Surface of tension

Ve must also determine the location of the surface of tension,... 

R. Although expressions for this parameter exist, they are derived by a hybrid of molecular mechanical and thermodynamic arguments which are not at present known to be consistent as droplet size decreases (8). An analysis of the size limitation of the validity of these arguments has, to our knowledge, never been attempted. Here we evaluate these expressions and others which are thought to be only asymptotically correct. Ve conclude, from the consistency of these apparently independent approaches, that the surface of tension, and, therefore, the surface tension, can be defined with sufficient certainty in the size regime of the critical embryo of classical nucleation theory. 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

Both the surface tension and the surface of tension may be expressed in terms of integrals of radial moments of the components of the pressure tensor (17-20)... 

Equations 12 and 13 are two different expressions for the surface tension ( ) which at the present time are not known to be consistent. Although Equation 12, and Equation 9, are independent of the form of the localization of the pressure tensor. Equation 13 is not (22). With the assumption of the equivalence of Equations 12 and 13 an expression for the surface of tension is obtained... 

The superscript (p) will be used to distinguish the surface of tension defined by Equation 14 from other definitions introduced below. 

Each pair of expressions for the surface tension and surface of tension was then used, through Equation 9, to obtain the liquid and vapor densities. This process was repeated until the surface tension was stationary. While values of the surface tension and surface of tension are sensitive to variations in the phase densities, these densities are insensitive to variations in y and R. Consequently, with initial estimates of the surface tension taken from the planar limit, convergence was obtained in seven, or fewer, iterations. 

That the degree of consistency observed for the various routes to the determination of the surface tension is not the result of a cancellation of errors is demonstrated by the agreement displayed by the surfaces of tension in Figure 4. Also shown is the limiting form whose slope is one and whose intercept defines the Tolman parameter (24)... 

Figure 4. The dependence of the surface of tension on droplet size...

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. 

In systems with surface of discontinuity there exists a surface of tension where all forces applied to the system act, both in tangential (surface tension) and normal (capillary pressure) directions. 

In the case of liquid films three surfaces of tension could be recognised. One of them is inside the film and the tension of the film y, expressed by the known Bakker s integral, refers to it. Since the main contribution to yis related to the monolayer regions at the surfaces of the film yean be split into two parts... 

So, the basic surface of tension divides the film thickness h (the distance between the surfaces of tension at the film surfaces where y and y2 are applied) into a ratio inversely proportional to the quantities y and y [e.g. 9, p. 306]. 

Let us imagine two non-interacting rigid plates with zero thickness situated at the surfaces of tension at the film surfaces, i.e. at z = h/2. Besides, the pressure pc the plates are subjected to a variable external force 11/1 (A is the area of the plane-parallel film). This fdrce counterbalances the forces causing the thinning of a non-equilibrium film by liquid drainage from it into the adjacent liquid meniscus. Thus, the film thickness h can change reversibly at fixed values of the independent variables. 

The quantity 11/ is a measure of the so-called disjoining action , introduced by Derjaguin in 1936 [12]. The disjoining pressure n [8] is determined by the long-range interaction forces between the surfaces of the film (normal to the both surfaces of tension there) and tends to zero when the film thickness is sufficiently large [5]. Eq. (3.15) proposes a more general definition of IT than that for the equilibrium case (Eq. (3.10))... 

Fig. 3.3 shows a symmetrical plane-parallel film in contact with a cylindrical meniscus L. The plane of symmetry of the film coincides with the basic film surface of tension. Let us assume that at the other surfaces of tension the surface tension everywhere is constant and equal to that of the meniscus [Pg.96]

The above equations describe a simple mechanical model of the film, its adjacent transition zone and the bulk meniscus. According to this model the force quantities, lAytosd and 2Ayf, are applied only on the basic surface of tension. The disjoining pressure and the capillary pressure act always and everywhere normally to the phase surfaces, identified as surfaces of tension. There are other two A) in0 force components related to the two phase surfaces. These components counterbalance each other at any point of the basic surface of tension. Eq. (3.33) coincides formally with the force balance condition of de Feijter and Vrij [22] if one would write... 

By extrapolating the meniscus profile far from the film to its intersection with the basic surface of tension, the macroscopic contact angle ft> is defined... 

Figure 33. Cross-sectional views of tension wood in a young quaking aspen stem. (Reproduced from Ref. 39. Copyright 1982, American Chemical Society.) (A) Light micrograph of a section that was selectively stained to differentiate the gelatinous layers in G-ftbers. (B) SEM of a surface of tension wood fiber zone. The Gravers, which are loosely attached to the rest of the fiber wall, were dislodged during specimen preparation and...

Here T is the temperature, p is the pressure, a is the surface tension, A is the surface area, V is the volume, 5 is the entropy density, Pi are the particle densities, and pi the chemical potentials of the different components, R is the radius of the critical cluster referred to the surface of tension, the index a specfies the parameters of the cluster while p refers to the ambient phase. The equilibirum conditions coincide with Gibbs expressions for phase coexistence at planar interfaces (R oo) or when, as required in Gibbs classical approach, the surface tension is considered as a function of only one of the sets of intensive variables of the coexisting phases, either of those of the ambient or of those of the cluster phase. In such limiting cases, Gibbs equilibrium conditions... 

It is noteworthy that the above treatment is only valid for flat interfaces. Things get more complicated if one deals with curved surfaces, for which it is necessary to consider a pressure gradient existing between two phases in contact. In such a case the surface tension becomes dependent on the position of the dividing surface. A position of the dividing surface that yields a minimum value of a is referred to as the position of the surface of tension , according to Gibbs. 

According to Gibbs, it is possible to chose a position of the dividing surface such that So = 0, the so-called surface of tension, for which one can write... 

In agreement with the Laplace equation, the action of the stress field of the curved interface on phases in contact is analogous to the action of an elastic film with tension o located at the surface of tension. It is, however, important to realize that the properties of the interfacial layer are significantly different from those of a film. Namely, the surface tension a is independent of the surface area S, while the tension of the elastic film increases with increasing deformation1. 

Gibbs main idea was to introduce a dividing (mathematical ) surface its position (inside of the s-phase of Bakker s model) is defined in such a way that the adsorption of one of the components is zero and with a surface energy independent of surface curvature. This surface is noted as the surface of tension. In essence we can restrict Gibbs thermodynamics to the lUPAC recommendation for colloid and surface chemistry, prepared by Everett (1971). Gibbs idea of a dividing surface agrees with the definition by Rusanov (1981) mention above. 

Surface of Tension An imaginary boundary having no thickness at which surface or interfacial tension acts. 

Surface Work The work required to increase the area of the surface of tension. Under reversible, isothermal conditions, the surface work (per unit surface area) equals the equilibrium or static surface tension. 

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