Disjoining pressure Hamaker constant

Fig. 4 illustrates the time-dependence of the length of top s water column in conical capillary of the dimensions R = 15 pm and lo =310 pm at temperature T = 22°C. Experimental data for the top s column are approximated by the formula (11). The value of A is selected under the requirement to ensure optimum correlation between experimental and theoretical data. It gives Ae =3,810 J. One can see that there is satisfactory correlation between experimental and theoretical dependencies. Moreover, the value Ae has the same order of magnitude as Hamaker constant Ah. But just Ah describes one of the main components of disjoining pressure IT [13]. It confirms the rightness of our physical arguments, described above, to explain the mechanism of two-side liquid penetration into dead-end capillaries. 

Aa being the Hamaker constant for the interaction of i through vacuum. When the film thickness Xp 7Ls, the disjoining pressure Hyw is given by... 

From the above equation, the variation of equilibrium disjoining pressure and the radius of curvature of plateau border with position for a concentrated emulsion can be obtained. If the polarizabilities of the oil, water and the adsorbed protein layer (the effective Hamaker constants), the net charge of protein molecule, ionic strength, protein-solvent interaction and the thickness of the adsorbed protein layer are known, the disjoining pressure II(x/7) can be related to the film thickness using equations 9 -20. The variation of equilitnium film thickness with position in the emulsion can then be calculated. From the knowledge of r and Xp, the variation of cross sectional area of plateau border Qp and the continuous phase liquid holdup e with position can then be calculated using equations 7 and 21 respectively. The results of such calculations for different parameters are presented in the next session. 

With respect to the molecular interactions the simplest asymmetric films are these from saturated hydrocarbons on a water surface. Electrostatic interaction is absent in them (or is negligible). Hence, of all possible interactions only the van der Waals molecular attraction forces (molecular component of disjoining pressure) can be considered in the explanation of the stability of these films. For films of thickness less than 15-20 nm, the retardation effect can be neglected and the disjoining pressure can be expressed with Eq. (3.76) where n = 3. When Hamaker s constants are negative the condition of stability is fulfilled within the whole range of thicknesses. 

The rupture mechanisms of thin liquid films were considered by de Vries [15] and by Vrij and Overbeek [16]. It was assumed that thermal and mechanical disturbances (having a wavelike nature) cause film thickness fluctuations (in thin films), leading to the rupture or coalescence of bubbles at a critical thickness. Vrij and Overbeek [16] carried out a theoretical analysis of the hydrodynamic interfacial force balance, and expressed the critical thickness of rupture in terms of the attractive van der Waals interaction (characterised by the Hamaker constant A), the surface or interfacial tension y, and the disjoining pressure. The critical wavelength, for the perturbation to grow (assuming that the disjoining pressure just exceeds the... 

In fact, Equation 5.170 is applicable to the dispersion contribution in the van der Waals interaction. When components 1 and 2 are identical, Ag is positive (see Equation 5.169), therefore, the van der Waals interaction between identical bodies, in any medium, is always attractive. Besides, two dense bodies (even if nonidentical) will attract each other when placed in medium 3 of low density (gas, vacuum). When the phase in the middle (component 3) has intermediate Hamaker constant between those of bodies 1 and 2, Ag can be negative and the van der Waals disjoining pressure can be repulsive (positive). Such is the case of an aqueous film between mercury and gas. ... 

The dimensional coefficient A that appears in (6 84) is known as the Hamaker constant. Its value depends on the materials involved, but a typical magnitude is 10 20 to 10 19 J. Generally A is positive, which corresponds to a positive disjoining pressure and attraction between the interface and the solid substrate. However, in some circumstances, A < 0, and the surfaces repel. 

As mentioned in Section 12.1, the disjoining pressure is given by pdisj = — d V/d h. Differentiating Eq. (12.2) then yields />disj = —4/6 nh3. The disjoining pressure thus is negative, as it should be for an attractive force. According to Table 12.2, the value of An, i-e., for water across air, equals 37 10 21 J, and the value for air across water is virtually the same. We thus obtain />disj = —2000 Pa, a considerable pressure. The value of the Hamaker constant for oil across water depends on the type of oil and is... 

Consequently, for symmetric films the molecular component of disjoining pressure is always negative, which corresponds to a tendency of dispersion medium layer separating identical phases to decrease its thickness. At the same time, one should emphasize that in such systems in the absence of non-dispersion interactions the lower the value of complex Hamaker constant is, the more similar in nature the interacting phases (dispersed phase and dispersion medium) are. If contacting phases are essentially similar in structure and chemical composition, the value of A may be as low as 10 21 J or even much lower. The so low Hamaker constants result in changes in the nature of colloidal stability. 

The first term is the disjoining pressure in the film with A the Hamaker constant describing the van der Waals interaction of the film with the surrounding media. The second term represents the electrostatic pressure exerted on the film by an electrostatic potential difference, U, between the conducting media cladding the film, with s being the dielectric constant of the film material. Finally the third term describes the Laplace pressure in the film with cr donating the interface tension between the film and the upper (liquid) medium. 

FIGURE 1.10 Plot of disjoining pressure isotherm for a plane-parallel air-water-air film with only van der Waals interactions. Here disjoining pressure, n LA, is given by IlALAffO =.Ah/6jc/i where Ah is Hamaker constant of 3.7 x 10" J. (From IsrealachviUi, J.N. Intermolecular and Surface Forces with Applications to Colloidal and Biological Systems, Academic F ress, London, 1985 [50].)... 

Consider now for, example, the likely disjoining pressure-thickness isotherm for a pseudoemulsion film of an oil drop in an aqueous surfactant solution. In these circumstances, and by analogy with the corresponding foam films, the shape of the isotherm is likely to be determined by a combination of electrostatic, van der Waals, steric/hydration, and the so-called hydrophobic forces. Here electrostatic forces favor positive disjoining pressures, stable films, and wetting of the oil. van der Waals forces with positive Hamaker constants, on the other hand, favor negative disjoining pressures and unstable films. 

It would, however, appear to be inappropriate for the systems considered by Denkov and coworkers [34, 35] to foUow Vrij and Overbeek [41] by only allowing for van der Waals forces in substitution of the relevant expression for dIlAwo(Wdfi in Equation 3.23. Kellay et al. [42] suggest that in the case where the oil-water inter-facial tension is extremely low, there may, for example, be a positive contribution to the disjoining pressure from the sterically frustrated amplitude of the fluctuations. The presence of anionic surfactant implies a repulsive electrostatic contribution and the Hamaker constants for alkane-water-air films are almost all negative [43], which also implies a positive contribution to the disjoining pressure from van der Waals forces. The latter would, however, mean that pure water should spontaneously wet hydrocarbons (in the absence of surfactant), which is clearly not correct It has been argued therefore that in this case, the so-called short range hydrophobic forces may be responsible for the instability of pseudoemulsion films of pure water on hydrocarbon oils [44]. 

When the phase in the middle (component 3) has intermediate a Hamaker constant between those of bodies 1 and 2, A can be negative and the van der Waals disjoining pressure can be repulsive (positive). Such is the case of an aqueous film between mercury and gas. 

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