13 CBE 341 Lecture 2 (9/15) – Surface tension Molecules in a liquid feel a mutual attraction, and thus have a natural tendency to minimize their surface area. The obvious case is a liquid drop on a non-wetting horizontal surface, see figure below. Small droplets become almost perfectly spherical (minimizes surface area per volume). However, larger droplets become flatter as the gravitational forces acting on the fluid become more significant. The fluid property that describes this phenomenon is called surface tension. The surface tension, γ, at an interface between two immiscible fluids is a material property which has the units of energy per unit area or force per unit length. (Note: s and g are the most commonly used symbols to denote surface tension. Surface tension is also referred to as interfacial tension). It is the energy required to create unit interfacial area, which is a definition often used in thermodynamics. The latter definition, namely force per unit length, is used in mechanics. Because, surface tension gives rise to forces at the interface, it affects force balance at the interface. To illustrate in a simple manner that these definitions are equivalent, consider a planar interface of length L and width W, see figure below. Let us assume that the interface is homogeneous and no spatial variation of properties occurs. We will begin by applying a mechanical definition for the surface tension. The forces acting to maintain the interface are shown. Now if we stretch this interface by moving the right side by a distance x,D then the work done xLx Lx.= g×D= gDxee The amount of new interface created L x. = D\ The energy required to create unit interfacial area is given by (Work done)/(area created) = xL/Lx .= gD × D = g Thus, we have recovered the definition of surface tension based on surface energy.14 In this course, I will not go into formal derivation of force balances at fluid-fluid interfaces; however, I will simply state without proof some main results, which are useful to know. Even though we considered a planar interface in the previous page, fluid-fluid interfaces need not be planar – instead, they can assume complex three-dimensional structures (cylinders and spheres being the simplest examples). In the presence of such a curved interface, the pressure in the fluid on either side of the interface will differ from each other even in an equilibrium state. Consider the surface in the above figure. Now if the surface is planar/flat the forces acting on either side of the interface are equivalent, and DP = 0. However, if DP ≠ 0 the surface will be curved and the force bending the surface will be balanced by the surface tension. Consider the liquid droplet in the figure below. A force balance on half of the droplet yields, 22() ()(2)ioPr P r rppgp=+ where Pi is the pressure inside the droplet, Po is the pressure outside the droplet, and r is the radius. -gWeygLex-gLexgWeyyxBlue region is the new area created15 After rearrangement: 2PrgD = Note that in this derivation we sliced the droplet in half and performed a force balance on the control volume. This control volume was selected for easy calculation of the forces; however, a smaller control volume which represents less than half of the spherical droplet could also be used (figure below). Performing the same procedure used to calculate DP for the half droplet, we get: 12cosPrgqD = We recognize that 1cosrrq=, with substitution we obtain the same expression: 2PrgD = rHalf&a&Spherical&DropletFg=&2prg =&DPpr2r1Smaller&Fraction&of&a&Spherical&DropletFg=&2pr1gcosq =&DPpr12qq2gDP&=&&r2gcosqDP&=&&r116 This also shows that the pressure throughout a spherical droplet is homogenous. Note that if the pressure is not homogenous within a droplet, the droplet will warp away from a spherical geometry. Analogously, how do we arrive at this expression from the thermodynamic definition of surface tension? Recall that the interfacial energy is given by, 24iErpg=. This is the amount of work done to give the droplet its spherical shape. Given that work, gdW P dV=, (note that here gP is the gauge pressure, and gPP= D) thus idW dE=. Assuming the surface tension is constant, 8idE r drpg=. Since for a spherical element 24dV r drp=, we obtain the following expression for P: 2824igdEdW r drPPdV dV r dr rpg gp= D === = This is the same expression derived using a mechanistic approach. Now for any surface, there is a pair of orthogonal curves at any point on the surface and if the curves are of differential dimensions, each curve may be approximated as the arc of a circle. The radii of the two circles are called the principal radii of curvature (R1 and R2). The curvature of a surface, at any point, is described in terms of R1 and R2 of the orthogonal curves corresponding to that point. On a complex surface the radii may vary from point to point. If the centers of the two circular arcs lie on the same side of the surface, then both R1 and R2 assume the same sign. If they lie on opposite sides, then one of them is positive and the other is negative.17 To decide when a positive sign should be assigned and when a negative sign should be used, we choose (arbitrarily) one side of the interface as “inner” side and the other as “outer” side. Let the pressure in the “inner” fluid at the point of interest (i.e. right next to the interface) be denoted as Pinner. Similarly, let the pressure in the outer fluid at the point of interest be Pouter. We define as capillary pressure, Pc, the difference between the two pressures. .CinnerouterPPP P= D = - The correct sign to assign for the radius of curvature depends on which side of the interface the center of the circular arcs lie. If the center of the circular arc lies on the side of the inner fluid, then the corresponding radius gets a positive sign. If it lies on the side of the outer fluid, it gets a negative sign. Thus, in the figure above, if the inner fluid is chosen as that lying above the interface, the radii R1 and R2 are both positive; if you had chosen the fluid above the interface as outer fluid, then both R1 and R2 are assigned negative signs. The Young-Laplace