Interface surface force balance

The equation was originally derived for the homogeneous solid-liquid interface (no air pockets) (Fig. 6a) using the surface force balance and empirical considerations however, it was later put in a proper thermodynamic framework. It is important that according to Wenzel model the inherently hydrophilic flat surface will be more hydrophilic when rough, and inherently hydrophobic surface will become more hydrophobic. 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

We now describe the conditions that correspond to the interface surface. Eor stationary capillarity flow, these conditions can be expressed by the equations of continuity of mass, thermal fluxes on the interface surface and the equilibrium of all acting forces (Landau and Lifshitz 1959). Eor a capillary with evaporative meniscus the balance equations have the following form ... 

In the second approach shown in Fig 3.12(b), a force is applied continuously using a Vickers microhardness indenter to compress the fiber into the specimen surface (Marshall, 1984). For ceramic matrix composites where the bonding at the interface is typically mechanical in nature, the interface shear stress, Tf, against the constant frictional sliding is calculated based on simple force balance (Marshall, 1984) ... 

Theoretical analyses of interfacial debonding and frictional pull-out in the fiber pull-out test were initially modeled for ductile matrices (e.g. tungsten wire-copper matrix (Kelly and Tyson, 1965, Kelly, 1966)) assuming a uniform IFSS. Based on the matrix yielding over the entire embedded fiber length, as a predominant failure mechanism at the interface region, a simple force balance shown in Fig. 4.19 gives the fiber pull-out stress, which varies directly proportionally to the cylindrical surface area of the fiber... 

Liquid-liquid interface At the interface between two immiscible liquids, the boundary conditions that must be satisfied are (a) a continuity of both the tangential and the normal velocities (this implies a no-slip condition at the interface), (b) a continuity of the shear stress, and (c) the balance of the difference in normal stress across the interface by the interfacial (surface) force. Thus the normal stresses are not continuous at the interface, but differ by an amount given in the following expression ... 

Experiments have shown that the bubbles are not always in thermodynamic equilibrium with the surrounding liquid i.e., the vapor inside the bubble is not necessarily at the same temperature as the liquid. Considering a spherical bubble as shown in Fig. 9-4, the pressure forces of the liquid and vapor must be balanced by the surface-tension force at the vapor-liquid interface. The pressure force acts on an area of nr2, and the surface tension acts on the interface length of 2irr. The force balance is... 

As a molecule approaches the solid surface, a balance is established between the intermolecular attractive and repulsive forces. If other molecules are already adsorbed, both adsorbent-adsorbate and adsorbate-adsorbate interactions come into play. It is at once evident that assessment of the adsorption energy is likely to become exceedingly complicated in the case of a multicomponent system - especially if the adsorption is taking place from solution at a liquid-solid interface. For this reason, in the numerous attempts made to calculate energies of adsorption, most attention has been given to the adsorption of a single component at the gas-solid interface. 

In the classical treatment of surface tensions, it is intuitively assumed that the surface tension of a solid, 7s, can be assigned as if it is a material constant. In a practical sense, Eq. (25.3) is valid if the surface tension of the solid does not change after the contact with the liquid (sessile droplet) is made. While Young s equation describes the force balance at the three-phase line, it does not give information relevant to the true interfacial tension at the interface that is beneath the droplet, which is the major concern of surface dynamics. In general cases, 7s and 7sl are... 

Aqueous film stability is dependent on the adhesive force or negative interfacial tension at the two-phase (i.e., solid/liquid) boundary. The force balance at the two-phase boundary may change independently from the three-phase force balance due to surface configuration change of interfacing surface state moieties, which occurs in order to minimize interfacial tension with water as described in previous chapters. 

To express the force equilibrium condition in a mathematical form, we can now consider a force balance on an arbitrary surface element of a fluid interface, which we denote as A. A sketch of this surface element is shown in Fig. 2-14, as seen when viewed along an axis that is normal to the interface at some arbitrary point within A. We do not imply that the interface is flat (though it could be) - indeed, we shall see that curvature of an interface almost always plays a critical role in the dynamics of two-fluid systems. We denote the unit normal to the interface at any point in A as n (to be definite, we may suppose that n is positive when pointing upward from the page in Fig. 2-14) and let t be the unit vector that is normal to the boundary curve C and tangent to the interface at each point (see... 

Although the expression (2-131) is a perfectly general statement of the force balance at an interface, it is not particularly useful in this form because it is an overall balance on a macroscopic element of the interface. To be used in conjunction with the differential Navier-Stokes equations, which apply pointwise in the two bulk fluids, we require a condition equivalent to (2-131) that applies at each point on the interface. For this purpose, it is necessary to convert the line integral on C to a surface integral on A. To do this, we use an exact integral transformation (Problem 2-26) that can be derived as a generalization of Stokes theorem ... 

Determination of the wetting tension of a liquid, a) schematic of an experimental set-up, 1 -force balance, 2 - plate, 3 - level of the liquid surface, 4 - temperature control jacket, 5 - lift b) rewetting at a liquid/liquid interface, after Richter (1994)... 

Surface tension. Assuming that a membrane stretches over each interface, the magnitudes of the interfacial tension between each pair of phases are the fluid-fluid interfacial tension Ogt, the wetting fluid-solid interfacial tension oft, and the nonwetting fluid-solid interfacial tension ogs. When in static equilibrium, the vectorial force balance at the line of contact (the law of Neumann triangle, Ref. 87) gives... 

Fig. 10.1.4. The liquid may spread freely over the surface, or it may remain as a drop with a specific angle of contact with the solid surface. Denote this static contact angle by 6. There must be a force component associated with the liquid-gas surface tension (t that acts parallel to the surface and whose magnitude is a cos 0. If the drop is to remain in static equilibrium without moving along the surface, it has to be balanced by other forces that act at the contact line, which is the line delimiting the portion of the surface wetted by the liquid, for example, a circle. It is assumed that the surface forces can be represented by surface tensions associated with the solid-gas and solid-liquid interfaces that act along the surface, and tr i, respectively. Setting the sum of the forces in the plane of the surface equal to zero, we have...

Thus the shear stress depends on the local surface tension gradient, in the absence of which Eq. (10.5.3) simply reduces to the usual fluid dynamic boundary condition that the tangential viscous stress is continuous at the interface of two different fluids. The normal force balance simply gives the scalar equation... 

As we have shown, the surface forces at an interface depend upon the surface tension gradients there. If adsorbed surface-active materials are distributed at an interface, then this distribution must be known to determine the surface forces, since the surface tension gradients depend on the local surface concentration of adsorbed material. The surface mass concentration of the adsorbed substance follows from an interfacial mass balance. 

For a liquid that rests on a smooth surface with a finite contact angle, one can determine the relationship between the interfacial tensions at the different interfaces from consideration of the balance of surface forces at the line of contact of the three phases (solid, liquid, and gas). Remembering that the interfacial tension always exerts a pressure tangentially along the surface, the surface free-energy balance (a... 

There remains the question of the physical-i.e., operational [9] -definition of the terms. It appears to the writers that the derivation as a force balance is merely intuitional, and, as a consequence, it leaves the quantities and yg o undefined operationally. Thus, if these be viewed as forces parallel to the solid surface, one must ask with what property of the solid they are to be identified. Unlike the case with liquids, there is for solids a surface or stretching tension (the work per unit stretching of the surface [20, 25, 28]), in general nonisotropic. If this is what is involved, liquid drops on a crystalline surface of low symmetry should not be circular in cross section this is apparently contrary to observation. From the thermodynamic derivation, however, we see that one is dealing with the work of exchanging one type of solid interface for another, and that surface free energies, not stretching tensions, are the proper quantities. 

If the interface is stationary, or if it translates without accelerating, then a steady-state force balance given by equation (8-180) states that the sum of all surface-related forces acting on the interface must vanish. Body forces are not an issue because the system (i.e., the gas-liquid interface) exhibits negligible volume. The total mass flux vector of an adjacent phase relative to a mobile interface is... 

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