Calculated interface tension

FIGU RE 9.3 Calculated interface tension, using Eqnations 9.8 throngh 9.10, of the interface between coexisting phases of two incompatible, bnt otherwise identical polymers, for five degrees of incompatibility. For further details, see the caption of Fignre 9.2. 

One possible explanation is as follows. If we calculate interface tension by the equation of Glrlfalco and Good, ... 

One molecular solid to which a great deal of attention has been given is ice. A review by Fletcher [74] cites calculated surface tension values of 100-120 ergs/cm (see Ref. 75) as compared to an experimental measurement of 109 ergs/cm [76]. There is much evidence that a liquidlike layer develops at the ice-vapor interface, beginning around -35°C and thickening with increasing temperature [45, 74, 77, 78]. 

To calculate the force created by this concentration gradient we must relate a difference in concentration to the interfacial tension (y) difference at each end on the Au segment. First, we note that the interfacial tension of a solution may be taken as the mol fraction-weighted average of the component interface tensions (Eq. (4))... 

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... 

Fig. 21. Ratio between the interface tension 7 and the simple expression for the strong segregation limit yssL in (54) as a function of inverse incompatibility. Symbols correspond to Monte Carlo results for the bond fluctuation model, the solid line shows the result of the SCF theory, and the dashed line presents first corrections to (54) calculated by Semenov. Also an estimate of the interface tension from the spectrum of capillary waves is shown to agree well with the results of the reweighting method. Adapted from Schmid and Muller [107]...

This estimate has been used to normalize the interface tension in Fig. 21. The collapse of the data for different chain lengths onto a common curve shows that the interface tension indeed only depends on the combination xN and the data are well described by numerical mean field calculations [107] and analytic predictions by Semenov [110]. 

A third variant of the VOF method calculates the interface tension force by the CSS method and perform an independent FLIC reconstruction of the interface to improve the design of the advection schemes. In this way the tailored advection discretization schemes prevent numerical smoothing of the interface [149]. 

Computer simulations have over the years contributed to our understanding of the liquid water/vapor interface. A fairly early study by Wilson et al. (1987) used the TIP4P water model with 342 molecules at a rather high temperature, 52 °C. They confirmed the preferred dipole orientation at the surface with the positive end (hydrogen atoms) towards the bulk liquid and considered the interfacial region to extend 0.75 nm into the liquid. However, the calculated surface tension, 132 46 mN m , is nearly twice the experimental value, so that the implementation of the model needed revision. Subsequent work by others improved on this situation. Taylor et al. (1996) used the SPC/E model of water with 526 molecules at several temperatures from —5 to 100 °C (and also with 1,052 molecules at 25 °C) at 0.1 MPa. A main conclusion is... 

Using a statistical treatment of the variation of local intermolecular forces as the interface is traversed from the Uquid to the vapor phase, it is possible to calculate the surface tension of a simple Uquid (e.g., argon) that agrees well with experiment. However, such exact methods become quite complex or (currently) impossible for calculating surface tensions in most practical systems. 

Comelisse, P.M.W., Peters, C.J., and de Swaan Arons, J. (1993) Application of the Peng-Robinson equation of state to calculate interfacial tensions and profiles at vapor-liquid interfaces, J uid Phase Equilibria 82, 119-129. 

The amplitude of the Fourier modes can he studied hy scattering experiments or in computer simulations, " and the spectmm of fluctuations has been utilized to extract the interface tension. Equation [43] can he used to calculate the fluauations of the local interface position, h x, y), averaged over the lateral length scale, L... 

Let us now consider the results of similar experiments when water contains nonionic surfactant EOio at a concentration Co = 0.025%, much higher than the CMC. Equilibrium interface tension decreases in this case to yi2 = 7 mN/m (see Fig. 7). The results shown in Fig. 9 do not differ much from that obtained for the same system without surfactant (Fig. 8) because they were obtained for a capillary that was not equilibrated previously with surfactant solution. As a result, concentration near the meniscus was 100 times lower than Co due to adsorption of surfactant on the capillary surface. Advancing contact angle 0a = 80°, calculated using the B(z) graph (two black points in Fig. 9b), is close to the value calculated from Fig. 8b in the absence of surfactant. 

The Pc values were recalculated into wetting tensions y cos 9 = Pcf/2. In Fig. 32 are shown calculated dependencies of the ratio y cos 0/yo on flow rates V (curves 1-4), where yo is the bulk interface tension. Each curve in Fig. 32 is bounded by two vertical dotted lines, which give the limiting values of yA cos a/To and yR cos 0R/yo obtained at high flow rates, v > 10 " cm/s, from curves 2 and 3 in Fig. 31, respectively. 

In order to compare the derived expression (40) with experimental data shown in Fig. 32, the Cm/Co values were recalculated into ym/To values using isotherms of surface or interface tensions y(C) for CTAB solutions in contact with air or silicon oil. Fig. 33 shows, calculated in this way, thedependence of ym/ yo on V for a particular case of 5 x 10 " M CTAB-air system (y = 43 mN/m),... 

Comparison of the results of calculations (Fig. 33) with experimental data shown in Fig. 32, demonstrates qualitative agreement between both. The quantitative differences may be associated with the fact that the theory starts from the condition y = 7o at v = 0. However, in experiments the conditions V = 0 corresponds to y cos 0< jo because cos Q > 1. In some cases, experimental values of y cos 0 at v = 0 are higher than 1. This suggests that in the case under consideration the interface tension, even near to v 0, has not yet relaxed to the equilibrium value yo. 

Kheifets and coworkers [107-112] showed the practical importance of the application of the potentials and Eq. (32) for the analysis and characterization of ion-exchange extraction. They also proposed an approach to calculate the Alf(pi distribution potentials from the interface tensions, namely using dependences of Gibbs isotherm type and the Lippmann equation. 

Lane, J. E., Correction terms for calculating surface tension from capillary rise, J. Colloid Interface Sci., 42, 145-149 (1973). 

In this chapter we consider the simultaneous equilibrium of three phases. We shall see that there are circumstances in which the phases meet in a line of three-phase contact. Macroscopically, this locus of points in which they meet is one-dimensional and locally linear it is analogous to the macroscopically two-dimensional and locally planar interface between two phases. There is an excess free energy per unit length, or line tension, associated with the three-phase line, and we should in prindple be able to calculate that tension from a microscopic theory. Such a line of three-phase contact should have a predictable, and in principle discernible, three-dimensional structure at the molecular level, and its structure and tension should be related, just as are the structure and tension of the interface between two phases. 

The CSF and CSS based versions of the VOF method have been used to calculate improved estimates of the single particle drag and lift coefficients and for simulating breakage and coalescence of dispersed flows containing a few fluid particles [20, 53, 54, 150, 232]. A third variant of the VOF method calculates the interface tension force by the CSS method and perform an independent PLIC reconstruction of the interface to improve the design of the advection schemes. In this way the tailored advection discretization schemes prevent numerical smoothing of the interface [160]. 

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... 

Acomp equals unity when the gradient A9 is zero for aU z, due to infinite incompatibility. A equals zero when c(z) is equal to c at all z, due to absence of solvent. It is also equal to zero in the case of infinite incompatibility. The interface tensions calculated from the profiles in Figure 9.2 are in Figure 9.3. The values found for the range of practical concentrations of 5-20% are roughly between 1 and 20 pN/m. A higher incompatibility, that is, a lower c, leads to a higher interface tension. 

FIGURE 9.8 Concentrations inside droplets after osmotic swelling calculated by solving Equation 9.16 for 5, using the interface tension from Figure 9.3. The radius of the droplets prior to swelling was 10 pm and the phase volume fraction of the droplet phase was 0.5. 

Smith [113] studied the adsorption of n-pentane on mercury, determining both the surface tension change and the ellipsometric film thickness as a function of the equilibrium pentane pressure. F could then be calculated from the Gibbs equation in the form of Eq. ni-106, and from t. The agreement was excellent. Ellipsometry has also been used to determine the surface compositions of solutions [114,115], as well polymer adsorption at the solution-air interface [116]. 

It was noted in connection with Eq. III-56 that molecular dynamics calculations can be made for a liquid mixture of rare gas-like atoms to obtain surface tension versus composition. The same calculation also gives the variation of density for each species across the interface [88], as illustrated in Fig. Ill-13b. The density profiles allow a calculation, of course, of the surface excess quantities. 

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