Chapter 3

Simple Models for

Molecule–Molecule Interactions

It is often the practice in the discussion of classical tip–substrate forces

to cite relevant equations obtained using assumptions that are not clearly
understood because the derivation is never discussed in any detail. Such
an approach seems to be the norm in today’s literature rather than the
exception. Students from some disciplines do not well tolerate the mathe-
matics required. Other disciplines of a more applied bent seem in a great
rush to move forward, blindly accepting what is evidently already known.
While acceptable up to a point, the price paid is a loss of insight and intu-
ition. If you are satisfied with this approach, then you can safely skip this
chapter and continue to refer to the standard results for the van der Waals
interaction which are summarized in Sec. 3.3.
In the following chapter, we attempt to systematically explain the essen-
tial points needed to understand the simple models of molecular interactions
and we systematically review the physics underlying the hierarchy of inter-
actions that lead to tip–substrate forces. We follow a pedagogical course,
much in the spirit of Israelachvili’s book on surface forces, in which many
simple models are first developed and discussed before they are used to
reach the next level of sophistication [israelachvili98].
While an effort is made to derive all the important results, the topics
covered in this chapter are most accessible to students who have completed
upper division courses in Electricity and Magnetism, Statistical Thermo-
dynamics and Introductory Quantum Mechanics.

49
50 Fundamentals of Atomic Force Microscopy, Part I Foundations

3.1 The Interaction of an Ion with a Dipole

While the force of interaction between two point charges (Sec. 2.2) is
known by all who attend lectures in any introductory level physics class,
the interaction between a point charge (ion) and a molecule is more inter-
esting. By representing the molecule electrically as an electric dipole, the
topic becomes tractable by recalling discussions often found in intermediate
courses on electricity and magnetism. A discussion of this interaction forms
an important first step toward understanding the tip–substrate interaction
in AFM.
A review of different electrostatic interactions with increasing complex-
ity is required to better understand the important issues. An overview to
the variety of possible ion-molecule interactions is sketched in Fig. 3.1.

3.1.1 An ion interacting with a fixed polar molecule

Conisder a point charge Q positioned a distance z away from a molecule
with a permanent dipole moment p = qd situated in a uniform medium
with dielectric constant κ. This situation approximates a dissolved ion in
a solvent that interacts with a nearby molecule having a permanent dipole
moment p as shown in Fig. 3.2. We use the notation z to designate the sep-
aration distance rather than the more common r in an attempt to develop
a consistent notation that will carry through to our discussion of van der
Waal’s forces at the end of this chapter.
The dielectric constant of the surrounding medium is usually treated
as a bulk quantity and book values of κ are used to estimate interaction

ion

fixed angle
Interacting polar molecule
ion
with

angle averaged

non-polar molecule polarization

induced dipole

Fig. 3.1 An overview of ion–molecule interactions from the strongest (ion/ion) to the
weakest (ion/non-polar).
 Simple Models for Molecule–Molecule Interactions 51

Fig. 3.2 A point charge Q placed a distance z away from an electric dipole p = qd. The
interaction occurs in a surrounding homogenous solvent with a dielectric constant κ.

energies when the interacting entities are sufficiently far apart. As evident
from Eq. (2.3), high values of κ for some liquids explains why they behave as
good solvents — attractive Coulombic interactions between ions of opposite
polarity are greatly reduced, thus allowing charged ions to remain dissolved
rather than agglomerating to form a solid crystal.
It should be clear that using bulk (continuum) values for κ and treating
κ as a constant under all circumstances is an approximation. The dielectric
constant can be expected to decrease from its bulk value as the number of
intervening solvent molecules decreases, a condition met when two interact-
ing molecules/atoms come into close proximity to each other. The result-
ing interaction should increase since the dielectric constant is expected
to decrease with separation, thereby enhancing the relevant electrostatic
forces. These effects are not accounted for in the discussion that follows.
A model for the interaction between a point charge and a dipole requires
the definition of a plane oriented to contain both these objects so that a
relevant angle θ can be defined as shown in Fig. 3.2. The equations derived
are coordinate specific to this definition of θ. At first glance, since the
dipole is electrically neutral, you might expect no electrostatic interaction.
However, each charge in the dipole separately experiences an interaction
force with the point charge Q. The resulting forces are nearly equal and
opposite as shown by F+ and F− in Fig. 3.2. A determination of the net
interaction force requires the vector summation of F+ and F− .
52 Fundamentals of Atomic Force Microscopy, Part I Foundations

As discussed in Sec. 2.2.2, it is easier to calculate the electrostatic

potential energy U (z). The relevant forces can be calculated by taking the
derivative of U (z) as required by Eq. (2.11).
If the zero of energy is taken when the charge Q is located very far from
the dipole, by inspection, U (z) for Fig. 3.2 can be written as


Q q (−q)
U (z) = + . (3.1)
4πκε0 r+ r−
When r+ > d, r− > d, (point charge Q far from the dipole), the square
of the relevant distances r+ and r− can be approximated to first order
in d by

2
2 d
r− = xo + + yo2
2

2

d xo d
= (x2o + yo2 ) + xo d +
z2 1 + 2 + · · · (3.2a)
2 z

2

d xo d
r+ = xo −
2
+ yo
z 1 − 2 + · · · ,
2 2
(3.2b)
2 z
 2
where z 2 ≡ x2o + yo2 , and the assumption that d2  z 2 has been made.
This assumption allows a Taylor series expansion when evaluating U (z)


− 12

− 12 
Qq xo d xo d
U (z) = z 1− 2
2
− z 1+ 2
2
4πκε0 z z



Qq 1 1 xo d 1 xo d
= 1+ + ··· − 1− + ···
4πκε0 z 2 z2 2 z2
Q q 1 xo d Qp 1

= cosθ, (3.3)
4πκε0 z z 2 4πκε0 z 2
where the angle θ (in radians) measures the relative orientation between the
point charge Q and the dipole with dipole moment p = qd and cos θ = x/d
from Fig. 3.2. Note that U (z) is really U (z, θ) and that it changes sign
depending on θ:
π
U (z, θ) > 0 when 0 ≤ θ ≤ ,
2
(3.4)
π
U (z, θ) < 0 when ≤ θ ≤ π.
2
−2
Also note that U (z, θ) varies as z . Recall that for two point charges,
the interaction potential varied as z −1 (see Eq. (2.14)). Note also that the
 Simple Models for Molecule–Molecule Interactions 53

electrostatic interaction potential energy (see Eq. (3.3)) does not contain
the electrostatic energy required to assemble the dipole located at the ori-
gin. This is a constant offset not included in Eq. (3.3) (or in any of the
discussions that follow).
The signed magnitude of the force between the point charge and dipole
(fixed θ) can always be obtained by applying Eq. (2.11)
 
∂ Qp 1 2Q p 1
|F (z, θ)|
−

2
cos θ = cos θ. (3.5)
∂z 4πκε0 z 4πκε0 z 3
The direction of the force is given by the sign associated with F — if
|F (z, θ)| > 0, then the force is repulsive while if |F (z, θ)| < 0, the force
is attractive. Clearly depending on θ, Eq. (3.5) can be either attractive or
repulsive in nature.

3.1.2 An ion interacting with a polar molecule free to rotate

The situation discussed above approximates the electrostatic interaction
between an ion and a polar molecule when the molecular dipole moment
is permanently fixed in space. Such a situation might arise if the molecule
is bound to an inert surface in a fixed orientation. What happens if the
relative angle between the dipole moment and the point charge varies? This
situation might arise when the position of the point charge fluctuates due to
thermal energy. Or, perhaps the orientation of the dipole moment fluctuates
in space but the position of the point charge is constant. Can we account
for these situations by appropriately calculating the “angle average” of the
electrostatic interaction potential energy? This calculation is equivalent to
varying the angle θ in Fig. 3.2 and asking what is the average value for
U (z, θ). What new results might come from this averaging procedure?
The answer to this question relies on computational techniques that are
usually discussed in courses on statistical thermodynamics. When using
the word thermodynamics, it is usually a short-hand way of asking “what
would nature do if left alone?” The answer to this question is generally that
a system tends to its lowest energy state.
Perhaps it is best to first discuss a general problem to review the appro-
priate techniques. Let us try to answer the question “What is the most
likely value you will measure for some quantity “y” in a system contain-
ing r particles that is in equilibrium with a reservoir at temperature T ?”
In principle, y can be any quantity and indeed, we will eventually substitute
U (z, θ) for y.
54 Fundamentals of Atomic Force Microscopy, Part I Foundations

Table 3.1 A representative list of all possible values

for a physical quantity y, the corresponding energies
E, and the weighting factors w for each state.

Some physical
quantity, y Energy, E Multiplicity, w

y1 E1 w1
y2 E2 w2
y3 E3 w3
... ... ...
yN −1 EN −1 wN −1
yN EN wN

To answer this question, the main task is to collect information about

how the energy of the system is related to the precise value of y. Suppose
the quantity y spans a countable number of possible energy states N , so it is
possible to tabulate all values of y (Table 3.1) along with the corresponding
energies E. It may be possible that one value of y is more heavily weighted
than other possible values. To account for this possibility, we must also
enumerate the multiplicity or weight (w) which specifies the number of
ways each value of y can be obtained.
In principle, the energies could describe discrete states — like elec-
trons confined to a nanometer-size region of space; or the energies could
be continuous — like gas atoms with different kinetic energies. How might
r particles in the system be distributed among the N energy levels? This
is a general question of considerable interest to all scientific disciplines.
Let the symbol y be used to define the thermal average (also called
the Boltzmann average) of y. If Pi is the probability that each value of y
will be obtained, then we have
y = yi P (yi ). (3.6)
i
We rely on statistical thermodynamics to estimate the probabilities
P (yi ). If yi specifies a possible state of the system having a weight or
degeneracy wi with an energy Ei that can be realized when the system is
in thermal equilibrium with a reservoir held at temperature T , the proba-
bility P (yi ) of finding the value yi is given by
−Ei
wi e kB T
P (yi ) = −Ei , (3.7)
i wi e
kB T

where the denominator is required for proper normalization.

 Simple Models for Molecule–Molecule Interactions 55

−E
The exponential factor e kB T is the Boltzmann weighting factor which
results when thermodynamics maximizes the configurational multiplicity
(entropy) for a system of r particles distributed among N energy lev-
els in thermal equilibrium at some temperature T . The quantity kB is
Boltzmann’s constant with the value kB = 1.38 × 10−23 J/K. The denom-
inator is known as the partition function in statistical thermodynamics
textbooks. If the variable y is continuous, then the energies Ei are likely
continuous and the sums in Eqs. (3.6) and (3.7) must be replaced by an
integral.
Estimating the weighting factors wi for each state of the system becomes
an important issue that depends on the details of the system under con-
sideration. For the specific problem of a point charge interacting with a
dipole, the proper weighting factor w comes from enumerating all possible
ways the charge Q can be oriented with respect to the dipole (see Fig. 3.3).
As shown in Fig. 3.3, Q is free to rotate in 3-dimensions, producing a strip
of width 1/2pdθ for a constant orientation angle θ.
Let w(θ) equal the number of ways that p and the point charge Q can
be arranged between angles (θ − dθ/2) and (θ + dθ/2) to give the same
interaction energy U (z, θ). We can estimate w(θ) by calculating the ratio of
the area of the shaded strip in Fig. 3.3 (which includes all possible relative

Fig. 3.3 When a point charge Q, a distance z from a dipole, is free to move with respect
to the dipole, a calculation of the factor w(θ) for the dipole-point charge interaction
U (z, θ) requires a proper weighting for each angle θ that gives the same interaction energy.
This is equivalent to integrating the position of the charge Q around the azimuthal angle
φ for each possible θ.
56 Fundamentals of Atomic Force Microscopy, Part I Foundations

orientations for a fixed θ) to the area of the entire sphere

2π    
area of shaded stripe ϕ=0
dϕ × p2 sin θ × p2 dθ
w(θ) ≡ = 2π π p   
surface area of sphere dϕ × sin θ × p dθ
ϕ=0 θ=0 2 2

sin θdθ sin θdθ 1

= π = π = 2 sin θdθ (0 ≤ θ ≤ π). (3.8)
0
sin θdθ −cosθ| 0

This procedure for defining w(θ) can be used to define the angle averaged
value for any continuous function f (θ) by
π π
1
f (θ)|angle = f (θ)w(θ)dθ = f (θ) sin θdθ. (3.9)
0 2 0

Example 3.1: Calculate the angle weighted value of a constant C

and various trigonometric functions f (θ) = cos θ, sin θ, cos2 θ, sin2 θ, and
sin θ cos θ.
It follows from the above discussion that we must evaluate Eq. (3.9)
for the various functions listed.
π π
1 C
C|angle = C sin θdθ = − cos θ =C
2 0 2 0
π
1
cos θ|angle = cos θ sin θdθ = 0
2 0
π
1 π
1 (θ − sin θ cos θ) π
sin θ|angle = sin2 θdθ = × =
2 0 2 2 0 4
π
1 π
1 (cos 3θ − 9 cos θ)
sin2 θ|angle = sin3 θdθ =
2 0 2 12 0

1 [(−1) − 9 · (−1)] − (1 − 9) 2
= =
2 12 3
π π
1 1
cos2 θ|angle = cos2 θ sin θdθ = cos2 θd(cos θ)
2 0 2 0

1 3 1
1 1 x 1
= x2 dx = × =
2 −1 2 3 −1 3
π π
1 1 sin3 θ
sin θ · cos θ|angle = 2
cos θ sin θdθ = =0
2 0 2 3 0
 Simple Models for Molecule–Molecule Interactions 57

To calculate the angle averaged value of U (z, θ) in Eq. (3.5), you might
be tempted to vary the angle θ defined in Fig. 3.3, following the procedure to
weight each angle as laid out above. From Eq. (3.8), you would include the
sin θ weighting factor (the factor of 1/2 can be dropped since it appears in
both the numerator and denominator) and you would obtain the following
2π π
dϕ U (z, θ) sin θdθ
U (z)|angle ≡ 0
2π
0
π ,
0
dϕ 0
sin θdθ
where
Qp cos(θ)
U (z, θ) = = Uo (z) f (θ),
4πκε0 z 2
with
Qp 1
f (θ) = cos θ and Uo (z) = (3.10)
4πκε0 z 2
π
Uo (z) cos θ sin θdθ
U (z)|angle = π
0

0
sin θdθ
 
− 12 cos2 θ|π0 Uo (z) [(−1)2 − (1)2 ]
= Uo (z) · = · = 0.
− cos θ|π0 2 (−1) − (1)
The result is zero because the calculation treats all angles as equally
likely. Instead, we must include a thermal average that additionally
favors those angles with lowest energy. This requires the inclusion of the
Boltzmann factor as given in Eq. (3.7) in addition to the angle weighting
factor discussed above.
Following this approach we have
−U (z,θ)
 −U (z,θ)  2π
dϕ
π
U (z, θ)e kB T
sin θdθ
U (z) ≡ U (z, θ)e kB T
= 0 0
−U (z,θ)
. (3.11)
2π π
0
dϕ 0
e kB T
sin θdθ
Since U (z, θ) = Uo (z) · f (θ) as before with f (θ) = cos θ, we have


Uo (z)f (θ)
π −
Uo (z) f (θ)e kB T
sin θdθ
U (z) = 0

Uo (z)f (θ)
 .
π −
0
e kB T
sin θdθ
Let
Uo (z)
β=− ,
kB T
58 Fundamentals of Atomic Force Microscopy, Part I Foundations

then π
f (θ)eβf (θ) sin θdθ
U (z) = Uo (z) π βf (θ)
0

0
e sin θdθ

π 
d
= Uo (z) ln eβf (θ) sin θdθ . (3.12)
dβ 0

The last

step follows from  the identity

π π
d 1 ∂
ln eβf (θ)
sin θdθ =  π βf (θ) × eβf (θ)
sin θdθ
dβ 0 0
e sin θdθ ∂β 0

π 
1
=  π βf (θ) × f (θ)e βf (θ)
sin θdθ .
0
e sin θdθ 0
(3.13)
Equation (3.13) implies that only the natural logarithm of the integral
must be evaluated (with f (θ) = cos θ) before a derivative with respect to
β is taken. This simplifies the discussion considerably since now a Taylor
series expansion of eβf (θ) is possible. If βf (θ)  1, we can write
π
eβf (θ) sin θdθ, (3.14)
0
π π  
β2 2
βf (θ)
e sin θdθ
1 + βf (θ) + f (θ) + · · · sin θdθ
0 0 2
π π
= sin θdθ + β f (θ) sin θdθ
0 0
π
β2
+ f 2 (θ) sin θdθ + · · ·
2 0
π
β2
= − cos θ|π0 + 0 + cos2 θ sin θdθ + · · ·
2 0

2

β 1 β2
= 2− + · · · . (3.15)
cos3 θ|πo + · · · = 2 +
2 3 3
This gives
π 
 U (z,θ)  d
− k T βf (θ)
U (z, θ)e B = Uo (z) ln e sin θdθ
dβ 0


d β2
= Uo (z) × ln 2 + + ···
dβ 3
1 2β
= Uo (z) ×  ×
2+ β2
+ ··· 3
3
 Simple Models for Molecule–Molecule Interactions 59


2β 1 β2
= Uo (z) × × × 1− + ···
3 2 6
β
= Uo (z) + · · · . (3.16)
3
Since
Uo (z) Qp 1
β≡− and Uo (z) = , (3.17)
kB T 4πκε0 z 2
we finally have the angle-averaged, thermal interaction energy between a
point charge Q and a thermally rotating dipole p


1 Uo (z) 1 (Qp)2 1
U (z) = − × Uo (z) = − , (3.18)
3 kB T 3kB T (4πκε0 )2 z 4
where the leading factor of (3kB T )−1 is a combination of the angle-weighted
value of cos2 θ = 1/3 with the factor kB T coming from the Boltzmann
thermal factor which preferentially favors dipole orientations with lower
energy states.
A comparison between the fixed angle point charge–dipole electro-
static interaction potential energy and the angle averaged thermal result in
Eq. (3.18) is given in Fig. 3.4. In contrast to the fixed angle situation which

2 p= 1 D
Fixed =30o Q=+1e-
=75o
Electrostatic Energy

T=300 K
(in units of 10-21 J)

1 =2.5
angle
averaged

-1 =150o
=105o

-2
0 1 2 3 4
Ion-Dipole Separation z (nm)

Fig. 3.4 A comparison between fixed angle (see Eq. (3.3)) and a thermal angle averaged
(see Eq. (3.18)) calculation of the interaction energy U (z, θ) between a fixed dipole with
dipole moment 1D and a moveable ion of charge Q = +1.6 × 10−19 C. For the fixed
angle calculation, only a few representative angles of 30◦ , 75◦ , 105◦ and 150◦ are shown.
The interaction can be attractive (negative) or repulsive (positive), depending on the
dipole orientation. The thermal angle-averaged calculation
U (z, θ) is always attractive
due to the thermal averaging introduced by the Boltzmann factor.
60 Fundamentals of Atomic Force Microscopy, Part I Foundations

can be either repulsive (positive) or attractive (negative) depending on the

location of the point charge, the thermal angle averaged result is always
attractive. In addition, the interaction energy now varies as 1/z 4 rather
than 1/z 2 for a fixed dipole-ion separation (see Eq. (3.3)). We conclude
that the thermal angle averaged interaction is a shorter range interaction
than the fixed angle result.

3.1.3 Induction — the polarization of a non-polar molecule

What happens when a non-polar atom or molecule comes into close prox-
imity to a point charge? Lacking a dipole moment, you might expect no
electrostatic interaction will take place. However it is possible to induce an
electric dipole moment in the non-polar molecule, thereby insuring that it
too can be attracted or repelled. The physics of this situation is outlined
in Fig. 3.5.
If a non-polar molecule with a diameter of ∼2ao finds itself in an applied
electric field, the charge distribution can rearrange and in the process, an
induced dipole moment pinduced can be formed. Experiment shows that to
first order in Eapp , the induced dipole is given by
 applied ,
pinduced = αE (3.19)
where α is called the electronic polarizability and has units of m · C2 /N.
In principle, α depends on the frequency of E  app . At this stage of the
discussion, we will not include the frequency dependence; furthermore, what
produces E app is not important.
As shown in Fig. 3.6, a simple model to estimate the polarizabil-
ity treats the molecule as a small positively charged nucleus (total
charge +q) surrounded by a larger spherical negative charge distribution

enlarged
non-polar Schematic
molecule electronic
Eapp
pinduced = αE applied charge
cloud
- +
- +
molecule
diam. 2ao

Fig. 3.5 A non-polar molecule with a diameter ∼2ao in an external applied electric
field Eapp . The applied field distorts the electron distribution and the molecule becomes
polarized, acquiring a net dipole moment pinduced . The proportionality constant between
pinduced and E is the electronic polarizability α.
 Simple Models for Molecule–Molecule Interactions 61

Eapp
Einternal
ao
d
d
+ +

(a) (b) (c)

Fig. 3.6 A simple model to estimate the electronic polarizability α. In (a), an assumed
spherically symmetric electron charge cloud with charge density ρ and radius ao sur-
rounds a positively charged nucleus. In (b), an external electric field Eapp displaces the
electronic charge cloud to the left, causing a net displacement d between the nucleus and
the electron distribution. In (c), an internal electric field Einternal develops at the nucleus
because of the displaced electronic charge. At the nucleus, Einternal opposes Eapp .

(total charge −q) with radius ao . The uniform electron charge density ρ (in
C/m3 ) is given by
−q
ρ = 4 3
. (3.20)
3 πao

In the presence of E app , the positive nucleus and the negative electronic
charge cloud will be shifted in opposite directions, producing a net charge
displacement d. The size of d will be determined by an equilibrium condi-
tion set by the internal electric field E  internal that develops because of the
displaced electron cloud.
At the position of the nucleus, E  internal must balance E  app for a stable
condition to result. If the electron cloud maintains its spherical shape, then
the internal electric field (opposite in direction to the applied field at the
positive nucleus) can be approximated using Gauss’ law. Because of the
assumed spherical symmetry,
1 q

 internal | =
|E , (3.21)
4πεo d2
where
4π 3
q
= ρd .
3
Using Eq. (3.20) for ρ gives
1 qd
 internal | =
|E . (3.22)
4πεo a3o
62 Fundamentals of Atomic Force Microscopy, Part I Foundations

Defining |p| = qd in Eq. (3.22) provides the following result

|p| = 4πεo a3o |E
 int | = α|E
 int |. (3.23)
With this simple model, the polarizability is given by
α = 4πεo a3o = 4πεo αo , (3.24)
where αo = (ao )3 is known as the “electronic polarizability volume” (units
of m3 ) and is comparable to an effective “volume” (ao 3 ) of the electronic
charge cloud in this simple model.
This model for polarizability is very simplistic and is only useful because
it provides some physical insight into the microscopic origin of induced dipole
moments. A more accurate model requires knowledge of the precise electron
wave functions and application of perturbation theory in quantum mechanics.

Example 3.2: Using Eq. (3.24), estimate the electronic polarizability

for a H2 O molecule. Compare to the measured value of α for water.
The structure of a H2 O molecule is

From this structure, the H-H distance is calculated to be around 150 pm.
This suggests an estimate for the “radius” of a water molecule might
lie between a lower limit of 150 pm/2 = 75 pm and an upper limit of
∼96 pm.
The polarizability can be estimated as
Lower limit:
α = 4πεo αo = 4π × 8.85 × 10−12 C2 /Nm2 · (0.075 × 10−9 m)3
= 4.7 × 10−41 m · C2 /N.
Upper limit:
α = 4πεo αo = 4π × 8.85 × 10−12 C2 /Nm2 · (0.096 × 10−9 m)3
= 9.8 × 10−41 m · C2 /N.
These estimates for α are low, giving a value that ranges between 28%
to 59% of the book value for H2 O which is listed as 1.66×10−40 m·C2 /N.
 Simple Models for Molecule–Molecule Interactions 63

Table 3.2 A table of common solvents with a listing of their dielectric properties.

Chemical K, dielectric p, αo = α/4πo

Material Formula Structure constant1 (in Debye)2,3 (in 10−30 m3 )

Acetone (CH3 )2 CO 21 2.9 6.3

Isopropanel (CH3 )CHOH 18 1.7 6.9

Ethanol (CH3 )2 CH2 OH 24 1.7 5.1

Methanol CH3 OH 33 1.7 3.3

Toluene C6 H5 CH3 2.4 0.4 12.3

Trichloro
fluoromethane CC3 F 2.0 0.45 8.5
(CFC-11,
Freon-11)

Water H2 O 80 1.8 1.5

1 Solventswith κ < 15 are generally considered non-polar.

2 Tabulated dipole moments can be different than those listed, depending on whether
they apply to the gaseous or liquid phase
3 1 Debye = 3.33 × 10−30 C · m

Table 3.2 lists relevant dielectric constants, dipole moments and

polarizability volumes for some common solvent molecules.

3.1.4 The interaction of a point charge with

a non-polar molecule
If a non-polar molecule encounters the electric field produced by a point
charge, then it is possible that an induced dipole is formed as shown in
Fig. 3.7. In Fig. 3.7(a), a non-polar molecule is located in a dielectric
64 Fundamentals of Atomic Force Microscopy, Part I Foundations

Fig. 3.7 In (a), a non-polar molecule interacts with a point charge separated by a
distance z from the molecule. In (b), the electric field produced by the point charge at
the molecule causes an induced dipole moment in the molecule that is always oriented
as drawn.

medium having a dielectric constant κ. The non-polar molecule has no

dipole moment as schematically illustrated. The point charge Q produces
an electric field E that polarizes the molecule as shown in Fig. 3.6(b). The
polarization is such that the angle between pinduced and E as defined in
◦
Fig. 3.7 is always 180 as shown. The electrostatic interaction potential
energy that develops between the point charge and a dipole has already
been derived (see Eq. (3.3)) and it must now be evaluated for θ = π
Q pinduced 1 Q pinduced 1
Uinduced (z) = cos θ =− (3.25)
4πκε0 z 2 θ=π 4πκε0 z 2
 where E
with pinduced = αE,  is the electric field generated by a point charge
Q. We therefore have
 1
Q α|E| αQ2 1
Uinduced (z) = − = − . (3.26)
4πκε0 z 2 (4πκε0 )2 z 4
 Simple Models for Molecule–Molecule Interactions 65

Example 3.3: A Li ion is 2 nm away from a non-polar molecule which

is dissolved in a liquid with a dielectric constant of κ = 2.3. The
molecule has a polarizability volume αo = 4.8 × 10−30 m3 . Calculate
(a) the electrostatic interaction potential energy (in eV) and (b) the
force between the non-polar molecules and the Li ion.
αQ2 1 4πε0 αo Q2 1 αo Q2 1 1
(a) Uinduced (z) = − = − = −
(4πκε0 )2 z 4 (4πκε0 )2 z 4 (4πε0 ) κ2 z 4
(4.8 × 10−30 m3 )(+1.6 × 10−19 C)2
=−
4π × 8.85 × 10−12 × Nm
C2
2

1 1
× ×
(2.3)2 (2 × 10−9 m)4
= −1.105 × 10−57 × 1.89 × 10−1 × 6.25 × 1034 J
1eV
= −1.31 × 10−23 J × = −8.2 × 10−5 eV.
1.6 × 10−19 J
This interaction energy is ∼300 times smaller than thermal energies
which are typically estimated as kB T = 4.21 × 10−21 J = 0.026 eV at
T = 300 K. In general, the ion and molecule will bind or “condense”
when their interaction energy is large compared to thermal energies.
Similarly, we might expect the ions will remain dispersed in solution
until the interaction energy becomes comparable to kB T .

∂Uinduced (z) αQ2 1

(b) F = − = (−4)
∂z (4πκε0 )2 z 5
4πε0 αo Q2 1 4αo Q2 1 1
= −4 × 2 5
=−
(4πκε0 ) z (4πε0 ) κ2 z 5
4(4.8 × 10−30 m3 )(+1.6 × 10−19 C)2
=−
4π × 8.85 × 10−12 Nm
C2
2

1 1
× ×
(2.3)2 (2 × 10−9 m)5
= −4.42 × 10−57 × 1.89 × 10−1 × 3.13 × 1043 N
= −2.6 × 10−14 N (attractive).
This force is small, about 350 times smaller than the smallest force
(∼10 pN = 10 × 10−12 N) that can be measured using an AFM.
66 Fundamentals of Atomic Force Microscopy, Part I Foundations

The important message from this discussion is that ion-dipole inter-

actions vary as z −4 , irrespective of whether the dipole is a permanent
characteristic of the molecule (see Eq. (3.18)) or induced by an external
electric field (see Eq. (3.26)).

3.2 Molecule-Molecule Interactions

Within the context of AFM, it is useful to have models that account for
the force that a tip of radius R experiences when positioned a distance d
above a flat sample as shown in Fig. 3.8. Any net force must result from
interactions between atoms that comprise the tip and sample. Usually, the
atoms in the tip and sample are electrically neutral, so ion-ion or ion-
molecule interactions are not often relevant. To understand the nature of
the tip–substrate interaction, it therefore becomes important to consider
the interaction of neutral atoms (or molecules) with other neutral atoms
(or molecules).
A number of distinct possibilities arise as shown in Fig. 3.9 and these
situations, when systematically developed, lead to the classic expressions
for the Keesom, Debye and London interactions. All three of these interac-
tions can be further grouped into a single category called a van der Waals
interaction.

Tip
Sample
d
R

Fig. 3.8 The ultimate goal is to use our understanding of molecule-molecule forces to
estimate the force between a sharp tip of radius R positioned a distance d from a flat
substrate.
 Simple Models for Molecule–Molecule Interactions 67

fixed angle
polar molecule
angle averaged
polar Interacting
(Keesom)
molecule with non-polar
molecule
polarization

induced dipole
(Debye)
non-polar
non-polar Interacting molecule
induced dipole-dipole
molecule with
(London)

Fig. 3.9 An overview of molecule–molecule van der Waal interactions ranging from the
strongest (polar/polar) to the weakest (non-polar/non-polar).

The important point that will develop is that electrically neutral

atoms/molecules can electrostatically interact with each other. In what
follows, we focus on how these interactions vary with separation distance,
since it is this quantity we control with an AFM.

3.2.1 The interaction of two polar molecules

Consider two polar molecules with permanent dipole moments p1 , p2 ,
embedded in a continuous dielectric with dielectric constant κ and sepa-
rated by a fixed distance r as shown in Fig. 3.10. The two molecules lie
in the z-y plane and can be rotated through arbitrary angles ϑ1 , ϑ2 with
respect to the z-axis as shown. The standard definitions for the spheri-
cal polar coordinates (r, θ, ϕ) are assumed. In addition, the orientation of
dipole p2 can be twisted with respect to p1 through an angle ζ in the x–z
plane as shown. The two dipoles will interact electrostatically because the
dipole p1 produces an electric field that exerts a force on the charges in
dipole p2 .
The interaction potential energy will now depend on three angles
(ϑ1 , ϑ2 , ζ) that specify the relative orientation of the two dipoles. There is
no standard definition of the angles ϑ1 , ϑ2 , so the form of any analytical
result is tied to the way these angles are defined.
To calculate the electrostatic interaction potential energy, we can make
use of the result given by Eq. (2.26)

U (r, ϑ1 , ϑ2 , ζ) = −
p2 · E
 1,
68 Fundamentals of Atomic Force Microscopy, Part I Foundations

Fig. 3.10 The geometry required to analyze the interaction of two dipoles. The dipole
p1 is tilted by an angle ϑ1 in z–y plane. The dipole p2 is tilted by an angle ϑ2 in the
y–z plane. In addition, p2 is twisted by an angle ζ in the x–z plane. The dipole-dipole
separation distance is r.

z z
p1,y=p1sin 1 p1,z=p1cos 1
p2
p2
x x
p2cos
y y

Fig. 3.11 The interaction potential energy between two permanent dipoles will contain
two contributions that depend on the components of p1 . In general, the rotation angle
of p2 is ζ as shown.

where

E
p1 | 1
 1 (r, θ) = | (2 cos θr̂ + sin θθ̂). (3.27)
4πκεo r3
Here E  1 is the electric field produced by dipole p1 .
When evaluating U (r, ϑ1 , ϑ2 , ζ) in Eq. (3.27), it is useful to first consider
the components of p1 parallel to y (p1,y ) and z (p1,z ) as limiting cases.
These two geometries are sketched in Fig. 3.11. These limiting cases
illustrate that U (r, ϑ1 = π/2, ϑ2 , ζ) cannot depend on ζ since p2 is always
perpendicular to p1 for any value of ζ. Similarly, U (r, ϑ1 = 0, ϑ2 , ζ) must
depend on the projection of p2 along the z-axis and hence must depend on
cos ζ. These two results are useful to understand the two contributions to
U (r, ϑ1 , ϑ2 , ζ) for any arbitrary value of ϑ1 .
 Simple Models for Molecule–Molecule Interactions 69

Fig. 3.12 Defining the coordinates and angles for the dipole–dipole interaction between
two dipoles separated by a distance z.

The final expression for U (r, ϑ1 , ϑ2 , ζ) includes two terms that represent
the relative contributions for the two components of p1 shown in Fig. 3.11.
A derivation of the final result requires some geometrical dexterity and
uses the standard approximation for the electric field E  1 (Eq. (3.27)) from
a “pure” or “point” dipole in which r  d, where d is the separation
between the charges comprising the dipole and r represents the dipole–
dipole separation.
After completing the calculation in a standard spherical-polar coordi-
nate system, it is convenient to re-label the separation distance r as z and
to redefine the angles as shown in Fig. 3.12.
The final result for the dipole–dipole interaction energy is
1 p1 p2
Utotal (z, ϑ1 , ϑ2 , ζ) = − (2 cos ϑ1 cos ϑ2 − sin ϑ1 sin ϑ2 cos ζ)
4πκεo z 3
p1 p2 1
= × f (ϑ1 , ϑ2 , ζ), (3.28)
4πκεo z 3
where
f (ϑ1 , ϑ2 , ζ) ≡ −(2 cos ϑ1 cos ϑ2 − sin ϑ1 sin ϑ2 cos ζ). (3.29)
The angular function f (ϑ1 , ϑ2 , ζ) is bounded such that
−2 ≤ f (ϑ1 , ϑ2 , ζ) ≤ 2. (3.30)
Plots of f (ϑ1 , ϑ2 , ζ) are useful to show the angular dependence of the inter-
action energy. Two special cases are provided in Fig. 3.13.
The relative orientation of two dipoles is sketched in Fig. 3.14 for a
few angles to show the configuration producing the minimum (attractive)
and maximum (repulsive) interaction. These orientations are also indicated
in Fig. 3.13 on the ζ = 0 plot. These alignments provide a quick refer-
ence (in conjunction with Eq. (3.28)) to estimate the interaction energy
70 Fundamentals of Atomic Force Microscopy, Part I Foundations

Fig. 3.13 Representative contour plots of f (ϑ1 , ϑ2 , ζ) (Eq. (3.29)) for ζ = 0 and ζ = π/2.
The labels a, b and c on the plot for ζ = 0 shows the locations for the three relative
orientations of two dipoles sketched in Fig. 3.14.

z
(a) 1=0
2=0; =0
U(z) is minimum p1 p2

(b) 1=0 z
2= ; =0
U(z) is maximum p1 p2

(c) 1= /2
z
2= /2; =
same as
1= /2 p1 p2
2=3 /2; =0

Fig. 3.14 Three representative configurations for two dipoles separated by a distance z.

between two polar molecules that have a fixed orientation with respect to
each other. Such a situation might arise if, for instance, two polar molecules
are chemically bound to a planar surface.

3.2.2 The angle-averaged interaction between

two polar molecules
If the orientation of the dipoles p1 and p2 can vary due to thermal effects,
a weighted angle average must be calculated in order to estimate the
net interaction energy. Following a similar procedure already outlined in
 Simple Models for Molecule–Molecule Interactions 71

Sec. 3.1.2 above, we define U (z) as

 
−U (z,Θ1 ,Θ2 ,ζ)
U (z) ≡ U (z, Θ1 , Θ2 , ζ)e kB T
, (3.31)

where now
p1 p2 1
U (z, Θ1 , Θ2 , ζ) = − [2 cos Θ1 cos Θ2 − sin Θ1 sin Θ2 cos ζ].
4πκεo z 3
(3.32)
Here   again represents the thermal average over all angles and implies
that the two objects are free to move by thermal motion as they interact.
Let

U (z, Θ1 , Θ2 , ζ) = Uo (z) · f (Ω), (3.33)

where
p1 p 2 1
Uo (z) = (3.34)
4πκε0 z 3
and

f (Ω) = −2 cos Θ1 cos Θ2 + sin Θ1 sin Θ2 cos ζ. (3.35)

To find the angle averaged value for U (z, Θ1 , Θ2 , ζ), we then must evaluate
−Uo (z)f (Ω)
2π π π
0
dζ Θ1 =0 Θ2 =0
Uo (z)f (Ω)e kB T
sin Θ1 dΘ1 sin Θ2 dΘ2
U (z) ≡ −Uo (z)f (Ω)
.
2π π π sin Θ1 dΘ1 sin Θ2 dΘ2
0
dζ Θ1 =0 Θ2 =0
e kB T

(3.36)
The discussion below follows closely that found in Sec. 3.1.2 for the case of
a point charge near a dipole. Defining
Uo (z)
β=− (3.37)
kB T
and realizing that

2π π π 
d
Uo (z) × ln dζ eβf (Ω)
sin Θ1 dΘ1 sin Θ2 dΘ2
dβ 0 Θ1 =0 Θ2 =0

2π π π
Uo (z) dϕ Θ1 =0 Θ2 =0 f (Ω)eβf (Ω) sin Θ1 dΘ1 sin Θ2 dΘ2
=  ζ=0

2π π π βf (Ω) sin Θ dΘ sin Θ dΘ
ζ=0
dϕ Θ1 =0 Θ2 =0
e 1 1 2 2

(3.38)
72 Fundamentals of Atomic Force Microscopy, Part I Foundations

allow us to focus on the integral

2π π π
I= dζ eβf (Ω) sin Θ1 dΘ1 sin Θ2 dΘ2
0 Θ1 =0 Θ2 =0
2π π π  
β2 2

dζ 1 + βf (Ω) + f (Ω) + · · · sin Θ1 dΘ1 sin Θ2 dΘ2 .
0 Θ1 =0 Θ2 =0 2
(3.39)

After some algebra, we find the above integral is given to order β 2 by


β2
I = 8π × 1 + + ··· (3.40)
3
U (z) can now be evaluated by taking a logarithmic derivative
 
−
U (z,Θ1 ,Θ2 ,ζ) d
U (z) ≡ U (z, Θ1 , Θ2 , ζ)e kB T
= Uo (z) ln(I)
dβ


d β2
= Uo (z) ln(8π) + ln 1 + + ···
dβ 3
1 2β
= 0 + Uo (z) β2
×
1+ 3
3


2β β2 2β

Uo (z) × × 1− + · · · ≈ Uo (z) . (3.41)
3 3 3
Since
Uo (z) p1 p 2 1
β=− and Uo (z) = , (3.42)
kB T 4πκε0 z 3
the angle-averaged interaction energy between two dipoles separated by a
distance z becomes

2
2 1 p1 p2 1
UKeesom (z) = − . (3.43)
3 kB T 4πκε0 z6
Equation (3.43) was obtained by allowing both dipoles freedom to rotate
since Eq. (3.38) integrates over both angles θ1 and θ2 . This double counts
equivalent configurations between the two dipoles. The correct answer is
therefore obtained by multiplying Eq. (3.43) by a factor of 12 , giving

2
1 1 p1 p2 1
UKeesom (z) = − . (3.44)
3 kB T 4πκε0 z6
 Simple Models for Molecule–Molecule Interactions 73

Another way of obtaining this result is to lock the orientation of dipole

p1 such that Θ1 = π/2. Then f (Ω) in Eq. (3.36) simplifies to sin Θ2 cos ζ.
Under these circumstances, it is relatively straightforward to show that the
2/3 value of the factor found in Eq. (3.43) is reduced to a value of 1/3.
The interaction energy acquires a characteristic z −6 dependence on
dipole–dipole separation and is inversely proportional to temperature. This
result was derived in 1921 by W.H. Keeson. He showed that two molecules
with permanent dipole moments undergoing thermal motion will on average
give rise to an attractive intermolecular force[keesom21].

Example 3.4: Calculate the Keesom angle-averaged electrostatic

potential energy for two polar molecules with dipole moments of 1.7 D
that are separated by a distance of 2 nm in a medium having a dielectric
constant of 24. Assume the temperature of the system is at 300 K.

2
1 1 p1 p2 1
UKeesom (z) =
3 kB T 4πkε0 z6


1 1
=−
3 (1.38 × 10−23 J/K)(300K)
  2 2
3.33×10−30 cm
 (1.7D) × 1D  1
× C2  (2 × 10−9 m)6
4π × (24) × 8.85 × 10 −12
Nm2


2
1 1 (5.66 × 10−30 )2
=− Nm4
3 1.24 × 10−20 J 2.67 × 10−9
1
×
6.40 × 10−53 m6
1
= − (8.06 × 1019 J−1 ) × (1.44 × 10−100 N2 m8 )
3
×(1.56 × 1052 m−6 )


1 eV
= −6.03 × 10−29 J ×
1.6 × 10−19 J
= −3.8 × 10−10 eV
(Continued)
74 Fundamentals of Atomic Force Microscopy, Part I Foundations

Example 3.4: (Continued )

This is very small interaction energy when compared with thermal

energies.
The force between the two dipoles is attractive and can be found
by taking the negative gradient of UKeesom (z) with respect to z
(Eq. (2.11)). The force F is found to be

2 
∂ UKeesom (z) 1 1 p1 p2 (−6)
F =− =− −
∂z 3 kB T 4πκε0 z7


6
= −6.03 × 10−29 J ×
2 × 10−9 m

= −1.8 × 10−19 N = −18 aN(attractive)

3.2.3 The interaction between a di-polar molecule

and a non-polar molecule
The electric field far from a polar molecule with a permanent dipole moment
is complicated, changing in orientation and strength depending on the exact
coordinates (r, θ) in a polar spherical coordinate system. Unlike an x–y
coordinate system in which the unit vectors always point in the same direc-
tion, the unit vectors (r̂, θ̂) in a spherical polar coordinate system change
direction as a function of (r, θ). Consider a polar molecule with a permanent
dipole moment p1 embedded in a uniform dielectric with dielectric constant
κ oriented along the z-axis, the far-field components (Er , Eθ ) of the dipole’s
electric field when r  d (d = separation between dipole charges) is known
from classical electrostatics[griffiths13].
p1 | 1 
 dipole (r, θ) = |

E 3
2 cos θr̂ + sin θθ̂ . (3.45)
4πκεo r
The magnitude of the dipolar electric field is given by
p1 | 1
| 1
|E
 dipole (r, θ)| = (4 cos2 θ + sin2 θ) 2
4πκεo r3
p1 | 1
| 1
= (3 cos2 θ + 1) 2 . (3.46)
4πκεo r3
 Simple Models for Molecule–Molecule Interactions 75

z
Non-polar
molecule
E

r
E r θ
p1
-

Edipole =
+
pinduced, 2 E

Fig. 3.15 Non-polar molecules located at a radial distance r in the far-field vicinity of a
polar molecule with a permanent dipole moment p1 . Induced dipole moments pinduced,2
on the non-polar molecules align with the local direction of the electric field as shown in
the inset.

At a fixed distance r from p1 , the magnitude of the electric field clearly

depends on the angle θ.
If a second, non-polar molecule is located a distance r from the polar
molecule, at first thought you might claim there will be no net interaction.
However, the second molecule can be polarized by Edipole as shown in
Fig. 3.15. This case was considered by P. Debye in 1921. He showed that
a net attractive force can exist between two molecules even if only one of
them has a permanent dipole moment. As shown schematically in Fig. 3.15,
the orientation of the induced dipole will depend on the precise direction
of Edipole at any point (r, θ).
If the second molecule has non-zero polarizability, then, according to
Eq. (3.19) it will acquire a dipole moment in the direction of the dipolar
electric field
 dipole ,
pinduced,2 = α2 E (3.47)
where α2 is the polarizability of molecule 2. Since pinduced,2 is always parallel
 dipole , the electrostatic interaction energy will then be given by
to E
U (r, θ) = − pinduced,2 · E
 dipole = −α2 E 2
dipole
 1
 2
p1 | (3 cos2 θ + 1) 2
| p21 α2 (3 cos2 θ + 1)
= −α2 3
=− .
4πκεo r (4πκεo )2 r6
(3.48)
76 Fundamentals of Atomic Force Microscopy, Part I Foundations

In this case, a thermal average involving the Boltzmann factor is not

required because the induced dipole is always parallel to the electric field.
Experimentally, we have no control over the position angle θ so we may
as well average over all possible values. As we showed in Example 3.1, the
weighted average of cos2 θ = 1/3. This gives an expression for the posi-
tion averaged interaction energy between a polar and non-polar molecule
separated by a distance r = z
p21 α2 1
UDebye (z) = −2 · (polar-induced-dipole interaction)
(4πκεo )2 z 6
(3.49)

Example 3.5: Calculate the Debye electrostatic potential energy for

two identical dipolar molecules having a dipole moment of 1.7 D that
are separated by a distance of 2 nm in a fluid having a uniform dielectric
constant of 24. Take the polarizability volume for each molecule to be
αo = 5.1 × 10−30 m3 .
Since the two molecules are dipolar, they will experience a Keesom
interaction in addition to a dipole/induced dipole interaction. The Debye
formula (Eq. (3.49)) is used to describe the polar/non-polar interaction.
UDebye (z = 2 nm)
p2 α 1 p2 4πεo αo 1 p2 αo 1 1
= −2 × 2 6
= −2 × 2 6
= −2
(4πκεo ) z (4πκεo ) z (4πεo ) κ2 z 6
  2 
3.33×10−30 Cm −30 3
 (1.7 D) × 1D (5.1 × 10 m ) 
= −2  
4π × 8.85 × 10−12 Nm C2
2

2
1 1
× 6
24 (2 × 10−9 m)
 
1.63 × 10−88 C2 m3  
= −2 × 1.74 × 10−3 × (1.56 × 1052 m−6 )
1.11 × 10 −10 C2
Nm2


1 eV
= −7.4 × 10−29 J ×
1.6 × 10−19 J
= −4.6 × 10−10 eV.
For the parameters chosen, the interaction potential energy is compa-
rable to the Keesom interaction discussed in Example 3.4.
 Simple Models for Molecule–Molecule Interactions 77

3.2.4 The interaction between two non-polar molecules

As indicated in Fig. 3.9, a third possibility for molecule–molecule inter-
actions involves two molecules (or atoms) that have neither a net charge
nor permanent dipole moments. We know that such non-polar/non-polar
interactions exist because inert gases like Ar condense into liquids at low
temperatures. The condensation from gas to liquid results from the inter-
action of transient electric dipoles that are momentarily produced by such
neutral atoms or molecules. The origin of this interaction is electron delocal-
ization around each atom, a quantum effect which requires wave functions
that are solutions to Schrödinger’s equation.
A consequence of the wave function description of electrons confined to
atoms and molecules is the possibility that at any instant of time, a slight
imbalance of electronic charge will be located on one side of an atomic-
scale object than the other, causing a momentary (fluctuating, flickering)
dipole moment that can induce a dipole moment in another nearby atom
or molecule that is correlated with the first. A slight attraction will result
while the momentary dipole moment exists and the electrostatic potential
energy of the atomic-scale pair is lowered. Because the two dipoles are
dynamically correlated, the attraction does not average to zero over time.
The likelihood that an atom or molecule produces a momentary fluctua-
tion in charge density increases with size. In addition, the interaction must
also be proportional to the polarizability of the second atom or molecule
involved. Since polarizability is also proportional to size (see Eq. (3.24)), we
might expect a priori that the interaction energy should be proportional
to the product of the two polarizabilities.
Exact calculations of this interaction are quite involved, but a sim-
ple model based on the quantum mechanical lowering of the ground state
energy of two charged-coupled oscillators provides considerable physical
insight into the problem [london37]. This quantum interaction is commonly
referred to as the London dispersion interaction. Consider the situation
shown in Fig. 3.16 which shows two non-polar molecules separated by a
distance z.
Assume that a time dependent charge fluctuation in molecule 1 induces
a dipole moment in molecule 2. How can we describe the effect of the
charge fluctuation? Because our model system has atomic-scale dimensions,
a quantum mechanical solution based on Schrödinger’s equation is required.
The model assumes the mass m of the charge that fluctuates is confined
to a parabolic potential that can be described by an effective spring con-
stant k (N/m). This gives expressions for the confining potential energy
78 Fundamentals of Atomic Force Microscopy, Part I Foundations

non-polar non-polar
molecule 1 molecule 2
+(t)
z
+ - + -
-
α1 (t) α2

z1, v1 z2, v2
+q -q +q
m m m m -q
fixed k z fixed k

Fig. 3.16 Two non-polar molecules separated by a distance z. A momentary fluctuation

in the charge distribution of molecule 1 induces a dipole moment in molecule 2. The
situation is modeled by two springs with masses that are charged as shown in the bottom
half of the figure. At some instant in time, the oscillating mass in molecule 1 is moving
at a velocity v1 and is separated from the fixed mass by a distance z1 while the moveable
mass in molecule 2 is moving at a velocity v2 and is separated from the fixed mass by
a distance z2 . When viewed within the context of a single molecule, q in this model is
likely to be a small fraction of the single electronic charge e− .

(Hook’s law) for the charge fluctuation in each molecule

1
U1 (z) = kz12 = mωo2 z12 , (3.50a)
2
1 2
kz = mωo2 z22 ,
U2 (z) = (3.50b)
2 2
where
" ωo (radians/s) is a characteristic oscillation frequency given by ωo =
k/m. If the two oscillating systems are uncoupled, Schrödinger’s equation
separates into two identical equations
 
h2 ∂ 2 1
− 2 Ψ1 + mωo2 z12 Ψ1 = E1 Ψ1 , (3.51a)
2m ∂z1 2
 
h2 ∂ 2 1
− Ψ2 + mωo2 z22 Ψ2 = E2 Ψ2 , (3.51b)
2m ∂z22 2
where Ψ1 , Ψ2 are the model-dependent wavefunctions describing the charge
fluctuations and E1 and E2 are the energy eigenvalues of each oscillator
respectively. The energy eigenvalues for these two equations are well known
from any introductory quantum course[morrison10]


1
E1 = n + hωo n = 0, 1, 2, . . . (3.52a)
2



1
E2 = n + hωo ; n
= 0, 1, 2, . . . , (3.52b)
2
 Simple Models for Molecule–Molecule Interactions 79

where n and n
are integer quantum numbers. The ground state energy is
just E1 + E2 with n = n
= 0.
When the fluctuation involves a charge, the situation is more compli-
cated since now the two oscillators can interact and become coupled. Using
the co-ordinates defined in Fig. 3.15, the corresponding electrostatic poten-
tial energy must contain four terms and can be written as
 2 
1 q q2 q2 q2
Uelectr (z) = − − + . (3.53)
4πκεo z z + z2 z − z1 (z − z1 ) + z2
The dielectric constant κ of the surrounding medium (if any) is included
to match the discussions found previously in this chapter. A consequence of
this interaction term is that when z  z1 and z  z2 , the exact electrostatic
interaction in Eq. (3.53) can be approximately written as
1 q2
Uelectr
− z2 z1 . (3.54)
2πκεo z 3
This gives a new expression for the potential energy in Schrödinger’s
Equation which is
1 2 1 2 1 q2
Utot = U1 + U2 + Uelect = kz1 + kz2 − z2 z1 . (3.55)
2 2 2πκεo z 3
The cross-term containing z2 z1 in Eq. (3.55) prevents a simple solution
for the new energy eigenvalues of the coupled oscillator problem. However,
if we rewrite the expression for Utot to have a separable form, something
like
1 1
Utot = ks (z1 + z2 )2 + ka (z1 − z2 )2 (3.56)
2 2
then progress can be made.
This can be accomplished by defining new effective spring constants ks
and ka such that
2 2
1 2 1 2 1 q2 1 (z1 + z2 ) 1 (z1 − z2 )
kz1 + kz2 − z2 z1 = ks + ka . (3.57)
2 2 2πκεo z 3 2 2 2 2
Some algebra gives


1 q2
ka = k+ , (3.58)
2πκεo z 3


1 q2
ks = k− . (3.59)
2πκεo z 3
80 Fundamentals of Atomic Force Microscopy, Part I Foundations

It follows that the new energy eigenvalues for the coupled system are
#
q2
k − 2πκε
1
o z
3
ωs = , (3.60a)
m
#
1 q2
k + 2πκε o z
3
ωa = . (3.60b)
m
The change in the quantum ground state energy will be
1 1
∆U (z) = Ucharged (z) − Uuncharged (z) = (hωa + hωs ) − 2 × hωo .
2 2
(3.61)
 
1 q2
Assuming that the electrostatic force 4πκε 2 is small compared to the
1  o z

restoring force of the spring 2 kz , we can justify a Taylor Series expansion

of ωs and ωa . After some algebra, we find that

2
1 1 q2 1
∆U (z) = − hωo . (3.62)
2 4πκεo k z6
Equation (3.62) unfortunately contains two model dependent parameters
k and q which are difficult to define in a physically relevant way. These
two parameters can be eliminated by realizing that a fluctuating electric
field from molecule 1 causes the charge separation in molecule 2 by exerting
a force which must be balanced by the “effective” spring that attempts to
restore the separated charges. Essentially, the spring will “stretch” until the
restoring force of the spring in molecule 2 matches the electrostatic force
produced by the electric field from molecule 1 (see Fig. 3.17).
In equilibrium, this simple argument implies that

|q|E = kz2 . (3.63)

Likewise, in equilibrium, the electric field E will induce a dipole moment

in molecule 2 given by

pinduce,2 = |q|z2 = αE, (3.64)

where α is the polarizability of the molecule in question. Evidently, by

combining Eqs. (3.63) and (3.64) and using Eq. (3.23), we must have
q2
α= = 4πεo αo . (3.65)
k
 Simple Models for Molecule–Molecule Interactions 81

Before charge In “equilibrium”, after

separation charge separation

E E
+q -q
m m m
fixed fixed k

z2

(a) (b)

Fig. 3.17 A simple model to estimate the parameters q and k in the London dispersion
force. The electric field E is produced by molecule 1 (not shown). The force of the
spring in molecule 2 matches the electrostatic force produced by the electric field from
molecule 1.

In this way, the two model-dependent parameters, the spring constant k

and the charge fluctuation q, can be related to the polarizability of the
molecule.

Example 3.6: Suppose a charge imbalance of 0.01e− momentarily

occurs in a He atom. Estimate the effective spring constant k required
to mimic the polarizability volume of He which is measured to be about
0.2 × 10−30 m3 .
From Eq. (3.65), we have
q2
= 4πεo αo
k
q2 [0.01 × 1.6 × 10−19 C]2
k= =  
4πεo αo 4π 8.85 × 10−12 NmC2
2 · 0.2×101−30 m3
2.6 × 10−42 1 N
= −10
· −30
= 0.12 .
1.1 × 10 0.2 × 10 m

The parameter hωo represents a characteristic “motional” energy of

the charge in a polarized atom/molecule. Lacking specific values for this
quantity, we perhaps can represent it by using the characteristic ionization
energy I for the atom or molecule in question. This is a well defied and
easy-to-measure quantity that for chemical elements lies in the range of
5–25 eV.
With these identifications, the simple model gives rise to a result often
quoted in the literature for the attractive interaction energy between two
82 Fundamentals of Atomic Force Microscopy, Part I Foundations

identical non-polar molecules

1 α2 I 1
∆U (z) = − . (3.66)
2 (4πκεo ) z 6
2

To do better, the motion of the charge should be considered in

three dimensions, using a spherically symmetric harmonic oscillator, solv-
ing Schrödinger’s equation in spherical-polar coordinates rather than in
1-dimension discussed above. The energy eigenvalues for this case are well
known and can be written as


3
En = n
+ hωo , (3.67)
2
where n
is again an integer quantum number. The net result is that pre-
factor of 12 in Eq. (3.66) is changed, giving
3 α2 I 1
∆U (z) = − . (3.68)
2 (4πκεo )2 z 6
For dissimilar atoms or molecules having ionization energies I1 , I2 and
polarizabilities α1 and α2 interacting with each other in a medium with a
uniform dielectric constant κ, we finally arrive at a model for the London
or dispersion interaction potential energy


3 α1 α2 I1 I2 1
ULondon (z) = − 2
, (3.69)
2 (4πκεo ) I1 + I2 z 6
where an effective ionization energy Ieff is often defined as


1 1 1 I1 I2
= + ⇒ Ieff = (3.70)
Ieff I1 I2 I1 + I2
to better approximate an “average” ionization energy when I1 and I2 are
vastly different.
Computationally-intensive calculations of these interaction energies go
well beyond the simple model discussed above and require implementations
of Hartree-Fock and density functional theories which require full knowledge
of the wavefunctions for the interacting atoms and/or molecules involved.
A review of these efforts has recently appeared [tkatchenko10].

3.3 The van der Waals Interaction

The total attractive interaction potential energy of two freely rotating

molecules separated by a distance z in a uniform dielectric medium with
Fundamentals of Atomic Force Microscopy Downloaded from www.worldscientific.com

Table 3.3 Dipole moments p, polarizabilities α, ionization energies I, and estimates for the various contributions to the van der
Waals interaction for representative gas phase atoms and molecules. A comparison to experiment is also made where CExper is

Simple Models for Molecule–Molecule Interactions

derived from the coefficients a, b found in the vdW equation of state (Eq. (2.1)). A value of T = 300 K was used to evaluate the
Keesom contribution. The units for the various values of C are 10−79 Jm6 . Adapted from [butt03].
by WSPC on 10/06/15. For personal use only.

αo Percent
p(D) (in 10−30 m3 ) I(eV) CKeesom CDebye CLondon CvdW CExper Diff.

He 0.00 0.20 24.6 0.0 0.0 1.2 1.2 0.9 37%

Ne 0.00 0.40 21.6 0.0 0.0 4.1 4.1 3.6 15%
Ar 0.00 1.64 15.8 0.0 0.0 51.0 5.0 45.3 13%
O2 0.00 1.58 12.1 0.0 0.0 36.2 36.2 46.0 −21%
N2 0.11 1.95 14.0 0.0 0.0 63.9 63.9 60.7 5%
CO 0.11 1.95 14.0 0.0 0.0 63.9 63.9 60.7 5%
HCl 0.00 2.59 12.5 0.0 0.0 100.6 100.6 103.3 −3%
HCl 1.04 2.70 12.8 9.4 5.8 112.0 127.2 156.8 −19%
CO2 0.00 2.91 13.8 0.0 0.0 140.2 140.2 163.6 −14%
NH3 1.46 2.30 10.2 36.4 9.8 64.7 110.9 163.7 −32%
H2 O 1.85 1.46 12.6 93.8 10.0 32.2 135.9 176.2 −23%
HBr 0.79 3.61 11.7 3.1 4.5 183.0 190.6 207.4 −8%
Hl 0.45 5.40 10.4 0.3 2.2 363.9 366.4 349.2 5%
CH3 OH 1.69 3.20 10.9 65.3 18.2 133.9 217.5 651.0 −67%
CHCl3 1.04 8.80 11.4 9.4 19.0 1059.4 1087.7 1632.0 −33%

83
84 Fundamentals of Atomic Force Microscopy, Part I Foundations

dielectric constant κ is comprised of three interactions: (i) dipole–dipole,

(ii) dipole–induced dipole, and (iii) induced dipole–induced dipole. Col-
lectively these interactions are often referred to as the van der Waals
interaction. From the previous sections, we have
UvdW (z) = UKeesom + UDebye + ULondon

2 
 
1 1 1 p21 p22 p21 α2 3 I1 I2 α1 α2
=− 6
+ 2 6
+
4πκε0 3 kB T z z 2 I1 + I2 z6
CdW
=− . (3.71)
z6
Different texts may have slightly different pre-factors in front of each
term, the exact value is determined by the constraints imposed at the begin-
ning of the calculation. For instance, if the interaction between two identical
atoms or molecules with dipole moment p, polarizability α, and ionization
energy I, then Eq. (3.71) reduces to

2  
1 1 1 p4 p21 α 3 I α2
UvdW (z) = − + 2 +
4πκε0 3 kB T z 6 z6 4 z6
CdW
=− . (3.72)
z6
−6
It is important to realize that each term varies as z and hence the
various terms can be grouped together with one overall coefficient that
depends on the properties of the interacting molecules. The London inter-
action is the most general of the three since it does not require either
molecule to have a permanent dipole moment; it always contributes.
Table 3.3 provides a listing of the various contributions to the van der
Waals interaction for a number of gas phase atoms and molecules. The table
includes both calculated and experimental values and shows that the cal-
culated values for CvdW are typically within ±20% of experiment.

Example 3.7: Consider the interaction of two electrically neutral

NH3 molecules separated by a distance of 2 nm in vacuum.
(a) Verify the calculation for CvdW for the molecule NH3 using the
values of p, αo and I listed in Table 3.3.
(b) Evaluate the intermolecular force that two NH3 molecules
experience?
(Continued)
 Simple Models for Molecule–Molecule Interactions 85

Example 3.7: (Continued )

a) We have from Eq. (3.71)

UvdW (z) = UKeesom + UDebye + ULondon

2 
 
1 1 1 p21 p22 p21 α2 3 I1 I2 α1 α2
=− +2 6 + .
4πκε0 3 kB T z 6 z 2 I1 + I2 z6
Since the two NH3 molecules are identical, we can set p = p1 = p2 =
1.46D, α = α1 = α2 = 4πεo αo = 4πεo (2.3 × 10−30 m3 ), and I1 = I2 =
I = 10.2 eV in what follows. Since the two molecules are in vacuum,
κ = 1. This gives

2
1 1 1 p21 p22 CKeesom
UKeesom = − · =− ,
4πε0 3 kB T z 6 z6

2
1 1 1 4
CKeesom = · p
4πε0 3 kB T
 2  
1 1 1
=
3 4π · 8.85 × 10−12 Nm
C2
2 1.38 × 10−23 K
J
· 300 K
 4
× 1.46D · 3.33 × 10−30 Cm/D

2 4


1 19 N m 20 1
= 8.08 × 10 2.42 × 10
3 C4 Nm
 
× 558.7 × 10−120 C4 m4

= 36.4 × 10−79 J m6 ,

2
1 p21 α2 CDebye
UDebye = − ·2 6
=− ,
4πκε0 z z6

2
1
CDebye = · 2p2 α
4πκε0
 2
1
= 2(1.46D · 3.33 × 10−30 Cm/D)2
4π · 8.85 × 10−12 Nm
C2
2

(Continued)
86 Fundamentals of Atomic Force Microscopy, Part I Foundations

Example 3.7: (Continued )

C2
×(4π · 8.85 × 10−12 · 2.3 × 10−30 m3 )
Nm2

2 4

19 N m
= 8.08 × 10 (2)(23.6 × 10−60 C2 m2 )
C4
 m
× 2.56 × 10−40 C 2
N
= 9.8 × 10−79 J m6 ,

2

1 3 I1 I2 α1 α2 CLondon
ULondon =− · 6
=− ,
4πκε0 2 I1 + I2 z z6

2

1 3 I
CLondon = · α2
4πκε0 2 2
 2

1 3 1.602 × 10−19 J
= 10.2 eV ·
4π · 8.85 × 10−12 Nm
C2
2
4 eV

2
C2 −12 −30 3
× 4π · 8.85 × 10 · 2.3 × 10 m
Nm2



N2 m 4 3 −18 −80 4 m
2
= 8.08 × 1019 (1.63 × 10 J) 6.55 × 10 C
C4 4 N2

= 64.7 × 10−79 J m6 ,

CvdW = CKeesom + CDebye + CLondon

= (36.4 + 9.8 + 64.7) × 10−79 J m6

= 110.9 × 10−79 J m6 .

b) The force between the two molecules is attractive and can be found
by taking the negative gradient of UvdW (z) with respect to

(Continued)
 Simple Models for Molecule–Molecule Interactions 87

Example 3.7: (Continued )

z (see Eq. (2.11)). The force F is found to be

 
∂UvdW (z) (−6)
F =− = − −CvdW 7
∂z z


−79 6
= −110.9 × 10 Jm × 6
(2 × 10−9 m)7


110.9 × 10−79 Jm6
= −6
1.3 × 10−61 m7

= −5.2 × 10−16 N = −52 fN(attractive).

Under what circumstances might such a weak force produce a mea-
sureable effect?
One possible answer is that when the vdW force is integrated over an
object with dimensions of ∼ 1µ m, the vast number of atoms/molecules
that must be taken into account serves as a force “multiplier” that will
produce nanoNewton (nN) forces. This line of thought is the central
topic of the next chapter and forms the basis for understanding the
operation of an AFM.
A second answer relates to soft biological molecules like proteins
which may contain thousands of polar and non-polar subunits that
are placed in close proximity to each other. The charge fluctuations
in such extended molecules are far greater than in individual gas-phase
molecules. The net sum of 100’s of tiny vdW interactions act on each
segment of a protein, causing it to structurally bend and twist into a
final shape that becomes biologically interesting.

3.4 Chapter Summary

In the early 1900s, understanding the origin of the non-zero interac-

tion between electrically neutral gas-phase molecules posed a formidable
challenge. An explanation for the interaction between electrically neutral
atoms and molecules was ultimately reached by considering the electri-
cal dipole moments that atoms and molecules possess. There are three
88 Fundamentals of Atomic Force Microscopy, Part I Foundations

origins for these dipole moments. Molecules containing atoms with differ-
ent electronegativities create permanent molecular dipole moments as the
atoms comprising the molecule produce small shifts in the electronic charge
distribution. Secondly, molecules and atoms with no net dipole moment can
develop an induced dipole moment when subjected to an external electric
field. Lastly, neutral atoms and molecules with no net dipole moment can
develop a fluctuating, time-dependent dipole moment that depends on the
size and shape of the atom or molecule under consideration. This effect is
strictly quantum mechanical in nature. The time-correlated fluctuations of
these spontaneous dipole moments conspire to produce an attractive inter-
action between all atoms and molecules.
The derivation of the interaction potential energy between gas-phase
molecules was discussed in some detail. The classical results obtained by
Keesom and Debye were derived for both static dipoles and for dipoles
induced by external electric fields. London’s discussion of the spontaneous
dipole moment in an electrically neutral object modeled as a quantum
spring-mass system is also derived. While London’s derivation leads to a
short-range interaction that is always attractive, the interaction between
permanent dipoles can be either attractive or repulsive depending on the rel-
ative dipole orientation. A thermal averaging procedure is discussed which
shows how molecules with a permanent dipole moment, free to rotate, can
take advantage of thermal motion to orient in such a way as to always pro-
duce a net attractive interaction. Taken together, the sum of these three
possibilities — permanent dipole, induced dipole and fluctuating dipole —
lead to short-range attractive interactions that all vary as 1/z 6 .

3.5 Further Reading

Chapter Three References:

[bell35] R. P. Bell, Trans. Faraday Soc. 31, 1557–1560 (1935).

[butt03] H.-J. Butt, K. Graf, M. Kappl, Physics and Chemistry of
Interfaces, Wiley-VCH, Weinheim (2003).
[debye21] P. Debye, Z. Phyzik 22, 302 (1921).
[griffiths13] D.J. Griffiths, Introduction to Electrodynamics, 4th ed.,
Pearson, (2013).
[israelachvili98] J.N. Israelachvili, Intermolecular and surface forces. Academic
Press, New York (1998).
[keesom21] W.H. Keesom, Z. Physik 22, 129 (1921).
[london37] F. London, Trans. Farad. Soc. 33, 8–26 (1937).
 Simple Models for Molecule–Molecule Interactions 89

[morrison10] John C. Morrison , Modern Physics for Scientists and Engineers,

Academic Press (2010).
[tkatchenko10] A. Tkatchenko, et al., MRS Bulletin 35, 435 (2010).

Further reading:

[atkins10] P.W. Atkins and J. de Paula, Physical Chemistry, 9th edition, W.H.
Freeman and Co., New York NY, USA (2010).
[jeffrey 97] G.A. Jeffrey, An introduction to hydrogen bonding. Oxford University
Press (1997).
[kotz06] J.C. Kotz, P. Treichel and G.C. Weaver, Chemistry and chemical reac-
tivity, 6th edition, Thomson Brooks/Cole, Belmont CA, USA (2006).
[rigby86] M. Rigby, E.B. Smith, W.A. Wakeham, and G.C. Maitland, The
forces Between Molecules, Oxford-Clarendon Press (1986).
[stone96] A.J. Stone, Theory of Intermolecular Forces, Oxford-Clarendon Press
(1996).
[nelson67 ] R.D. Nelson, D.R. Lide, A.A. Maryott, “Selected Values of Elec-
tric Dipole Moments for Molecules in the Gas Phase”, Nat. Stand.
Ref. Data Set., Natl. Bur. Stand. (NSRDS-NBS) No. 10 (1967).

3.6 Problems

1. Two identical dipoles p1 and p2 are separated by 1 nm in a dielectric

medium with κ = 5. The dipoles are aligned in four different configu-
rations as shown below. Complete the table below in order to specify
their orientation.
90 Fundamentals of Atomic Force Microscopy, Part I Foundations

Configuration Θ1 Θ2 ζ

(a)
(b)
(c)
(d)

Calculate the interaction potential energy (in eV) for each of the config-
urations shown above. Which of the four configurations is most stable?
Which of the four configurations produces the largest force between the
two dipoles?
2. Calculate the interaction potential energy between two fixed water
molecules in vacuum whose centers are 10 nm apart, and whose dipole
axes make an angle of 120◦ with respect to each other. Assume the water
molecules lie in the same plane and they are located in free space. How
does the interaction potential energy change if the two molecules can
orient themselves by thermal motion? Assume T = 300 K.
3. A water molecule (dipole moment 1.85 D) approaches a singly charged
ion. Describe the most energetically favorable orientation of the water
molecule with respect to the ion. Calculate the potential energy of
the interaction at a distance of 1.0 nm and compare this to the ther-
mal energy (3kB T /2) at 20◦ C based on the equipartition theorem.
Assume the dielectric constant of liquid water is 80. At this tempera-
ture, will the water molecule “lock-in” to a fixed position with respect to
the ion?
4. The discussion in this chapter often uses distances between molecules
that are typically less than 1 nm. Is this realistic? On average, what
is the distance between molecules in 1 mole of gas at P = 100 kPa
pressure and T = 300 K? Assume the ideal gas law P V = N kB T is
valid. What is the value of V (in m3 ) when the number of particles N is
equal to NA = 6.02 × 1023 ? Divide V by NA to obtain an estimate for
the volume per gas atom. Take the cube root of this value to estimate
the distance between atoms (or molecules) in the gas.
5. An ion with a charge q is situated a distance z from a non-polar molecule
with a polarizability volume α. Both the ion and molecule are in a
dielectric medium with a relative dielectric constant κ. (a) What is the
induced dipole moment? (b) What is the electric field generated by this
induced dipole at the location of the point charge? (c) What is the force
of attraction between the charge q and the molecule?
 Simple Models for Molecule–Molecule Interactions 91

6. LiCl is dissolved in water. A multitude of forces can develop as listed

in the table below. Between what atomic components (Li ions, Cl ions
or water molecules) do these various forces interact?

Force Component Acts between

ion-dipole
hydrogen bonding
ion-ion
ion-induced dipolar
dipole-dipole
dipole-induced dipolar
induced dipole-induced dipole

7. What do molecules of H2 , N2 , CO2 and CH4 have in common?

8. The interaction between two dipoles of fixed orientation that lie in the
same plane is often written as
1 1
Utotal (z, θ1 , θ2 , ζ = 0) = p1 · p2 − 3 (
[ p1 · r̂) (
p2 · r̂)]
4πκεo z 3
1 p1 p2
= [cos(θ12 ) − 3 cos θ1 cos θ2 ]
4πκεo z 3

where the angles are defined in the figure. Show this is equivalent to
the expression given by Eq. 3.28.
9. Calculate the potential energy of interaction between two Ar atoms
separated by 0.5 nm in vacuum. The polarizability volume of Ar is
about 1.6 × 10−30 m3 , the ionization energy of Ar is about 15.7 eV per
atom.
92 Fundamentals of Atomic Force Microscopy, Part I Foundations

10. Two methane molecules interact via a 12-6 Lennard-Jones potential

characterized by the parameters ε = 2.1 × 10−21 J and σ = 0.42 nm.
Derive an equation for the force on a methane molecule as a function of
its distance from another methane molecule. Make a plot of the force
vs. separation. Make sure the force has units of nN and the separation
has units of nm. What is the separation between the two molecules
when the net force acting on the two methane molecules becomes zero?
11. A water molecule has a polarizability volume of 1.5 × 10−30 m3 and
a permanent dipole moment equal to 1.85 D. At what separation dis-
tance will a Na ion induce a dipole moment in a water molecule equal
to its permanent dipole moment? Assume the dielectric constant of
liquid water is 80, a constant independent of the molecule-molecule
separation.
12. The coefficients appearing in the vdW equation of state for a gas
(see Eq. (2.1)) can be used to define an experimental value for CvdW .
The coefficients a, b are determined by fitting P –V data for a gas. Ide-
ally, you might expect P V = nRT . But careful experiment shows that
P is reduced from the value predicted by the ideal gas law. By fitting
the measured P –V data, accurate values for the vdW coefficients a and
b can be obtained.
As an example, measurements performed on CH4 show that a =
0.225 Nm4 /mol2 and b = 4.28 × 10−5 m3 /mol. You can use these values
to define
9ab
CExper = ,
(2π)2 NA3
where NA is Avagadro’s number. Evaluate CExper using the coefficients
a and b for CH4 and compare to the value of CvdW found using the
appropriate values of p, α, and I for CH4 listed in Table 3.3.
Make a plot of P vs. V for one mole of CH4 at some constant tem-
perature T near 300 K. Use the ideal gas law and then use the van der
Waals equation (Eq. (2.1)) with the coefficients a and b given above.
Roughly at what pressures do you find a 0.5% pressure reduction from
the predictions of the ideal gas law?
13. From P –V measurements, it is known that water vapor does not strictly
obey the ideal gas law. This implies that water molecules attract each
other when in the gas phase. Roughly, what percentage of this attrac-
tion is due to the London dispersion force?
14. The heat of vaporization (or the enthalpy of vaporization) measures
the amount of energy required to turn a fixed amount of liquid at
 Simple Models for Molecule–Molecule Interactions 93

its normal boiling temperature into a gas at the same temperature.

Ultimately, the value of the heat of vaporization must be related to
the strength of interaction between nearest neighbor atoms. Use the
value of CvdW from Table 3.3 to roughly estimate the energy required
to convert one mole of liquid neon into gas at its boiling temperature
of 27 K. Assume the inter-atomic Ne-Ne distance is about 0.3 nm. At
first, assume a Ne atom only interacts with another Ne atom. How does
your estimate compare to the measured value of 1.7 kJ/mole? In order
to better match your estimate to measurements, roughly how many Ne
atoms must interact in the liquid state?
15. Noble gas atoms are inert. They are odorless, colorless, non-flammable,
and have low chemical reactivity. If atoms cannot interact, are they
electronegative? The inert behavior demonstrated by all noble gasses
is a common characteristic suggesting that any noble gas atom-atom
interaction might be dominated by the London dispersion force. If sep-
arate containers of the noble gasses were filled to atmospheric pressure
at 25◦ C, which noble gas would you predict might be the most strongly
interacting? For reference, the boiling points and atomic radii of the
noble gases are listed in the table below. Make a plot of boiling point
vs. atomic size. Is there a correlation?

Noble Gas Chemical Symbol Boiling Point Atomic radius

helium He 4.2 K 31 pm
neon Ne 27.3 K 38 pm
argon Ar 87.4 K 71 pm
krypton Kr 121.5 K 88 pm
xenon Xe 166.6 K 108 pm
radon Rn 211.5 K 120 pm