Fl id S
Fluids,
Surface
f
T
Tension,
i
Capillaries
Foundation Physics
Lecture 2.12
MH
Motivation
Small Dimensions:
• Surface tension
dominates over other
forces ...
• Trees know it !
• Transpiration up to
200l/h
• Velocities up to
15m/h
Evaporation:
approx. 50 lilter
approx
per day
Photosynthesis:
solar energy
convertswaterand carbon
dioxide to sugar and starch
Water
storage:
in the porous
soil
Water
transport:
p
from the roots to the
leaves
Motivation
Small Dimensions:
Surface tension dominates
over other forces ...
Technologists use it!
Filling of channels
Bubble Jets
Membranes
Layering of Polymers
Soldering
Wetting of interfaces
Depending on the application you would like to have a
surface
f
either
ith hydrophobic
h d h bi (does
(d
nott like
lik water)
t ) or
hydrophilic (does like water)
Lotus effect (I)
New glass coating based on Nanotechnology:
Th second
The
d glass
l
shown
h
iis iimplemented
l
t d iin
expensive cars today.
Lotus effect (II)
Dirt resistant, self cleaning fabric (nano sphere company Schoeller (CH)
Motivation
Ink jet spotting to save
precious molecules
Functionalizing
cantilever arrays
Surface Tension
Macroscopic
p Phenomena:
The liquid film wants to contract as a result of the surface tension
(). The force F affecting from the outside redresses the balance.
Definition of the surface tension:
liquid film
F

L
Unit: 1 N/m
L
movable bar
The surface tension is a material constant.
constant
Surface Tension (2)
Microscopical
p
examination
The forces on the atoms respectively the molecules in a liquid
are different within or at the surface.
Within the forces cancel each other out:

Fk  0
At the surface arises a resulting force
directed
d
ected inwards:
a ds
' 
 Fk  Fres
i.e. to transport a molecule to the surface, work is required
against the opposing force Fres. This work increases the surface
energy of the liquid.
liquid Macroscopically the increase in surface
energy appears as a force F parallel to the surface of the liquid.
Surface Energy
If we want to increase the area of the liquid
q
film in the
experiment, work ∆W is necessary.
W  F  s
 2L  s  
L
W

A
Relating to the increase in surface area:
W

A
Unit: 1 J/m2
The surface
Th
f
tension
t
i can be
b regarded
d d
as energy per surface.
Surface Tension & Energy
• Surface Energy: Energy needed to extend surface
W = F.dx = 2 . L.dx
• Surface tension : Force per unit length [N/m]
• Systems always search to minimize Energy = minimize
S f
Surface/Interface
/I t f
with
ith highest
hi h t Energy
E
Table of Surface Tensions
Surface tension  of various liquids at 18 ºC for the
boundary layer to air in N/m:
Mercury
0.4355
Water (20°C)
0.076
Water (50 ºC)
0 0679
0.0679
Water (80 ºC)
0.0626
Benzene
0.029
Diethyl ether
0.017
Glycerol
0.0625
Olive oil
0.033
Ethyl Alcohol
0.023
Bl d whole
Blood,
h l
0 058
0.058
Soapy water (surfactant)
0.037
Drops of liquid
Surfaces of liquids are minimized areas. i.e. the amount of liquid tries to
minimize
i i i th
the potential
t ti l energy. Thi
This explains
l i th
the spherical
h i l shape
h
off a ffreely
l
falling water drop.
There is an equilibrium between the surface tension and
the pressure p inside the drop.
The resulting
Th
lti force
f
on a
circumference c
by surface tension
=
The counterforce by
pressure inside the drop
2  r      r 2  P
The pressure inside the drop is:
The pressure inside a bubble is:
2
P
r
4
P
r
c
Example of drop of liquid
Calculate the pressure within a drop of mercury
Drop of Mercury, r = 10-4 m, Hg = 0.471 N/m
(th pressure outside
(the
t id the
th drop
d
p0 is
i negligible)
li ibl )
2 2  0.471
0 471 N
3 N
4
P

 9.42 10 2  10 Pa
4
r
10
mm
m
Capillarity effect
Drop counter
(stalagmometer)
We calculate the volume
of the drop separating
from the pipette.
2  r   lal    g V
la = liquid-air surface tension
2  r   la
V 
g
V is dependent on
r la ,  and g
r,
Problem
Calculate the Drop volume of
a droplet as used in the inkink
jet spotting device at
CRANN?
= 0.076 N/m;
capillary diameter = 60 µm;
water= 998.59
998 59 kkg/m
/ 3
Capillarity
Dipping a tube into different liquids can show two different
wetting behaviours:
concave
Elevation of the liquid
g water
e.g.
convex
Depression of the liquid
e.g.
g mercury
y
Array Functionalization using Fluidics
Glass capillary device
Microfluidic device
PDMS cover
Filling
g with
Pipette tip
Reservoirs
2 cm
1 mm
Cantilever array
1 mm
5 mm
Principle of operation: microfluidic and capillary devices
filling and reflow
evaporation
capillary pressure p << 8 mbar
P ~8 mbar
reservoir
capillary
cantilever chip
Capillaries
p
are aligned
g
and connected to reservoir with
neutral pressure
=>Autonomous flow control by capillarity
Equilibrium
q
of forces at the liquid
q
interface
θ
Young s equation:
Young‘s
FSL-FSV+FLVcosθ = 0
θ
Three Phases: Contact angle
g at equilibrium
q
The atoms respectively the molecules at the liquid interface must be in
equilibrium. Z direction forces per length are:
 SV   SL   LV cos   0
 SV   SL
cos  
 LV
Contact Angle
Law of capillarity
Contact Angle
• Young’s equation is a useful starting point
•
for predicting contact angles, but
unfortunately, two of the three surface
tensions in equation from last slide are
extremely hard to measure
measure.
Values of LV are fairly simple to measure
and there are tables of these values, but
values
a ues for
o SV a
and
d SL a
are
ed
difficult
cu to
o
measure and one often resorts to a model
along with contact angles to predict these
values.
Contact angle
Calculation of the Capillary Rise
The force of the wall on the liquid Fw
FW  2    r   1,3   1,2 
 2    r   2,3  cos 
is in equilibrium with the weight of the liquid
column Fl.
Fl    r 2 h    g  2  r   2,3 cos   Fw
h
2   2,3 cos 
rg

2( 1,3   1,2 )
rg
2,3: liquid-vapor surface tension
Applies to capillary depression as well ( /2<< ; cos <0 ; h<0 ).
The above
Th
b
equation
ti ffor th
the column
l
h
height
i ht h applies
li ffor complete
l t
wetting too (=0 ; cos =1 ).
Column height
g by
y energy
gy consideration
The above equation
q
can be obtained by
y energy
gy consideration
as well.
The change in energy by wetting dEwet is equal to the change
of potential energy dEpot of the liquid column
column.
dEwet  dA   1,3   1,2 
 2  r  dh   2,3  cos     g  dV  h
   g  r 2    dh  h  dE pot
h 
2 
2 ,3
 cos 
r g
 h  
 r
Next Lecture
• To Be Covered: Biological and Medical
•
Applications of pressure and fluids
Reading: Chapter 7
 Section 7
7.1
1 and 7
7.2
2
Examples of pressure in humans, Molecular
phenomena and biological processes
 Solve the problems sets which have been
distributed by email today till Monday 18
18. Feb