Supplementary Materials for Teachers and Workshop Presenters
Dr. Rachel Levy
Mathematics Department
Harvey Mudd College
Creative Commons License CC BY-NC-SA 3.0 by Rachel Levy
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Challenge questions:
What is a surfactant?
What is the difference between
Buoyancy
Surface tension
What is soap’s job?
How are SOAP and SLOPE related?
What is Soap?
Soap is a surfactant
http://commons.wikimedia.org/wiki/File:Surfactant.jpg
The head is hydrophilic (loves water) and the tail is hydrophobic (hates water).
Fancy term for soap: surfactant
 Surface-active-agents lower surface tension
 With soap the surface tension of water is lower because the
bonds between soap and water on the surface are weaker than
the bonds on the surface of water with no soap.
 Surfactants are used in detergents to surround grease in balls
called micelles. The hydrophobic tails attach to the bits of dirt
and grease. The hydrophilic heads stay on the outer surface of
the micelle and help it move into the water and off clothes and
dishes.
Surfactants can attack dirt
Carlota Oliveira Rangel-Yagui1, Adalberto Pessoa Junior, Leoberto Costa Tavares,
J Pharm Pharmaceut Sci (www.cspscanada.org) 8(2):147-163, 2005
Your First Breath
Air inflates the alveoli of lungs
Process is like blowing up balloons
Natural surfactants make
it easier to breathe by lowering
surface tension in the alveoli
http://hyperphysics.phy-astr.gsu.edu/Hbase/ptens2.html#alv
Your lungs need surfactant!
http://www.valuemd.com/usmle-step-1-forum/21404-alveoli-surfactant.html
Surfactants lower surface tension.
(Answer to challenge question:
Soap’s job is to lower surface tension.)
What is surface tension?
Surface tension is an attractive force
between molecules on the surface of a
fluid.
Wikipedia:WassermoleküleInTröpfchen.svg
Surfactants lower surface tension
by weakening the attraction
between surface molecules
A water strider is a bug that uses surface
tension to walk on water.
Water Strider
http://www.everythingabout.net/articles/biology/animals/arthropods/i
nsects/bugs/water_strider/
Agnes Pockels (1862 – 1935) was one of the first
people to carefully study surface tension.
Lord Rayleigh to Nature magazine (1891):
I shall be obliged if you can find space for the
accompanying translation of an interesting letter
which I have received from a German lady, who
with very homely appliances has arrived at
valuable results respecting the behaviour of
contaminated water surfaces.
http://cwp.library.ucla.edu/Phase2 Pockels,_Agnes@871234567.html
Agnes Pockels: Nature Magazine
I will describe a simple method, which I have
employed for several years, for increasing or
diminishing the surface of a liquid in any
proportion, by which its purity may be altered at
pleasure.
A rectangular tin trough, 70 cm. long, 5 cm.
wide, 2 cm. high, is filled with water to the brim,
and a strip of tin about 1 1/2 cm. laid across it
perpendicular to its length, so that the underside
of the strip is in contact with the surface of the
water, and divides it into two halves.
By shifting this partition to the right or the
left, the surface on either side can be lengthened or
shortened in any proportion, and the amount of
the displacement may be read off on a scale held
along the front of the trough.
What is Slope?
Definitions of slope
Slope =
rise
run
Slope:
change in y
change in x
Slope:
y2-y1
x2-x1
What is the slope of this line?
What is the slope of this line?
What is the slope of this line?
What is the slope of this line?
What is the “slope”
of this curve?
What is the “slope” of this curve?
Consider tangent lines along the
curve -- at each point you can
measure a “slope” using the slope
of the tangent line.
Where is the slope of this curve…
positive? negative? zero?
Color gradient graph
Let x = position (left to right)
y= intensity (darkness) of the blue
y
x
Color gradient graph
Let x = position
y= intensity of the blue
y
x
Definition of slope
Slope:
change in one quantity
change in another quantity
blue intensity
position
Definition of slope
Slope:
change in intensity of blue
change in position
blue intensity
position
Experiments
1
Divide into teams of three.
2
Give each team member a number: 1, 2, 3.
3
Possible roles: experimenter, recorder, reporter
Paperclip Experiment
Supplies:
Clean hands (no soap, lotion)!
One paper plate
Cup of water
One large paperclip and one small paperclip
Piece of paper (optional)
Soap
Paperclip Experiment
Float a paperclip (or two) on the surface of the
water. If this is tough, float the paperclip on a scrap of paper,
then sink the paper, allowing the paperclip to remain on the
surface.
Put a drop of detergent near it.
What happens? Why?
What does it mean for something to float? Sink?
Hint: there are two possible answers
What does it mean for an object to
“float”?
Float could refer to buoyancy…
Float could refer to surface tension…
Buoyancy
 Objects less dense than the water will rise to the
surface.
 But metal ships (more dense than water) float! Why?
 When do metal ships sink?
Sinking
Gravity pulls the mass of the boat down.
The mass of the boat is black.
Buoyancy
Buoyancy pushes the boat up.
Archimedes (~250BC): Any object, wholly or partially immersed in a fluid, is
buoyed up by a force equal to the weight of the fluid displaced by the object.
Displaced water is green.
Buoyancy
If the boat springs a leak and takes on water,
how much water can it hold before it sinks?
Why do you feel light when you are floating in the water?
Can you explain why it is easier to float
in salt water than fresh water?
Hints: weight = mass* gravitational
constant
mass = density *volume
In your experiment, did the paper clip “float”
because of
(a) buoyancy or
(b) surface tension?
Surface Tension!
Boat Experiment
Supplies:
Clean hands (no soap, lotion)!
One paper plate
Cup of water
Paper boat
Soap
Boat Experiment
Float a paper boat on one side of the bowl.
Put a drop of detergent behind it
(between the boat and the edge of the bowl).
What happens? Why?
Plotting Surface Tension
 What is happening to the surface tension of the water
in the boat experiment?
Plotting Surface Tension
Graph
x = position
y = surface tension
(you can also draw your boat!)
Plotting Surface Tension
Graph
x = position
y = surface tension
(you can also draw your boat!)
Time 0: Before you put the soap in the water
Plotting Surface Tension
Graph
x = position
y = surface tension
(you can also draw your boat!)
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in
(before the boat has moved much)
Plotting Surface Tension
Graph
x = position
y = surface tension
(you can also draw your boat!)
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in
(before the boat has moved much)
Time 2: After the boat has stopped moving
Graph x = position y = surface tension and pic of boat
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in (before
the boat has moved much)
Time 2: After the boat has stopped moving
Time 0
Surf
Tens.
Position
bowl
View of bowl from top
Graph surface tension along this line
Graph x = position y = surface tension
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in (before
the boat has moved much)
Time 2: After the boat has stopped moving
(reminder: soap lowers surface tension)
Time 0
Surf
Tens.
Time 1
Surf
Tens.
Position
across
Bowl
Time 2
Surf
Tens.
Position
across
Bowl
Position
across
Bowl
Time 0
Time 0 before soap is added: zero slope  no motion.
Time 1
Time 2
No slope (zero slope)  no more motion.
How does the sign of the slope
relate to the direction of the
boat motion?
Time 0 before soap: zero slope  no motion.
How does the sign of the slope
relate to the direction of the
boat motion?
Time 1 after soap: positive slope  motion to right.
How does the sign of the slope
relate to the direction of the
boat motion?
Time 2 after soap: zero slope  no motion
Surface tension can be
high (time 0) or low (time 2),
but if there is no change, the surface
tension does not cause the fluid on the
surface (and the boat) to move.
When there is a change in surface
tension (time 1) across the bowl,
there is surface motion.
Pepper Experiment
Supplies:
Clean hands (no soap, lotion)!
One paper plate
Pepper or Tea leaves
Soap
Pepper Experiment
Put some water on a plate
Sprinkle pepper on the water
Put a drop of soap in the middle
Graph your results at
time 0: before you added the soap
time 1: right after you added the
time 2: longer after you added the soap
Time 0
Time 1
Time 2
The big idea:
To get the motion you saw in the experiments, there had to
be areas with different surface tension.
Change in amount of soap  change in surface tension
 motion!
Slope:
rise
run
or
change in y
change in x
Slope:
change in surface tension
change in position
Agnes Pockels can help you find s2 and s1!
x2-x1
s2-s1
Challenge questions revisited:
What is a surfactant?
What is the difference between
Buoyancy
Surface tension
What is soap’s job?
How are SOAP and SLOPE related?
Dr. Levy’s research
 Thin liquid films and surfactants
 Surfactant moves the fluid
 Fluid moves the surfactant
 Changes in space and time
 Two partial differential equations
one equation for height of the film
one equation for surfactant concentration
Levy’s research with
Harvey Mudd College undergraduate
math majors:
Students solve these partial differential equations modeling a thin liquid film
and surfactant using computer programs
How does the film height and surfactant concentration evolve
in space and time?
How do solutions of this model compare to experiments?
Height equation
Surfactant concentration equation
The upside down triangle is a fancy sign for slope!
The other symbol in yellow is a function that describes
how surfactant affects surface tension.
Collaborator Dr. Karen Daniels, Physics, North
Carolina State University
In the left picture, you are looking down at an experiment. The red line
is from a laser that indicates the glycerine film height. The yellow areas
show the variation in the concentration of surfactant on the thin film
of glycerine (slope!). The right picture is an enlarged horizontal slice of
the image on the left. We compare this experimental data to
mathematical solutions of the two equations.
Thank you very much!
 Dr. Rachel Levy
levy@hmc.edu