjoining pressure on film thickness, the n(/t) isotherm. The thermodynamic properties of... 

The DLVO-theory considers only the molecular van der Waals and electrostatic interactions. A complete analysis of the theory can be found in several monographs [e.g. 3-6] where original and summarised data about the different components of disjoining pressure in thin liquid films, including in foam films are compiled. 

According to DLVO-theory the disjoining pressure in thick films is considered as a sum of the electrostatic and van der Waals component... 

Two theories, macroscopic and microscopic, are involved in the calculation of the van der Waals component of disjoining pressure in thin liquid films. According to the microscopic theory, first treated by Kallman and Willstatter [145], de Boer [146] and Hamaker [147], the... 

A general formula for calculation of the dispersion molecular interactions in any type of condensed phases has been proposed in [148], The attraction between bodies results from the existence of fluctuational electromagnetic field of the substance. If this field is known in a thin film, then it is possible to determine the disjoining pressure in it. The more strict macroscopic theory avoids the approximations assumed in the microscopic theory, i.e. additivity of forces integration extrapolation of interactions of individual molecules in the gas to interactions in condensed phase. The following function for IIvw was derived in [148] for thick free films... 

A number of works are dedicated to the experimental verification of DVLO theory to foam films. As shown above, the disjoining pressure is given as a sum of ne/ and nvw, i.e. 

The relation between film thickness h and electrolyte concentration Cei obeys the DLVO-theory of electrostatic disjoining pressure (see Eqs. (3.71) and (3.72)). At film equilibrium and known h and Cei it is possible to calculate at the solution/air interface [95,155-157,169,170-173], Hence, a new area in the study of electrosurface forces at this interface has been developed on the basis of determining potential. 

Formation and stability studies of black foam films can be summarised as follows 1) surface forces in black foam films direct measurement of disjoining pressure isotherm DLVO- and non-DLVO-forces 2) thin foam film/black foam film transition establishing the conditions for the stability of both types of black films and CBF/NBF transition 3) formation of black foam films in relation to the state of the adsorption layers at the solution/air interface 4) stability of bilayer films (NBF) theory and experimental data. 

In Section 3.3.1 it was shown that the state of thin foam films is described by the Fl(/ ) isotherm of disjoining pressure. For relatively thick films, stabilised by surfactants, this isotherm is consistent with the DLVO-theory. However, black foam films exhibit a diversion from the DLVO-theory which is expressed in the specific course of the disjoining pressure isotherm. 

At equilibrium film thickness hi the disjoining pressure equals the external (capillary) pressure, n = p This is a common thin film and its equilibrium is described by the DLVO-theory. If h < hcr, at which the film ruptures (see Section 3.2.2), the film is common black (schematically presented in Fig. 3.42). Such a film forms via black spots (local thinnings in the initially thicker non-equilibrium film). The pressure difference nmax - pa is the barrier which hinders the transition to a film of smaller thickness. According to DLVO-theory after nmax the disjoining pressure should decrease infinitely. Results from measurements of some thermodynamic parameters of foam films [e.g. 251,252] show the existence of a second minimum in the 17(6) isotherm (in direction of thickness decrease) after which the disjoining pressure sharply ascends. 

Other discrepancies between the black film behaviour and DLVO-theory are related to the difference in the critical electrolyte concentration, corresponding to the transition between the two black films types (see Section 3.4.2) the existence of a second minimum in the 11(A) isotherm the sharp rise in the disjoining pressure (after the second minimum). All this is evidenced by the measurements of contact angles between the film and bulk phase. 

Another option to reach an agreement between theoretical and experimental isotherms is provided by the assumption that the shift observed is due to structural interactions in the film which determines the structural component of disjoining pressure ns, [5,312], In that context it is interesting to estimate the function ln(nexp - ITiheor) on h, presented in Fig. 3.60. It is plotted at different NaCl concentrations under the assumption that at constant ( -potential and at Cei = 10 4 and 10 3 mol dm 3, the DVLO-theory is conformed with. 

The CBF/NBF transition has already been considered in Section 3.4.1 with respect to the experimental n(/i) isotherms of disjoining pressure obtained with the Thin Liquid Film-Pressure Balance Technique. Theoretical concepts and comparison with the DLVO- and contemporary theories describing surface forces acting in this range of film thicknesses have also been discussed. 

Ceicr of CBF/NBF transition can be calculated on the basis of the DLVO-theory from the equation of disjoining pressure in thin liquid films. If the approximate Eq. (3.74) is employed... 

If we substitute the expression Eq. (2.4), for pa in Eq. (3.95), a relation about Cei cr as a function of the film radius r can be derived. Calculations from the latter yield a value of Cei,Cr about 5 times lower than the experimentally determined in the range of film radii studied. Probably this is caused by the properties of CBF that deviate considerably from the DLVO-theory (see Section 3.4.1.3). The improvement of DLVO-theory by introducing additional components of disjoining pressure will specify the Ceicr r) dependence. It should be bom in mind that in films of large areas the probability for transition from metastable to stable state is... 

The experimentally observed fact that a definite surfactant concentration is needed to overcome the barrier nmax of disjoining pressure deserves attention [323,332], This is one of the main conditions in the CBF/NBF transition which is not considered by the DLVO-theory (see Section 3.4.3). 

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. 

Non-equilibrium liquid films formed in the process of spreading have been considered in some early works, especially in the test of the theory of interfacial tension and the rule of Antonov [204], A review on the rule of Antonov and its interpretation on the basis of isotherms of disjoining pressure in wetting films is presented in [532]. However, these works do not deal with precise measurement of film thickness and the studies confined only the kinetics of spreading and lens formation. 

Another point of view concerning foam stability appeared in relation to the development of the general theory of stability of colloid systems (DLVO-theory). It has already been noted that this theory was verified for the first time with foam films [35]. This gave rise to the concept of foam stabilisation on the account of the electrostatic component of disjoining pressure [e.g. 24, 32, 36],... 

---