equation is a measure of change in Gibb’s
energy due to surface tension, and the
second term describes the mechanical work
done against the pressure difference across
the bubble surface.
Use the equilibrium condition dG/dr = 0 to
derive the Laplace equation:
ΔP =
2γ
r
The Laplace equation essentially gives the
pressure inside a stable bubble.
a. The Young–Laplace equation
can be applied to nonspherical bubbles.
In this expression, R 1 and R 2 represent
the two principal radii of curvature.
Show that the Young–Laplace equation
reduces to the Laplace equation for a
spherical bubble.
b. Calculate the Laplace pressure (ΔP) in
units of bars for bubbles of radius 1 nm,
2 nm, 10 nm, and 1000 nm. Comment on
how ΔP changes with the size of these
“nano-bubbles.”
c. Using your understanding of Laplace pressure, explain why small boiling chips are
often added to hot reaction mixtures.
References and recommended reading
Clint, J. H. Surfactant Aggregation, 1992, Springer Science +
Business Media, New York. This book contains some
excellent chapters on micelles and microemulsion. It is
easy to follow and is highly recommended for anyone
interested in surfactant science.
Evans, D. F. and Wennerström, H. The Colloidal Domain,
2nd ed., 1999, Wiley-VCH, New York. pp. 99–153, 217–295.
This is one of the finest books written on colloidal systems
and includes a thorough discussion of surfactant chemistry, monolayers, and microemulsions. Chapters 3 and 5
are particularly relevant to the intermolecular interactions
in nanomaterials.
Hartland, S., Ed. Surface and Interfacial Tension: Measurement, Theory, and Applications, 2004, Surfactant
Science Series, Volume 119. Marcel Dekker, Santa
Barbara, CA. This book provides a rigorous treatment of
interfacial tension, film stability, and wetting phenomena.
This book is recommended only for the serious student.
Holmberg, K., Jönsson, B., Kronberg, B., and Lindman, B.
Surfactants and Polymers in Aqueous Solution, 2nd ed.,
2003, John Wiley & Sons, Chichester, West Sussex, England.
This is one of the most thorough and comprehensible
books on surfactant chemistry. It is light on mathematical rigor but provides very thorough descriptions and
examples.
Roach, P., Shirtcliffe, N. J., and Newton, M. I. “Progress in
Superhydrophobic Surface Development,” Soft Matter,
2008, 4, 224–240. This is an excellent review article on
superhydrophobic surfaces and their relation to hard
nanomaterials. The review contains many examples and
references to methods used to construct superhydrophobic surfaces.
Rosen, M. J. Surfactants and Interfacial Phenomena, 3rd
ed., 2004, John Wiley & Sons, New York. This book is an
excellent source of data on many surfactants, including
industrial synthetic surfactants, “green” surfactants, and
amphiphilic molecules relevant to living systems.
Tóth, J., Ed. Adsorption: Theory, Modeling, and Analysis,
2002, Surfactant Science Series, Volume 119. Marcel
Dekker, Santa Barbara, CA. This book contains many
papers describing adsorption isotherms, including the
adsorption behavior of biological molecules. The book is
mathematically rigorous and is recommended only to
those seriously interested in adsorption phenomena.
CHAPTER 7: Fundamentals of Surface Nanoscience
254
energy due to surface tension, and the
second term describes the mechanical work
done against the pressure difference across
the bubble surface.
Use the equilibrium condition dG/dr = 0 to
derive the Laplace equation:
ΔP =
2γ
r
The Laplace equation essentially gives the
pressure inside a stable bubble.
a. The Young–Laplace equation
can be applied to nonspherical bubbles.
In this expression, R 1 and R 2 represent
the two principal radii of curvature.
Show that the Young–Laplace equation
reduces to the Laplace equation for a
spherical bubble.
b. Calculate the Laplace pressure (ΔP) in
units of bars for bubbles of radius 1 nm,
2 nm, 10 nm, and 1000 nm. Comment on
how ΔP changes with the size of these
“nano-bubbles.”
c. Using your understanding of Laplace pressure, explain why small boiling chips are
often added to hot reaction mixtures.
References and recommended reading
Clint, J. H. Surfactant Aggregation, 1992, Springer Science +
Business Media, New York. This book contains some
excellent chapters on micelles and microemulsion. It is
easy to follow and is highly recommended for anyone
interested in surfactant science.
Evans, D. F. and Wennerström, H. The Colloidal Domain,
2nd ed., 1999, Wiley-VCH, New York. pp. 99–153, 217–295.
This is one of the finest books written on colloidal systems
and includes a thorough discussion of surfactant chemistry, monolayers, and microemulsions. Chapters 3 and 5
are particularly relevant to the intermolecular interactions
in nanomaterials.
Hartland, S., Ed. Surface and Interfacial Tension: Measurement, Theory, and Applications, 2004, Surfactant
Science Series, Volume 119. Marcel Dekker, Santa
Barbara, CA. This book provides a rigorous treatment of
interfacial tension, film stability, and wetting phenomena.
This book is recommended only for the serious student.
Holmberg, K., Jönsson, B., Kronberg, B., and Lindman, B.
Surfactants and Polymers in Aqueous Solution, 2nd ed.,
2003, John Wiley & Sons, Chichester, West Sussex, England.
This is one of the most thorough and comprehensible
books on surfactant chemistry. It is light on mathematical rigor but provides very thorough descriptions and
examples.
Roach, P., Shirtcliffe, N. J., and Newton, M. I. “Progress in
Superhydrophobic Surface Development,” Soft Matter,
2008, 4, 224–240. This is an excellent review article on
superhydrophobic surfaces and their relation to hard
nanomaterials. The review contains many examples and
references to methods used to construct superhydrophobic surfaces.
Rosen, M. J. Surfactants and Interfacial Phenomena, 3rd
ed., 2004, John Wiley & Sons, New York. This book is an
excellent source of data on many surfactants, including
industrial synthetic surfactants, “green” surfactants, and
amphiphilic molecules relevant to living systems.
Tóth, J., Ed. Adsorption: Theory, Modeling, and Analysis,
2002, Surfactant Science Series, Volume 119. Marcel
Dekker, Santa Barbara, CA. This book contains many
papers describing adsorption isotherms, including the
adsorption behavior of biological molecules. The book is
mathematically rigorous and is recommended only to
those seriously interested in adsorption phenomena.
CHAPTER 7: Fundamentals of Surface Nanoscience
254