Surface Tension Effects

3. Surface Tension Effects

Surface tension can be illustrated through Figure 4 . Figure 4a shows a vapor bubble surrounded by boiling liquid (during evaporation), while Figure 4b illustrates a drop sur- rounded by vapor (during condensation). The fluid molecules at the surface of the bubble in Figure 4a are under tension. We will call the bubble the embryo phase and the boiling liquid the bulk mother phase. Beyond the interface MN, the molecules are faced by liquid molecules (L) on one side (i.e., molecules at closer intermolecular spacing with stronger intermolecular forces), while on the other side they are faced by vapor molecules (G) which are at a larger intermolecular spacing with weaker intermolecular forces. Thus, the molecules at the surface of vapor embryo are pulled towards L due to the very strong intermolecular forces. However, such a pull results in an increase in the intermolecular spacing along MN that decreases the liquid density at surface along MN, resulting in stretching forces. These effects (i.e., the de- creased liquid density) persist over a small distance δ. Three regions are formed, namely, (1)

vapor of uniform density separated from (2) liquid of uniform density by (3) a layer of nonuni- form density of thickness δ. Tensile forces exist on the surface that lies normal to this thick- ness. At larger distances from the interface, liquid molecules are surrounded by other like vapor of uniform density separated from (2) liquid of uniform density by (3) a layer of nonuni- form density of thickness δ. Tensile forces exist on the surface that lies normal to this thick- ness. At larger distances from the interface, liquid molecules are surrounded by other like

Consider a vapor bubble of radius a v in a liquid. The net tensile force exerted by the vapor at the mid plane of the bubble normal to the area A =

v equals (P v –P liq )A. This force pulls the molecules against the attractive forces within the layer δ´. At equilibrium,

πa 2

(P v –P liq )A = σ´ 2πa v δ, i.e., σ´ δ´ = (P v –P liq ) A/C, (45) where σ´ denotes the attractive force per unit area that counterbalances the pressure forces and

C the circumference. As δ´→0, σ'→∞. The thickness δ´→0 is a surface discontinuity and σ´δ´ = σ, which denotes the surface tension. Therefore,

(P v –P liq ) = A/(C σ), i.e., (46) (P v –P liq )=2 σ/a v .

(47) which is known as the Laplace equation.

This discussion has considered the mother phase to be liquid enclosing a vapor em- bryo phase. We can also develop the relations for the other scenario, e.g., for a condensing water droplet, in which case the mother phase is vapor having its molecules farther apart with weaker intermolecular forces, while the liquid embryo molecules exist at closer intermolecular spacing. Mechanical equilibrium exists if the embryo (liquid) phase pressure is higher than that of the mother phase (converse to the above example, cf. Eqs. (46) and (47)). For a condensing drop of radius a liq that is surrounded by vapor at P v , at equilibrium,

(P liq –P v )=2 σ/a liq . (48) More generally (P embroyo –P mother )=2 σ/a where a denotes the embryo radius.

In expanding a bubble or increasing drop radius, surface tension work must be per- formed. For e.g., consider a film of liquid contained within a rectangular wire whose sides are L and W. If one of the the sides of width is pulled by dX , the film the film area A increases by L dx. The work done on the film

δW = –F dx = –σ L dx. Likewise, as a bubble of radius “a” expands, work is performed to stretch its film surface from

A=4 πa 2 to A+ dA. δW = – σ dA.

(49) where dA = 8 π a da or dA = 2 dV/a where V = (4 /3) π a 3