Biomedical Engineering Reference
In-Depth Information
ε
s
that are connected by a simple inverse relationship.
κ
λ
=-
1
A
∂
A
∂Π
,
(7)
T
=-
A
∂Π
∂
-1
λ
ε
S
=
κ
.
(8)
A
T
These are the 2D analogs of the bulk isothermal compressibility and bulk
modulus, respectively. The scaling concepts introduced above clearly make
sense in terms of the compressibility/elasticity as well. For polymers where
the interface is a good solvent, the lateral modulus, sometimes called static di-
lational elasticity, is small whereas it becomes larger as the interface becomes
poorer.
3
Capillary Wave Dynamics
An earlier episode in the history of surface wave effects is the observation
by Benjamin Franklin [3] of calming ripples on Clapham Pond of nearly
half an acre with a teaspoonful of oil. Although he did not explicitly esti-
mate the thickness of the oil film, it turned out to correspond to roughly
a monomolecular thickness [4]. This was no surprise, as seamen for centuries
knew of this effect as well. The presence of the oil provides loss mechanisms
in addition to the viscosity of water, which damp out the energy of these prop-
agating gravitational waves. However, gravitational waves are not the only
ones present. On the surface of any liquid, there is a continuous distribu-
tion of wavelengths associated with roughening due to spontaneous density
fluctuations in the underlying liquid. These are called capillary waves or “rip-
plons”, and their amplitude was determined only two decades ago to be rather
small, 3-5A [36]. Thus, the interface of air/water is known to be molecularly
smooth averaged over a macroscopic length scale (1 cm
1 cm) and a time
scale of hours. We now turn to examine in detail the dynamics of these waves.
The theoretical underpinnings of the modern examination of the capillary
waves are traced to a monograph by Levich [37] and the experimental efforts
are traced to the pioneering laser light scattering studies by Katyl and In-
gard [38, 39], simultaneously by Bouchiat et al. [40]. The capillary wave origin
of surface laser light scattering was well established by Langevin and cowor-
kers in the early 1970s [41-45], and she has subsequently lead the community
in the study of surface-active substances on the interface, both experimentally
and theoretically [46]. She coined the term, surface light scattering (SLS), and
we will adhere to the designation.
One of the earliest theoretical studies in this area was work by Lord
Kelvin [8, 47] who derived an expression for the frequency of a propagating
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