Surface tension calculation

Note that the external field vanishes in the normal surface tension calculation. In the fully local approximation there is no surface tension. Thus we can obtain the surface tension y associated with a planar interface of area A by the expression... 

We recall however that a much better, actually quantitative, agreement with experimental data of surface tension has been obtained with SPC/E [173] and TIP4P [174]. Also, surface tension calculations are less straightforward than for other properties, so that a careful evaluation of simulation details such as run length is required before the role of the potential model can safely be assessed. 

It was established that the surface was in local equilibrium the surface tension calculated for the system with a gradient was indeed the same as for the system in global equilibrium at a given temperature see Figure 2. The fluxes for heat and mass across the surface can be written as... 

Experimental results obtained in isothermal conditions for light and heavy water are presented in Fig. 5. For light water they cover a range of nucleation rates of 5 orders in the interval of supercoolings from 33.9 to 37.8 K. In this interval the nucleation rate increased 10 times when the temperature decreased by 0.8 K. The effective value of the surface tension calculated from experimental data is ct, = 28.7 mN/m, and the value of the preexponential... 

If the radius of the capillary is large, so that (r/a) > 0.05, then the Basforth-Adams equation (Equation (481)) or the Lane equations (Equations (482) and (483)) can also be used in the surface tension calculation from the maximum bubble pressure method. This method can also be used to determine the surface tension of molten metals. It has been a popular method in the past, but now it is not very common in surface laboratories because of its poor precision. 

Solid Surface Tension Calculations from Contact Angle Results... 

Experimentally it is found that the assumptions of perfect wetting, etc., are not exactly correct and that the surface tension calculated by Eq. 17.8 is too large by a factor of about 1.1. However, with this kind of apparatus, the cleaning problem is much easier than the case of the other two , and the dimensions are easily measured. For these and other reasons of convenience, this type of apparatus is the most common laboratory device for measuring surface tension. 

Experiments [3, p. 45] indicate that the surface tension calculated from drop weights on burettes by means of Eq. 17.7 must be multiplied by a factor which ranges up to 1.5 to agree with those obtained by the most reliable methods. Assuming that this is entirely due to the shape of the droplet just before breakaway, calculate the angle that the drop s surface makes with the vertical. 

Surface tension calculated from Eq. (8.33) is depicted in Figure 8.16 (open markers). Good agreement is observed between 77 estimations and experimental data points for EVOH and SAN, with one exception — a discrepancy near 20% appears for PE between the y value estimated from parachor and the value measured. Nevertheless, for EVOH there is a linear dependence of y on composition, as expressed by Eq. (8.28). For PVAl the extrapolated value is y = 44.2 mN/m. Parachor estimates for SAN follow a second-order equation. We note that the extrapolation of experimental values to Pan =1 agrees quite well with the parachor estimate ypAN-parachor = 43.7 mN/m. For EVAc the 77 estimations are much lower than the experimental results, probably due to too-low densities. A similar tendency is shown in Fig. 8.17. 

Relatively new are the developments of microfluidic surface or interfacial tension methods. Such methods offer the potential for rapid, online measurements on small volume samples. Typically, use is made of a property change (such as droplet deformation or pressure drop) associated with fluid flow through some kind of constriction in a microchannel. The fundamental principles are usually the same as in the examples just given above, such as shape or pressure changes. If the device is used to generate bubbles, then the bubble formation frequency can be used as the basis for surface tension calculation. In a multiple-channel device, multiple bubbles or droplets are usually sensed simultaneously. These kinds of approaches have been applied to the determination of both surface and interfacial tensions [20-22]. 

Surface Tension Calculation in VOF Method Modeling interfacial tension effects is important because it is a potentially large force which is concentrated on the interface. There are two different approaches to modeling surface tension forces. The first mie is continuum surface force (CSF) defined as... 

We see from these data that the values of surface tension calculated from known forces of adhesion (which are equal to the capillary forces) are low in comparison with the actual values. The point is that these investigators failed to account for the disjoining pressure of the thin layer of liquid (see Fig. IV.6.c), which weakens the capillary interaction. This is why their method for determining surface tension by measurement of adhesive force did not give accurate results. 

Afanas ev, B.N. and Akulova, Y.P. (2000) A correlation between the hydrophilic-ity of a metal and its surface tension. Calculation of the bond energy of water molecules adsorbed on an uncharged metal surface. Prot. Met, 36 (1), 25—30. 

The accuracy of the surface tension calculated from (6.6) is low, perhaps 10 per cent for a MC simulation of 250 molecules and 10 steps. A more accurate residt can be obtained from a MC calculation of the free energy needed to cut a block of liquid into two parts, separate them, and allow the profiles to relax to their equilibrium shapes. Bennett devised a MC scheme for the calculation of the diange in free energy in the first step, and the whole operation was carried out by Miyazaki et al. The accuracy in Lennard-Jones liquid near its triple point is improved to about 2 per cent. 

The time dependent surface tension decay was measured according to the drop-volume technique as outlined by Tornberg [18,19]. The automatization procedure according to Arnebrant and Nylander [20] was employed. In this method surface tension reduction by macromolecules during adsorption at the air-water interface is measured by formation of drops of certain volumes. Time for detachment of the droplets is recorded. Surface tension calculated [19, 20] was plotted against detachment time and the value attained after 2000 seconds was set as the equilibrium value. The surface tension of the solutions is still decreasing after this period of time, but the rate of decrease is small, less than 0.05 mNm" per 100 seconds. The maximum error in surface tension values is 1.5%. 

Now, if we think that the increase of the interfacial area involved in the emulsification process is of a 10 order of magnitude and that even the most efficient surface-active agent can reduce the interfacial tension by a factor of 5-10, we cannot understand why emulsifiers can stabilize the emulsions on the base of surface-tension calculations. In order to answer this question, the modem theory of Deijaguin, Landau, Verwey, and Overbeek (DLVO) will be used, but from a qualitative point of view because this chapter is devoted to the formulation job. 

Figure 8 expresses the dependence of the surface tension, calculated by the parachor method, on the distance from the dew point. The values of surface tension vary significantly with this distance in the region of retrograde condensation and less significantly (but still noticeably) in the region of normal condensation. Variation of the surface tension is due to the fact that... 

A bacterium modeled as a cylinder (diameter A = 0.6 nm) shown below excretes surfactant at location h, leading to the surface tension difference Asurface tension. Calculate the velocity U of the bacterium if the drag offered by the fluid to the bacterium motion is equal to 37TfiAU. The viscosity of the fluid is fi = 0.001 Pa-s. 

Molecular dynamics simulations have been used to test the validity of the CW theory down to distances comparable to 4b- Equation [14] predicts a specific dependence of the interface width on the temperature. Simulations at different temperatures can be used to determine (C ) (by fitting the density profile to Eq. [13]). This, combined with surface tension calculations (see below), can be used to verify that V(C ) s proportional to - T/y. Figure 3 shows this plot generated using the data published in the very recent million particles simulation of the Lennard-Jones liquid/vapor interface. As can be seen, the relation in Eq. [14] holds quite well at low T. Another simple approach is to obtain 4b from the bulk radial distribution function (g(4b) 1) and confirm the validity of Eq. [14] using the independently calculated surface tension and (C ), as has been done for several liquid/liquid interfaces. Alternatively, if several simulations with different surface areas are performed, Eq. [14] suggests that a plot of straight line with a slope of... 

A. Ghoufi, F. Goujon, V. Lachet, and P. Malfreyt,/. Chem. Phys., 128, 154716 (2008). Multiple Histogram Reweighting Method for the Surface Tension Calculation. 

The harmonic mean equation is generally considered to be applicable to low surface tension materials such as organic polymers and liquids. If y and y are known for two liquids, and the contact angles of those liquids on the solid of interest are measured, equation 36 produces two simultaneous equations that can be solved to find the surface tension and polarity of a solid polymer surface. Numerous assumptions have been made in developing the theory of fi actional polarity. For example, it ignores the possibility of induced polarity at the interface between polar and nonpolar materials (82). These assumptions limit the application of equation 36 to systems where at least one and preferably both of the components are relatively nonpolar. The theory breaks down when interfacial interactions lead to molecular rearrangements at the interface between solid and liquid. In addition, it was foimd that pairs of liquids with similar dispersive and polar components of surface tension gave umeasonable results for the substrate surface tension calculated by the harmonic mean method (83). 

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