Pure_Fluid_Surface_Tensions - Motivation The surface...

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Motivation The surface tension plays a major role in interfacial phenomena. It is the fundamental quantity that determines the pressure change across a surface due to curvature. This in turn is the basis for the stability analysis of both thin films and liquid jets. While experimental data exists for many pure fluids, it is the goal of this project to develop an approximation that describes most pure fluids. This would allow the determination of the surface tension even when experimental data is not readily available. Density-Gradient Theory The surface tension of an interface can be thought of as an excess energy due to the presence of the interface. In the density gradient theory, the interface is not thought of as a perfect dividing plane between the two phases. Instead there is a region in which the density changes from that of the high-density phase to that of the low-density phase. The energy in the interface region is a function of the density at each point in the interface. So, the surface tension represents the excess energy present in the interface region due to the “gradient” in density. So to derive a relationship between the energy in the interface and the surface tension, we start with the energy balance for the interface region: σ∗ S = A + PV – N µ Where A is the Helmholtz energy of the system, S is the interfacial area, P is the pressure, V is the volume, N is the number of molecules and µ is the chemical potential. Dividing the energy equation by the area gives (derived in Appendix A, part 1): σ = (A +P – ρ µ) dx Where the limits of integration are from infinity to minus infinity. This must be integrated over the interface. The density gradient model assumes a planar interface in the sense that the density is only changing in the direction perpendicular to the interface. For this derivation, we will call this the x-direction. Using
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the expansion for the Helmholtz energy derived in Appendix A, part 2, the expression for
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