Curved interface, surface tension calculation

Takahashi et al.67) prepared ionene-tetrahydrofuran-ionene (ITI) triblock copolymers and investigated their surface activities. Surface tension-concentration curves for salt-free aqueous solutions of ITI showed that the critical micelle concentration (CMC) decreased with increasing mole fraction of tetrahydrofuran units in the copolymer. This behavior is due to an increase in hydrophobicity. The adsorbance and the thickness of the adsorbed layer for various ITI at the air-water interface were measured by ellipsometry. The adsorbance was also estimated from the Gibbs adsorption equation extended to aqueous polyelectrolyte solutions. The measured and calculated adsorbances were of the same order of magnitude. The thickness of the adsorbed layer was almost equal to the contour length of the ionene blocks. The intramolecular electrostatic repulsion between charged groups in the ionene blocks is probably responsible for the full extension of the... 

The determination of and pM experimentally is shown in Figure 11-1. Surface tension-log surfactant concentration curves for each of the two individual suractants in the system and at least one mixture of them at a fixed value of a must be determined. For calculating (the molecular interaction parameter for mixed monolayer formation at the aqueous solution-air interface), Cj, C2 and C°2 are required for pM, the CMCs, Cf, C2, and Cf2, are needed. 

Surface tensions of the soluble alkali salt of di- and tri-hydroxy bile salts have been widely employed [5,11,12,33,70-74] to measure CMCs of bile salts (see Section VI.l). Employing Gibbs adsorption isotherm equation and the steep slope of the experimental surface tension versus bile salt concentration curve, the surface excess, i.e. concentration of bile salt molecules/cn of interface, can be calculated accurately in high bulk ionic strength [12,70], Using this value and Avogadro s number, the area per molecule at the interface can be calculated [6]. These values (Table 3b),... 

As a consequence of surface tension, there Is a balancing pressure difference across any curved Interface. Thus, the vapor pressure over a concave liquid surface will be smaller than that over a corresponding flat surface. This vapor pressure difference can be calculated from the Kelvin s equation ... 

We have thus far restricted our discussion to plane interfaces. However, because of the existence of surface tension, there will be a tendency to curve the interface, as a consequence of which there must be a pressure difference across the surface with the highest pressure on the concave side. The expression relating this pressure difference to the curvature of the surface is usually referred to as the Young-Laplace equation. It was published by Young in 1805 and, independently, by Laplace in 1806. From a calculation of the p-V work required to expand the curved surface and so change its surface area, it is relatively straightforward to show that this equation may be written... 

There will actually also be a torque term in the surface tension, just as there is for curved dislocations, but this term is usually ignored. To make this force more physical to us, we will regard this surface energy as a surface tension. The pressure difference across a curved interface is real and occurs in all materials. In small gas-hlled voids created by implanting Xe into MgO (or Si or Al), the Xe will be crystalline if the void is small enough because the internal pressure is so high the Xe can interact with other defects just like any small crystal inside a crystalline matrix, unless the defect is a crack A simple example is calculated in Table 13.2. 

Figure 16 Djmamic surface tension during the adsorption with transfer of surfactant of C DMPO at a water/hexane interface. A drop of hexane initially free from surfactant is formed in the aqueous solution contaming the sur ctant. The water/oil volume ratio is g = 1000. The sohd curves are calculated from the model. The given concentrations are the initial values in water (A) C]3DMPO Cq = 1.5 x 10" (a), 2.3 X 10 (b), 5.3 x lO- mol/cm (c) (B) C12DMPO Cq = 1 10" (a) 2 x 10 (b), 3 x l(y8 mol/cm3 (c), (C) CioDMPO Cq = 3 x 10- (a), 5 X 10- (h), 8 X 10- mol/cm (c).

Considering capillary dynamics, the pressure drop term is often described by the Laplace equation, AP = 2yH, where y represents liquid surface tension and H represents the mean curvature of the liquid-gas interface associated with aU curves, C, passing through the surface. Furthermore, the character of a sufficiently smooth surface is through the invariant from differential geometry, the principal curvature of each curve, kj. The radii of curvature are the inverse of each principal curvature, A , = 1/r,. Considering the maximum and minimum radii of curvature at a point on a three-dimensional surface, the mean curvature can be calculated explicitly (see Appendix for more thorough derivation of the mean curvature parameter) ... 

From the surface tension/concentration curves at 25°C (Fig. la, b) the saturation adsorption (Fm) at the air-water interface and the area per molecule (.<4n,in) values were calculated. The smaller the the more effective it is its adsorption at the interfaces. We found that the values for series 2 and 3 (62-114 x 10 nm, and 96-122 x 10 nm, respectively) were higher than that for series 1 with the same alkyl chain length (67-62 x 10 nm ). This result indicates that the new molecules are less packed at the interface than those of series 1. The two charged groups in series 2 and 3 tend to spread them out on the interface due to an increase in the inter-intramolecular electrostatic repulsion forces [26],... 

Selve et al. [32] synthesized fluorinated nonionic surfactants with a two-chain polyoxyethylene hydrophilic head linked to the hydrophobe via an amide bond, F(CF2)i(CH2), C(0)N[(C2H40) CH3]2. They calculated the area A per surfactant molecule adsorbed on the air-water interface from the slope of the surface tension curve using the Gibbs equation [Eq. (13)]. The area A increases with increasing number of oxyethylene units for both fluorinated and hydrocarbon sur-... 

The adsorption of the anionic surfactant sodium dodecyl sulphate (SDS), probably the most frequently studied surfactant and often used as model substance at the air/water and at the decane /interface is given in Fig. 1.5. The surface and interfacial tension have been plotted as a function of SDS concentration in the aqueous phase. From the slope of the tangents to the curves in Fig. 1.5 the interfacial excess concentration (adsorption density) F at different interfacial tensions can be calculated directly using Gibbs fundamental adsorption isotherm (see section 2.4.1),... 

Using a drop time method for the determination of interfacial tension and a four-electrode potentiostat to polarize the interface, Kakiuchi and Senda measured electrocapillary curves for ideally polarized systems, in particular for the interface between an aqueous solution of lithium chloride and a solution in nitrobenzene of TBATPB. They showed that the surface charge density, Q, obtained by differentiation of the electrocapillary curve was equal to that calculated from the integration of the corresponding differential capacity versus potential curves. This demonstrated the validity of the Lippmann equation for the polarized ITIES ... 

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