Surface Tension
The effects of surface tension are easily seen in daily life. Surface tension allows a meniscus of
water to hover above the level of a glass, and a water bug to walk along the surface of a lake.
Earlier, we considered EMF as a form of non-PV work. Similarly, we can consider the energy
contributions due to surface tension.
Forcesurface tension = σ l
where l is the length of the boundary.
dwnon-PV = σ dA
= (surface tension) (change in area)
The units of σ are J/m2 or N/m, and σ > 0.
For a droplet of water in air, or for a bubble in a liquid,
dG = -S dT + V dP + σ dA + µwater dnwater
The positive sign in front of of σ means that increasing the area increases the energy. Hence, an
energy cost due to surface tension will be decreased by minimizing the area surrounding any given
volume. Surface tension effects are most easily observed on small length scales. For very small
bubbles, gravity plays almost no role and the bubbles are circular (as in your bottle of beer or
Coke). The same thing happens for a small liquid droplet in air. However, if the force due to
gravity is very small (for example, in the “microgravity” of outer space), even very large bubbles
can be spherical. For example, large gas bubbles in the stomachs of astronauts are spherical and
surrounded by liquid. For an astronaut, all belches are wet belches.
A water-air interface has a large surface tension. Almost anything added to water decreases the
surface tension (except, for instance, ionizing salts).
Values of surface tension of the substance with air
(Most taken from TS+W Table 5.2)
Mercury
Water
“
Benzene
Acetone
Ethanol
n-Hexane
σ (N/m)
0.487
0.072
0.059
0.029
0.024
0.023
0.018
Temp.
15ûC
25ûC
100ûC
20ûC
20ûC
20ûC
The “Laplace pressure” describes the difference in pressure between the inside of a bubble and the
outside of a bubble due to the surface tension, σ. We will derive the Laplace pressure for a sphere:
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Pin
Poutside
Imagine using an eyedropper to make a spherical bubble just under the surface of some water. The
difference in pressure between the inside of the bubble (Pinside) and the atmospheric pressure outside
(Poutside) will determine the radius r of the drop. Assume that the volume of water is big compared
to the volume of the bubble so that the height of the water and the temperature stay approximately
constant. Considering just the bubble:
dwsurface tension = dwPressure-volume
σ dA = (Pinside − Poutside )dV
For a sphere, Area = 4πr2.
Volume = 4/3 πr3.
dA = 8πrdr.
dV = 4πr2dr.
Substituting above,
σ (8πr dr ) = (Pinside − Poutside )(4πr 2 dr)
2σ = (Pinside − Poutside )r
So then, (Pinside − Poutside )=
2σ
r
The signs of Laplace pressures are such that pressure is higher on the concave side of the surface.
For small radii, the difference in pressure can get quite large. For a small air bubble of radius of 0.1
micron submerged in water, with σ = 0.07 N/m, then Pinside - Poutside = 1.4 x 106 Pa = 14 atm!
If the bubble were an ellipse (or another shape with two radii of curvature) rather than a sphere,
1
1
PLaplace = Pinside − Poutside = σ  +  .
 R1 R2 
(I’m not going to prove this here, but I’ve included a proof of it on the class website under the
“tutorial” section.)
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Example A: Capillary Action
If a capillary that has a small radius and a hydrophilic surface (e.g. glass) is put into water, the water
will climb the capillary. The meniscus will be concave toward the top.
Patm
• Pout
h
Patm
(Pinside − Poutside )= 2σ
r
Pressure on the inside of the interface, Pinside = Patm.
Pressure on the outside of the interface (just barely into the water side), Poutside = Patm - ρgh.
(We’ve seen this previously in the discussion of osmotic pressure).
Substituting in,
Patm − (Patm − ρgh) =
2σ
r
2σ
ρ gr
Notice that you could turn the equation around and use it as a way to experimentally determine σ.
ρ grh
σ=
2
Therefore, capillary action will make the water rise to a height h =
Example B: Capillary Action Revisited
Consider the same example, but for a capillary with a hydrophobic surface.
Patm
Patm
h
Pin
Pout = Patm
Pin = Patm − ρgh
Pin − Pout = − ρ gh =
so then h = −
2σ
r
2σ
ρgr
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Example: “The Hotdog Effect”
Imagine heating a sausage. The sausage skin doesn’t expand much and as the meat inside expands
the skin will eventually rip. Along which direction will it split? The hotdog has two radii of
curvature:
•
R1
Flong
Fshort
R2
The force pulling on the sausage skin in each direction is:
dFlong axis = Plong axis dA =
σ
Rlong
dFshort axis = Pshort axis dA =
dA = a small number
σ
Rshort
dA = a big number
The force is much bigger around the short axis of the sausage than along its length. Therefore, it
will split as shown below.
Example: Coarsening
Consider two air bubbles of liquid with the same surface tension, σ, one with a large radius R1, and
one with a small radius R2. There is a uniform pressure outside the bubbles.
R1
R2
What happens (can you guess?) The Laplace pressures are:
P1 =
2σ
R1
and
P2 =
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2σ
R2
The pressure inside bubble 2 (with respect to some common exterior pressure) is greater than in
bubble 1, so air will flow from bubble 2 to bubble 1 until the second bubble disappears entirely. In
foams this is called “coarsening” or “Ostwald Ripening” (especially in crystals).
Airflow
Aveoli, the small air sacs in lungs, act as connected bubbles of radius 0.1-0.5mm in a liquid.
If the aveolar surface were simply an air-water interface, the change in Laplace pressure would
cause small aveoli to coarsen, leading to lung collapse. Nature solves this problem by adding a
layer of surfactant (lipids and protein) to the aveolar surface. The surfactant layer greatly reduces
σ, which is proportional to the concentration of surfactants at the aveolar surface.
ta
e r nt
∆P =
∆P is small if σ (c1 ) ≈ σ (c2 )
ue
S u rf a c t
wa
s
Tis
2σ (c1 ) 2 σ (c2 )
−
R1
R2
R1
R2
Lung surfactant is compromised is smokers and in premature babies. The age limit at which
premature babies live is related to when they can produce lung surfactant. This is an active area of
research, especially to produce a synthetic lung surfactant. Currently, many therapies for premature
babies use lung surfactant derived from cows and there are concerns about transferring diseases
among species. I don’t know if any therapy is currently available for adults who have degraded their
lung surfactant by smoking (other than advising the patient to stop smoking).
References to Current Literature:
Enhancement of dendrimer-mediated transfection using synthetic lung surfactant exosurf neonatal in vitro,
KukowskaLatallo JF et al. BIOCHEM BIOPHYS RES COMM , v. 264(#1) pp. 253-261 OCT 14, 1999
Synthesis and structural characterization of human-identical lung surfactant SP-C protein
MayerFligge P et al. JOURNAL OF PEPTIDE SCIENCE , v. 4(#5) pp. 355-363 AUG 1998
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