Surface tension curve discontinuities

What characterizes surfactants is their ability to adsorb onto surfaces and to modify the surface properties. At the gas/liquid interface this leads to a reduction in surface tension. Fig. 4.1 shows the dependence of surface tension on the concentration for different surfactant types [39]. It is obvious from this figure that the nonionic surfactants have a lower surface tension for the same alkyl chain length and concentration than the ionic surfactants. The second effect which can be seen from Fig. 4.1 is the discontinuity of the surface tension-concentration curves with a constant value for the surface tension above this point. The breakpoint of the curves can be correlated to the critical micelle concentration (cmc) above which the formation of micellar aggregates can be observed in the bulk phase. These micelles are characteristic for the ability of surfactants to solubilize hydrophobic substances in aqueous solution. So the concentration of surfactant in the washing liquor has at least to be right above the cmc. 

The capillary-rise method was employed to measure the surface tension of aqueous solutions of disodlum alkyl phosphate at 25 °C. The cmc values of the solutions were obtained from the discontinuity in the surface tension - concentration curves(7). 

Figure 14 shows the knee-shaped curve of solution surface tension vs. concentration characteristic of aqueous surface-active agents the steeper the curve, the more efficient the wetting agent. It is generally assumed that the bend of the curve coincides with the critical micelle concentration (c.m.c.) of the respective compound in the aqueous medium. Since the discontinuities in the slopes of the individual curves of Figure 14 occur in the region of the c.m.c. values reported by various... 

The correctness of the preceding interpretation of the slope discontinuities in the curves of yLv was supported by experiments on solutions of the following pure polar compounds which are not able to form micelles ethanol, 1-butanol, 1,4-dioxane, propylene carbonate, diacetone alcohol, dipropylene glycol, 2-butanone, and tetrahydrofuran. Surface tensions for these compounds agreed well with the available literature values [87]. None of these polar solutes form micelles in water and, as can be seen in Figures 15, B, and 16, B, no discontinuities in the slopes of the curves of cos 6 vs. y observed for either... 

Since discontinuities at the c.m.c. occur in curves of surface tensions vs. concentration and of either cos 0 vs. concentration or cos 6 vs. surface tension, the generalization is proposed that micelle-forming compounds will produce discontinuities in the slopes of other surface properties of surface active agents when plotted against either the concentration or the surface tension of the solution. For example, it may be that such will occur in plots of the interfacial tension vs. the concentration of the surface active solute in the aqueous phase. 

In order to properly solve (17.5), sharp changes in the properties as well as pressure forces due to surface tension effects have to be resolved. In particular, surface tension results in a jump in pressure across a curved interface. The pressure jump is discontinuous and located only at the interface. This singularity creates difficulties when deriving a continuum formulation of the momentum equation. The interfacial conditions should be embedded in the field equations as source terms. Once the equations are discretized in a finite-thickness interfacial zone, the fiow properties are allowed to change smoothly. It is therefore necessary to create a continuum surface force (CSF) equal to the surface tension at the interface, or in a transitional region, and zero elsewhere. Therefore, the surface integral term in (17.5) could be rewritten into an appropriate volume integral... 

Since the pioneering work conducted by Schenck [39] and Jaeger [6], numerous surface tension measurements have been performed on low molar mass nematics by using various experimental techniques [7-28,40-46] (short reviews on this subject are available in [15] and [23]). Attention has been mainly paid to the change of surface tension at the nematic-isotropic phase transition. In contrast to common liquids, most of the temperature-scanning measurements (y(T)-curves) show, with increasing temperature, positive values or discontinuities in the temperature coefficient dy/dT. This... 

Figure 4.5. Some important manifestations of micelle fomiation abrupt changes in solution conductivity, a discontinuity in the surface tension-concentration curve a sudden increase in solution turbidity.

The sixth paper proposes a mathematical model of the computer-aided design of technical systems, based on notions of increasing sequence of macromodels, block frontal algorithms in the hypermatrix algebra, and a database approach. The next paper focuses on solid modeling, namely on using curves and surfaces with tension control to build complex surface models with geometric discontinuities. 

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