the bulk solution, not on the surface charge or surface potential. For
example, for an aqueous 100-mM NaCl solution at 25°C, the Debye length is
0.96 nm, independent of the charge density or potential of the surface itself.
5.2.3 Interactions between charged surfaces in a liquid
In general, we can solve for the pressure due to the presence of ions at a
position x between two charged surfaces as
P x
ð Þ = kT
X
i
r i x
ð Þ
(5.29)
where r i (x) is the number density of the ith electrolyte at x (measured in
molecules per cubic meter). Since the distribution of ions at any given
point must be calculated using the Poisson–Boltzmann equation, then
solving the Poisson–Boltzmann equation for the system must precede a
calculation of the pressure between two surfaces. However, as mentioned
previously, the Poisson–Boltzmann equation is rather complicated to
solve for most systems of practical interest. It is beyond the scope of this
text to discuss the solutions to this equation, but students should be
aware of this general approach for calculating the pressure between two
surfaces. We limit our discussion to qualitative descriptions of the forces
operating between two interacting surfaces, descriptions resulting from
the application of the approach described previously.
In the most elementary example, suppose we have a flat, neutral surface
approaching a flat, charged surface in a parallel orientation, such as that
shown in Figure 5.13. Before the approach of the neutral surface, the
charged surface has associated with it a diffuse electrical double layer
extending out into solution. As the neutral surface approaches, however,
the counterions in the double layer must become confined to a smaller and
smaller volume, resulting in a decrease in entropy of the system. The
approach of the neutral surface likely causes some of the counterions to
bind to the charged surface, resulting in a slight decrease in energy.
However, this favorable decrease in energy is offset by the much larger
decrease in the entropy of the system. For this reason, the interaction
between a neutral and charged surface in a liquid must always be repulsive.
Now let’s consider the interaction between two charged surfaces of like
charge, as shown in Figure 5.14. As the surfaces approach each other,
their electrical double layers begin to overlap, resulting in an effective
decrease in the entropy of the system and making the interaction unfavorable. Hence, the interaction between two surfaces of like charge is
CHAPTER 5: Intermolecular Interactions and Self-Assembly
160
example, for an aqueous 100-mM NaCl solution at 25°C, the Debye length is
0.96 nm, independent of the charge density or potential of the surface itself.
5.2.3 Interactions between charged surfaces in a liquid
In general, we can solve for the pressure due to the presence of ions at a
position x between two charged surfaces as
P x
ð Þ = kT
X
i
r i x
ð Þ
(5.29)
where r i (x) is the number density of the ith electrolyte at x (measured in
molecules per cubic meter). Since the distribution of ions at any given
point must be calculated using the Poisson–Boltzmann equation, then
solving the Poisson–Boltzmann equation for the system must precede a
calculation of the pressure between two surfaces. However, as mentioned
previously, the Poisson–Boltzmann equation is rather complicated to
solve for most systems of practical interest. It is beyond the scope of this
text to discuss the solutions to this equation, but students should be
aware of this general approach for calculating the pressure between two
surfaces. We limit our discussion to qualitative descriptions of the forces
operating between two interacting surfaces, descriptions resulting from
the application of the approach described previously.
In the most elementary example, suppose we have a flat, neutral surface
approaching a flat, charged surface in a parallel orientation, such as that
shown in Figure 5.13. Before the approach of the neutral surface, the
charged surface has associated with it a diffuse electrical double layer
extending out into solution. As the neutral surface approaches, however,
the counterions in the double layer must become confined to a smaller and
smaller volume, resulting in a decrease in entropy of the system. The
approach of the neutral surface likely causes some of the counterions to
bind to the charged surface, resulting in a slight decrease in energy.
However, this favorable decrease in energy is offset by the much larger
decrease in the entropy of the system. For this reason, the interaction
between a neutral and charged surface in a liquid must always be repulsive.
Now let’s consider the interaction between two charged surfaces of like
charge, as shown in Figure 5.14. As the surfaces approach each other,
their electrical double layers begin to overlap, resulting in an effective
decrease in the entropy of the system and making the interaction unfavorable. Hence, the interaction between two surfaces of like charge is
CHAPTER 5: Intermolecular Interactions and Self-Assembly
160