Example calculations surface tensions

It is illustrative now to compare surface tensions determined experimentally on a variety of surfaces For example, the surface tension of PE was found to be in the range 31-36 mN/m, while the value of PTFE has been determined as about 24 mN/m ° As an example of materials with surfaces dominated by CH, and CF, groups, one can consider the measured surface tensions of hexane (18.4 mN/m) compared to perfluorohexane (11.9 mN/rn) respectively. It is interesting to note that the ratios of the surface tensions of the members of these two classes of materials are similar to the ratios of their respective surface densities. More sophisticated theories, based on densities, have been invoked to calculate surface... 

Couchman and Karasz (11) recently have made some calculations indicating that spherical microphases should exhibit increased glass-transition temperatures because of an increased pressure inside such microphases attributed to the surface tension between microphases. Since there is some doubt about the existence of a surface of tension in the Gibbs sense (12) between chemically linked microphases, we shall simply note that these calculations are the only ones in existence that indicate a possible reason for an increase in the Tg of a glassy microphase and, in addition, that these calculations also postulate differences in Tg with differences in morphology. For example, this surface-tension-dependent effect would not be expected in samples with lamellar morphology, no matter how small the width of each lamella. 

The simplest possible approach for designing potential energy functions suitable for liquid interfacial simulation is to use the potentials developed to fit the properties of bulk liquids. Surprisingly, in many cases this provides a reasonable description of the interface (for example, the calculated surface tension of the pure liquid is in reasonable agreement with experiments). However, one may improve the potentials by relaxing the condition in equation (3). For example, in simulations of the interface between two immiscible liquids, one may still keep the relation in equation (3) for the interactions between molecules belonging to the same liquid, but have the parameters e, Cy (for the... 

As an example, the surface tension isotherms for alkyl dimethyl phosphine oxides (CnDMPO) for Cg to C16 at 25 °C are shown in Fig. 5. The Frumkin and reorientation mcxlels agree both well with the experimental data. Small differences between the calculated isotherms exist only for n > 13, while for lower n the curves for the two models perfectly coincide. Thus, neither of the two models can be preferred if one takes into account only the agreement between the experimental and theoretical data. However, negative values of the Frumkin constant a are obtained for lower homologues, and an unexpected dependence of a(n) are obtained, which together indicate that the coincidence between the Frumkin model and the experimental data is only formal. 

Group contribution methods are also useful because they can be applied to calculate surface tensions for a portion of the polymer molecule, for example a pol5mier chain end, for which there is no reliable method for direct experimental measurement. For end groups, one of several empirical relationships developed for low molecular weight compounds of various types can be employed, for example (8). 

Attempt of correlating the molecular structures and experimental data, for example, cmc, and the thermodynamic parameters of micellization (enthalpy, entropy, and free energy), rests on the assumption that they have been calculated by a consistent procedure this point needs further consideration. At the outset, it should be noticed that there are systematic differences between the results, for example, the cmc, obtained by using distinct experimental techniques. The reason is that the function plotted (absorbance of micelle-solubilized dye, conductivity, surface tension, light scattering intensity, etc.) versus [surfactant] measures different averages of the various species in solution. Examples are surface tension that primarily depends on monomer concentration and solubilization of (water-insoluble) dye that depends mainly on the total amount of micelles present. The consequence is that cmc measured from surface tension will always be lower than cmc measured by dye solubilization [28]. In fact, values of the cmc of the same surfactant, determined by different groups, by the same technique show differences. For example, fifty-four erne s determined by the same technique for Cj NMe Br (measurements at 25°C) differ by 22% [29]. 

From the steepest slope of the surface tension versus concentration isotherm (which generally occurs just before the critical micelle concentration—refer to Sec. I.C), one can easily calculate the area occupied by a surfactant molecule at the surface. Practically, for experiments carried out at 25 C, the area is given by dividing 943.5 by the drop of surface tension (in mN/m) per decade of concentration. The result is in square angstroms. Figure 3 shows, as an example, the surface tension-concentration isotherm of Dobanol 91-5 (C9—C11) fatty alcohol ethoxylated from Shell, with an average of five molecules of eth-... 

An example of surface tension is the formation of a meniscus at the orifice of a capillary filled with liquid, such as the nozzle of an inkjet print head. As shown in Figure 6.4, the liquid is held to the orifice by the surface tension x around the periphery of the capillary. The pressure drop AP across the meniscus can be calculated by equating the force r holding the meniscus to the capillary to the pressure pushing the fluid out of the capillary ... 

General hydrodynamic theory for liquid penetrant testing (PT) has been worked out in [1], Basic principles of the theory were described in details in [2,3], This theory enables, for example, to calculate the minimum crack s width that can be detected by prescribed product family (penetrant, excess penetrant remover and developer), when dry powder is used as the developer. One needs for that such characteristics as surface tension of penetrant a and some characteristics of developer s layer, thickness h, effective radius of pores and porosity TI. One more characteristic is the residual depth of defect s filling with penetrant before the application of a developer. The methods for experimental determination of these characteristics were worked out in [4]. 

One fascinating feature of the physical chemistry of surfaces is the direct influence of intermolecular forces on interfacial phenomena. The calculation of surface tension in section III-2B, for example, is based on the Lennard-Jones potential function illustrated in Fig. III-6. The wide use of this model potential is based in physical analysis of intermolecular forces that we summarize in this chapter. In this chapter, we briefly discuss the fundamental electromagnetic forces. The electrostatic forces between charged species are covered in Chapter V. 

The uncertainties in choice of potential function and in how to approximate the surface distortion contribution combine to make the calculated surface energies of ionic crystals rather uncertain. Some results are given in Table VII-2, but comparison between the various references cited will yield major discrepancies. Experimental verification is difficult (see Section VII-5). Qualitatively, one expects the surface energy of a solid to be distinctly higher than the surface tension of the liquid and, for example, the value of 212 ergs/cm for (100)... 

The monolayer resulting when amphiphilic molecules are introduced to the water—air interface was traditionally called a two-dimensional gas owing to what were the expected large distances between the molecules. However, it has become quite clear that amphiphiles self-organize at the air—water interface even at relatively low surface pressures (7—10). For example, x-ray diffraction data from a monolayer of heneicosanoic acid spread on a 0.5-mM CaCl2 solution at zero pressure (11) showed that once the barrier starts moving and compresses the molecules, the surface pressure, 7T, increases and the area per molecule, M, decreases. The surface pressure, ie, the force per unit length of the barrier (in N/m) is the difference between CJq, the surface tension of pure water, and O, that of the water covered with a monolayer. Where the total number of molecules and the total area that the monolayer occupies is known, the area per molecules can be calculated and a 7T-M isotherm constmcted. This isotherm (Fig. 2), which describes surface pressure as a function of the area per molecule (3,4), is rich in information on stabiUty of the monolayer at the water—air interface, the reorientation of molecules in the two-dimensional system, phase transitions, and conformational transformations. 

In order that the programme can be used to calculate both the resonance frequency (RF) and angular resonance frequency (WR) for a chosen bubble (radius, br) it will be necessary to input the solvent density (d), the surface tension of the solvent (a), the solvent viscosity (q) and the solvent vapour pressure (pv). The programme can correct the data to the appropriate SI unit. For example, if a bubble radius of 10 cm was entered the correct response to the question Type in the initial (equilibrium) bubble radius in cm - line 280 - would be either 0.01 or lE-2 followed by return. The programme will convert the value to metres - (i.e. BR=.01 br). 

The above set of equations can be solved numerically given input parameters, including surface tension a, temperature, solubility relation, D and p as a function of total H2O content (and pressure and temperature), initial bubble radius ao, initial outer shell radius Sq, initial total H2O content in the melt, and ambient pressure Pf. For example. Figure 4-14 shows the calculated bubble radius versus time, recast in terms of P versus t/tc to compare with the Avrami equation (Equation 4-70). 

EXAMPLE 6.4 Surface Tension and the Height of a Meniscus at a Wall. Calculate the height to which an n-octane surface will climb on a Teflon wall (this is the same system used in Example 6.3) if y is 22 mj m 2J = 30°, and p = 0.70 g cm 3. Comment on the ease or difficulty of making this measurement. 

EXAMPLE 7.4 Determination of Surface Excess Concentration from Surface Tension Data. The slope of the 25°C line in Figure 7.15 on the low-concentration side of the break is about -16.7 mN m 1. Calculate the surface excess and the area per molecule for the range of concentrations shown. How would Figure 7.15 be different if accurate measurements could be made over several more decades of concentration in the direction of higher dilution Could the data still be interpreted by Equation (49) in this case ... 

The strategy for scaling up the van der Waals attraction to macroscopic bodies requires that all pairwise combinations of intermolecular attraction between the two bodies be summed. This has been done for several different geometries by Hamaker. We consider only one example of the calculations involved, namely, the case of blocks of material with planar surfaces. This example serves to illustrate the method and also provides a foundation for connecting van der Waals forces with surface tension, the subject of the next section. 

The second method is to calculate a stress that is appropriate to a particular situation of interest. An example of this would be the stress acting on a drop of material due to its own weight as it rests on a support medium. The force of gravity tends to make the drop spread out into a film, while its surface tension tends... 

Table 4.3 shows the surface tensions and the interfacial tensions against water at 20°C. Based on the interfacial tension or surface tension measurement, it is possible to calculate the water-water dispersion and hydrogen bonding forces. The value of the surface tension is the sum of the combined dispersion and the hydrogen bonding forces. For example, for the water-n-octane system,... 

This hydrostatic approach also yields a formal closed formula for y in terms of the components of the stress tensor. When the stress tensor is expressed in terms of molecular variables, the resulting statistical mechanical formula for y provides a direct means for the calculation of surface tension. For example, it may be used directly to compute the surface tension of dilute ionic solutions (6). It also illustrates in molecular detail the iterative subtractive procedures that lead to the excess functions of the familiar phenomenological approach. 

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