Consequences of surface tension in homogeneous liquids:

Size: px

Start display at page:

Download "Consequences of surface tension in homogeneous liquids:"

Spencer Gray
6 years ago
Views:

1 Colloid science 3

2 Consequences of surface tension in homogeneous liquids: Pressure change of curved surfaces: Laplace equation Vapor pressure change of curved surfaces: Kelvin equation

3 There are two easily recognizable consequences of the existence of surface tension, even in clean, one-component liquids. Both phenomena manifest itself in case of curved surfaces, that is in case of a bubble, a droplet or in a thin capillary. The droplet and the bubble appear because the surface tension draws together the surface, in a small capillary there is curved surface because of the wetting or non-wetting behaviour of the liquid towards the solid.

4 surface tension curved surface pressure change change of vapor tension These changes appear only in case of curved surfaces: droplets, bubbles and capillary meniscus. The later depends on the wettability of the surface of the capillary. 4

surface tension pressure, vapor pressure change water

5 surface tension minimal surface curved surface cause effect relations leading to the final consequences of surface tension pressure, vapor pressure change water droplets bubbles in water surfaces in thin capillaries

6 Consequences of surface tension in homologeus liquids Capillary pressure: Laplace equation p 2 r Vapour pressure change: Kelvin equation RT ln p g p 2 V r m

7 Laplace pressure

8 Laplace pressure, capillary pressure The origin of the pressure and vapour pressure change in both cases is the curved surface. The flexion of the surface curvature in both cases has a direction. It means that if the flexion is positive from the liquid s point of view, the pressure is positive as well. Negative flexion results in negative pressure. Pressure is always higher inside a bubble or droplet. (This means a negative pressure in case of a bubble, from the liquid s point of view.) The cause of the pressure is the contractive force of surface tension.

9 Surface tension always wants to lessen the surface, because the surface means surplus energy, in the form of surface energy. (Gibbs free energy) The result of this contraction is a counter-force inside the bubble or droplet. Without any counter-force, the droplet would disappear under the constant contraction, and this is nonsense. So inside the droplet there will be a higher pressure. The smaller the droplet, the higher the pressure in it. If a surface can be characterized by two radiuses, the equation changes to p 2 p r r 1 r 2

10 Curvature with two different radii. This pressure change is the cause of the breakup of streams into drops.

11 Deduction of capillary pressure enlarging the radius by dr causes da enlargement on the surface The contractive force tries to reduce the volume, this is the cause of the pressure inside. The surface enlargement increases the surface Gibbs free energy, which is covered by the pressurevolume work. Δp dv = γ da V=A r Δp A dr = γ da Δp A = γ(da/dr) Δp dv = Δp A dr A= 4r 2 Π da/dr = 8rΠ Δp4r 2 Π = γ8rπ Δp=2 γ/r derivative of A with respect to r

12 Δp dv = γ da the pressure-volume work (pv) covers the energy needed for the surface enlargement (γa= specific surface energy area of surface) volume=area radius (V=A r) so the left side of the equation (Δp dv ) will be Δp A dr Δp A dr = γ da dr positioned to the right side: Δp A = γ(da/dr) for expressing da/dr we need the expression of the surface of a sphere: A= 4r 2 Π

13 the derivative of A with respect to r: da/dr = 8rΠ by the displacement of A and da/dr in the equation Δp A = γ(da/dr) Δp4r 2 Π = γ8rπ we gain the final form of the equation: Δp=2 γ/r

14 Curved surfaces can be found in thin, wettable capillaries, in bubbles and droplets. meniscus in a capillary convex droplet in air concave contact angle bubble in liquid

Capillary elevation x radius of capillary r radius of curvature for the meniscus α contact angle for the meniscus from the equation of the Laplace pressure: p 2 r 2 cos x from the equation of the

15 Capillary elevation x radius of capillary r radius of curvature for the meniscus α contact angle for the meniscus from the equation of the Laplace pressure: p 2 r 2 cos x from the equation of the hydrostatical pressure: Δp = ρgh from these equations: gh 2 cos x h 2 cos x g The capillary elevation is dependent from the diameter and quality of the capillary, the surface tension, special gravity and quality of the liquid. The two quailties are hidden in the contact angle which depends on the quality of solid and liquid partners.

16 The smaller the radius, the higher is the capillary elevation.

17 Capillary elevation in capillary systems. The soil is a capillary system too, the finer the particles are in it, the higher is the capillary elevation. It is the case in the walls in wet environment as well.

Lower surface activity means that the same delta p can be achieved in smaller bubbles, than in Coca Cola.

18 The difference between bubbles of sparkling water and beer. In beer there are much smaller bubbles than in Coca Cola, because of the surface active agents in beer. Lower surface activity means that the same delta p can be achieved in smaller bubbles, than in Coca Cola. So the bubbles can manifest themselves in much smaller size. (The bubble needs a smaller pressure to detach from the wall, because the surface tension of beer is smaller than that of the water.)

19 Pressure difference (Δp) in water droplets with different radii droplet radius 1 mm 0,1 mm 1μm 10 nm Δp (atm) 0,0014 0,0144 1, ,6

20 Capillary elevation can be experienced in the plant tissues. The fine capillaries of the xylem can transport water even to the highest of the trees. Concrete makes a capillary structure prone to absorb water

21 Kelvin equation

22 Vapour pressure change of curved surfaces It is harder to detach a molecule of water from a concave surface, than from a planar or convex surface. The simple cause is, the more neighbours a molecule has, the harder is to detech it from the grip of its neighbours.

24 The changed vapour pressure is expressed by the Kelvin equation. where R T p c p V m r γ RT ln p c p 2 is the Regnault constant is the temperature r V is the vapour pressure of the curved surface is the vapour pressure of a planar surface in the same conditions is molar volume of the liquid is the radius of the curvature m is surface tension of the liquid

Let s regard the following: liquid phase vapor phase If we detach a droplet of the size of dn mol from the liquid, the difference between this droplet and an

25 Let s regard the following: liquid phase vapor phase If we detach a droplet of the size of dn mol from the liquid, the difference between this droplet and an imaginary droplet (same size, that is, dn mole) inside the bulk phase, will be only the surface. (The real droplet will have surface, the imaginary will not.)

26 da = new surface made γ = surface tension (=Gibbs free energy, enthalpy) μ = chemical potential of liquid (partial molar free enthalpy) μ g = chemical potential of droplet (in gase phase) μ = μ 0 +RT ln x x fugacity, can be replaced by pressure The difference in enthalphy is needed for the surface enlargement, which makes surplus surface free energy: (μ g μ) dn = γ da

27 ( μ g μ) dn = γ da m g - m da dn da/dn can be written in the form of: da dn da dr dr dn the transformation is necessery because the derivative of surface with respect to molar number cannot be calculated otherwise

28 We need da/dr, the derivative of area surface of a sphere: A= 4r 2 Π da/dr= 8r Π

29 dr/dn, can be calculated regarding: n of moles = volume/molar volume volume of a sphere = molar vol. = V m n r p 3 V 4r 3 P m V m 4 r 3 3 P

30 invert of dr/dn will be dn/dr, it is the derivative of the previous equation: dn 12r 2 P dr 3 V m the converse of it (numerator and denominator exchanged, supposing the later is not zero): dr dn V m 4r 2 P

31 the fraction can be simplified by pi, r and 4 da/dn= 8r Π V m 4r 2 P 2V m = r put it into the original m g - m da dn equation m g - m 2V m r

32 The chemical potential: A kémiai potenciál egyenlete: μ=μ o +RT ln x i a fugacitás helyett a parciális nyomást írhatjuk, a μ o pedig mindkét esetben azonos mert azonos anyagról van szó. μ=μ o +RT ln p fugacity replaced by pressure 2V m (μ o +RT ln p g ) ( μ o +RT ln p)= r RT ln p g p 2 r V m μ o is the same, being the vapor and liquid phase identical quality (a két logaritmus különbsége a hányadosok logaritmusa) difference of logarithms is the logarithm of the ratio 32

33 RT ln p g p 2 V r m RT ln p g p or. V m 1 1 ( r 1 r 2 ) The smaller is the radius of a capillary, the more difficult is the evaporation of the liquid from it. This wapor pressure change of curved surfaces is the cause of the Ostwald-ripening. In a heterodisperse system, the smaller droplets will disappear and will be deposited on the bigger ones. Coarsening of the system.

35 The direction of the curvature matters too. Direction is seen from the liquid s point of view. The liquid surface convex positive curvature faster evaporation The liquid surface concave negative curvature slower evaporation r > 0 r = r < 0 p < p o p = p o p >p o

36 Ostwald ripening of colloid systems. The bigger the droplet, the slower is its evaporation. Smaller droplets evaporate faster, their content is deposited on the bigger ones. Bigger droplets grow, smaller droplets vanish. Heterogeneous phases appear. (The same happens if you leave the ice cream in the fridge for too long.)

38 Ageing of sols means the particle size grows, dispersity decreases. Smaller droplets (bubbles, crystals) disappear, bigger ones are growing. What happens if we open the valve between the two bubbles? Which one of them will blow up the other?

Surface chemistry. Liquid-gas, solid-gas and solid-liquid surfaces.

Surface chemistry. Liquid-gas, solid-gas and solid-liquid surfaces. Levente Novák & István Bányai, University of Debrecen Dept of Colloid and Environmental Chemistry http://kolloid.unideb.hu/~kolloid/