10 - Unifying concepts in intermolecular forces

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10
Unifying Concepts in Intermolecular
and Interparticle Forces
10.1 The Association of Like Molecules or Particles
in a Medium
In this second part of the book, we shall be looking at the physical forces between
particles and surfaces. While the fundamental forces involved are the same as those
already described (i.e., electrostatic, van der Waals, solvation forces), they can manifest
themselves in quite different ways and lead to qualitatively new features when acting
between large particles or extended surfaces. These differences will be discussed in the
next chapter. In this chapter we look at the similarities and see how the ideas developed in
Part I also apply to the interactions of macroscopic particles and surfaces. We shall find
that, independently of the type of interaction force involved, certain semiquantitative
relations describing molecular forces—known as combining relations—are applicable
quite generally to all systems—that is, to the interactions of molecules, particles, surfaces,
and even complex multicomponent systems.
Let us start by noting that for any type of interaction between two molecules A and B,
the interaction energy at any given separation is always, to a good approximation,
proportional to the product of some molecular property of A times some molecular
property of B (rather than, say, their sum). Let us denote these properties by A and B.
Referring to Table 2.2, we find, for example, that for the charge-nonpolar-molecule
interaction we may write A f Q2
A and B f aB; for the dipole-dipole interaction, A f uA or
uA
2 and Bf uB or uB
2, while for the dispersion interaction we have Af aA and Bf aB. Note
that even for the gravitational interaction (Eq. 1.1), we may put A f mass of A, and B f
mass of B.
Thus, for many different types of interactions, we may express the binding energies of
molecules A and B in contact as
WAA ¼ �A2; WBB ¼ �B2 ðfor like moleculesÞ (10.1)
and
WAB ¼ �AB ðfor unlike moleculesÞ; (10.2)
where only for the purely Coulombic charge-charge interaction are the signs reversed—
for example, positive for two like charges (Table 2.2). Let us now consider a liquid con-
sisting of a mixture of molecules A and B in equal amounts. If the molecules are randomly
dispersed (Figure 10.1a), then on average an A molecule will have both A and B molecules
Intermolecular and Surface Forces. 3rd edition, DOI: 10.1016/B978-0-12-375182-9.10010-7 191
Copyright � 2011, Elsevier Inc. All rights reserved.
N
N
A
ΔW
ΔW = −n(A − B)2
ΔW = Wass 
− Wdisp = −9(A − B)2
B
A
B
A
N
 − n
N − n
An
Dispersed Associated
Dispersed state
Associated state
ΔW
B
B
B
B B B
B
B
B
B
BB
BB
A
A
A A
A
A A
A
A
A
A A
AA
(a)
(b)
ΔW = −(A − B)2
ΔW
Dispersed Associated
A A A AB
BB
B
BB
(c)
(d)
FIGURE 10.1 (a) Two central molecules A and B surrounded by an equal number of A and B molecules in a solvent.
Since there are three AeA “bonds,” three BeB “bonds,” and 18 AeB “bonds,” we may write Wdisp ¼ –(3A2 þ 3B2 þ
18AB). (b) Seven A molecules and seven B molecules have associated in two small clusters. There are now 12 AeA
“bonds” and 12 BeB “bonds,” so that Wass ¼ �12(A2 þ B2). The net change in energy on going from the dispersed to
the associated state is therefore DW¼Wass�Wdisp¼�9(A� B)2. Note that there is no change in the number of A and
B molecules exposed to the solvent. Thus, DW does not involve any term due to the interaction with the surrounding
medium if it is unchanged during the redistribution of the A and B molecules. (c) Two A molecules associating in
a medium of B molecules. HereWdisp¼�2AB, andWass¼�(A2þ B2), so that DW¼�(A� B)2. (d) Two large particles A
associating in a medium of small solvent molecules B. This involves the replacement of 2n AeB “bonds” by n AeA
“bonds” and n BeB “bonds.”Wdisp¼�2NAB,Wass¼ �2(N� n)AB� nA2�nB2, so that DW¼ �n(A – B)2. Note that n is
proportional to the adhesion area or effective “contact area” of the two particles, which is proportional to their radii
(see Worked Example 10.1).
192 INTERMOLECULAR AND SURFACE FORCES
Chapter 10 • Unifying Concepts in Intermolecular and Interparticle Forces 193
as nearest neighbors, and similarly for molecule B. However, if the molecules are asso-
ciated, then the molecular organization of nearest neighbors around molecules A and B
will be as in Figure 10.1b. The difference in energy between the associated and dispersed
clusters will therefore be DW ¼ –9(A � B)2, where we note that in this two-dimensional
case nine AeA “bonds” and nine BeB “bonds” have replaced 18 AeB “bonds.” In three
dimensions with 12 nearest neighbors around each central molecule, and starting with six
A and six B molecules around each A and B molecule, we would find (Problem 10.1) that
DW ¼ �22(A � B)2 and that 22 AeA and 22 BeB “bonds” have been formed on associ-
ation. For the simplest case of two associating A molecules (Figure 10.1c), we have DW ¼
�(A � B)2.
Thus in general we may write
DW ¼ Wass �Wdisp ¼ �nðA� BÞ2 (10.3)
where n is always equal to the number of like “bonds” that have been formed in the
process of association, irrespective of how many molecules are involved or their relative
sizes (Figure 10.1d). Further, since (A� B)2 must always be positive, we see that in general
DW< 0—that is,Wass <Wdisp. We may therefore conclude that the associated state of like
molecules is energetically preferred to the dispersed state. In other words, there is always
an effective attraction between like molecules or particles in a binary mixture.
n n n
Worked Example 10.1
Question: Two rigidmacroscopic spheres of radiusR are in adhesive contact as in Figure 10.1(d).
What is their “effective” contact area—that is, the area that determines the number of inter-
molecular bonds n between them?
Answer: This can be a subtle problem. The answer depends on whether the intermolecular
forces are long-range or short-range compared to the sizes of the particles. However, sincewe are
here considering the adhesion ofmacroscopic spheres, wemay assume the forces to be of short-
range, effectively acting over a distance of the size of the molecules of the solvent or particles.
Thus, referring to Figure 10.2 (left), we need to determine the area that excludes solvent mole-
cules, of radius a, between the surfaces. That is, we need to determine pr2 in terms of R and a.
From the geometric construction of Figure 10.2 (right) we apply Pythagoras’s theorem:
AC2¼ AB2þ BC2¼ AD2þ BD2þ BD2þDC2. Thus: 4R2¼ a2þ 2r2þ (2R – a)2, which simplifies to
r2 ¼ ð2R� aÞaz 2Ra for R » a: (10.4)
2
B
Rr
A
RDa
a
a Cr
RR
FIGURE 10.2
194 INTERMOLECULAR AND SURFACE FORCES
This relation, known as the chord theorem, is important for deriving many of the equations in
later chapters. Thus, for two large spheres in contact their effective “area of contact” or
effective “interaction area” is given by pr2 z 2pRa, where a is a measure of the range of the
forces (usually of the order of molecular dimensions). Note that the effective area of interaction
is proportional to R.
It is a simple matter to show that for two contacting spheres of different radii, R1 and R2,
their effective interaction area is 4pR1R2a/(R1 þ R2), and that for two parallel cylinders it is
2[R1R2a/(R1 þ R2)]
1/2 per unit length.
n n n
Equation (10.3) can be developed further to provide more general insights into the
interactions of like solute molecules and particles in a medium. First, Eq. (10.3) may be
expressed in a number of different forms:
DW ¼ �nðA� BÞ2 ¼ �nðA2 þ B2 � 2ABÞ (10.5)
¼ �nð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
�WAA
p
�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
�WBB
p
Þ2 ¼ þnðWAA þWBB � 2WABÞ; (10.6)
where DWmay be readily seen to be the same as the interaction pair potentialw(s) in the
medium (at contact). Second, since –nWAA and –nWBB are roughly proportional to the
respective molar cohesion energies UA and UB, we find that
DWf� nð
ffiffiffiffiffiffiffi
UA
p
�
ffiffiffiffiffiffiffi
UBp
Þ2 (10.7)
which is essentially the same as Eq. (6.39), previously derived for the specific case of
dispersion forces. If WAA and WBB are sufficiently different (i.e., if the molecules are very
different; for example, A polar, B nonpolar), then DW will be large enough to overcome
the entropic drive to disorder resulting in a low solubility (immiscibility) or phase
separation. The immiscibility of water and hydrocarbons and the “like-dissolves-like”
rule, previously discussed in Sections 6.7 and 6.8, are examples of this phenomenon.
Furthermore, since DW f n, larger particles or polymers of higher molecular weight are
more likely to phase separate than smaller particles or molecules, and indeed the vast
majority of polymers are immiscible with each other.
Third, Eq. (10.3) shows that the value of DW/n for molecules of type A coming into
contact in medium B is the same as for the inverse case of molecules of type B associating
in medium A. This reciprocity property was previously noted for the specific case of van
der Waals forces (Section 6.7).
Finally, Eqs (10.3) and (10.5) clearly showthat the interactionof twosolutemoleculesA in
a solvent medium B is intimately coupled to the strength of the solvent-solvent interaction.
Thus, the attraction between two particles in water or between two protein molecules in
a lipid bilayer is not independent of the surrounding water-water or lipid-lipid interactions.
While the preceding semiquantitative criteria have broad applicability, there are two
very important exceptions: First, for the Coulomb interaction between charged atoms
or ions, since the sign of DW is reversed, the dispersed state (Figure 10.1a) is the favored
Specific (complementary) associations of like molecules
Complementary associations of unlike molecules
(e) Geometric
(c) Geometric
(f) Complementary ionic
(acid-base) and H-bonds
(d) Geometric and chemical
Water
Hydrophilic (polar)
Hydrophobic (non-polar)
Non-specific associations
Dimer
Cap
(a) Ionic
Solvent Solvent
(b) Van der Waals, hydrophobic
FIGURE 10.3 Nonspecific (a, b) and specific interactions (c to f) between molecules. Specific interactions can lead
to orientationally specific associations of like molecules or to preferential association of unlike molecules. Such
effects can be due to molecular geometry (molecular shape or “topology”) and/or chemical effects (specific bonds),
and they are particularly common in the interactions of biological molecules as discussed in Chapters 21 and 22.
Chapter 10 • Unifying Concepts in Intermolecular and Interparticle Forces 195
one. Thus, in ionic crystals (e.g., NaþCl–), the cations and anions are nearest neighbors in
the lattice. However, once oppositely charged ion pairs associate to form electroneutral
dipoles, these can then be treated as units that do obey the preceding relations.
There are certain classes of molecules and interactions that do not readily fall into this
simple pattern because the strength of the bond between two different molecules cannot
be expressed simply in terms of WAB ¼ �AB. Such interactions are called specific inter-
actions and, in the case of biological molecules, complementary or lock-and-key inter-
actions (Chapter 21). Figure 10.3 shows examples of how such specific interactions arise.
They can result from the complementary shapes of molecules, in which like molecules
cannot fit together, whereas unlike molecules can; or they can be the result of an
inherently specific interatomic bond, such as the hydrogen-bond (Figure 8.2). In the latter
case, a molecule such as acetone,
O
CH3 – C – CH3
, cannot form H-bonds with another
similar molecule, but it can do so with water via its C¼O group, and for this reason,
acetone is miscible with water. Such strongly hydrophilic groups (see Table 8.2) repel
each other in water due to their strong binding to water, and their specific interactions
cannot be described in terms of the simple equations just presented.
196 INTERMOLECULAR AND SURFACE FORCES
10.2 Two Like Surfaces Coming Together in
a Medium: Surface and Interfacial Energy
The previous approach may be readily applied to the interaction of two macroscopic
surfaces in a liquid. Let us start with two flat surfaces of A, each of unit area, in a liquid
B. We may equate DW of Eq. (10.6) with the (negative) free energy change of bringing
these two surfaces into adhesive contact in the medium. This energy, or work, is
defined as twice the interfacial energy gAB of the A-B interface, which is positive by
convention. Thus,
DW ¼ �2gAB or gAB ¼ �
1
2
DW ¼ 1
2
nðA� BÞ2: (10.8)
The factor 2 arises because by bringing these two surfaces into contact, the two initially
separate media of A have merged into one, so we have effectively eliminated two
unit areas of the A-B interface. Now, let there be n bonds per unit area. In Eq. (10.6)
nWAB is therefore the energy change of bringing unit area of A into contact with unit
area of B in a vacuum. This is known as the adhesion energy or work of adhesion per
unit area of the A–B interface. Likewise, nWAA is the (negative) energy change of
bringing unit areas of A into contact in a vacuum, known as the cohesion energy or
work of cohesion. Note that two unit areas of A are eliminated in this process. By
convention, the cohesion energy is related to the (positive) surface energy gA by nWAA ¼
�2gA—that is,
gA ¼ �
1
2
nWAA ¼ 1
2
nA2; (10.9a)
or simply,
g ¼ �1
2
W per unit area; (10.9b)
and similarly for gB. Note that the interfacial energy gAB ¼ 1
2n(A � B)2 of the A–B interface
is very different, both phenomenologically and quantitatively, from the adhesion energy
WAB ¼ �nAB of surfaces A and B.1 These different surface energies will be discussed in
more detail in Chapter 17; for the moment, we simply note that for two surfaces Eq. (10.6)
may be expressed in the form2
gAB ¼ gA þ gB � jWABj per unit area: (10.10)
1For example, the interfacial energy of two similar surfaces in contact (A ¼ B) is zero even as their surface
energy is not.
2In some conventions the sign of W for the work of adhesion or cohesion is positive (i.e., W>0) because
the reference state of zero energy is taken as the contact state (D ¼ 0). This is in contrast to the negative values
for W(D) and w(r) where, again by convention, the reference states are the fully separated states at D ¼ N or
r ¼ N
Chapter 10 • Unifying Concepts in Intermolecular and Interparticle Forces 197
This important thermodynamic relation is valid for both solid and liquid interfaces. It
gives the free energy (always negative3) of bringing unit areas of surfaces A into contact in
liquid B, and vice versa, since gAB ¼ gBA.
All the preceding equations belong to an important class of expressions known as
combining relations or combining laws. They are extremely useful for deriving relation-
ships between various energy terms in a complex system, and are often used for obtaining
approximate values for parameters that cannot be easily measured. For example, if we
return to Eq. (10.6), we may also write it as
DW ¼ nðWAA þWBB � 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
WAAWBB
p
Þ (10.11)
so that Eq. (10.10) becomes
gAB ¼ gA þ gB � 2
ffiffiffiffiffiffiffiffiffiffiffi
gAgB
p ¼ ð ffiffiffiffiffiffi
gA
p � ffiffiffiffiffiffi
gB
p Þ2; (10.12)
a useful expression that is often used to estimate the interfacial energy gAB solely from the
surface energies or surface tensions of the pure liquids, gA and gB, in the absence of any
data on the energy of adhesion WAB. We shall encounter these and other combining
relations again later.
10.3 The Association of Unlike Molecules, Particles,
or Surfaces in a Third Medium
Let us now proceed from two-component to three-component systems, starting with
a consideration of themixture shown in Figure 10.4. For twounlikemolecules or particles A
andBcoming together in the solventmediumcomposedofmolecules of typeC (Figure 10.4
a/b), we find that the processcan be split up into four elementary steps as follows:
DW ¼ Wass �Wdispf� AB � C2 þ AC þ BC ¼ �ðA� CÞðB � CÞ: (10.13)
This is a very interesting result because it shows that the energy of association can now be
positive or negative. If positive, the particles effectively repel each other and therefore
remain dispersed in medium C.
It is instructive to consider how an effective repulsion has arisen from interaction
potentials that are all purely attractive to begin with. The phenomenonmay be thought of
as Archimedes’ principle being applied to intermolecular forces. In the case of gravitational
forces we all know that iron sinks while wood rises in water. Thus, wood is effectively
experiencing a repulsion from the earth when in water. This is because it is lighter than
water, and if it were to descend, it would have to displace an equal volume of water and
therefore drive it upward to replace the spacepreviously occupiedby thewood. Sincewater
is denser and thus more attracted to the earth than wood, the whole process would be
3Actually, the interfacial energy can be positive, which implies that two surfaces or molecules of A repel each
other in medium B, and vice versa. The separation of molecules A will continue until no aggregates
remain—that is, until the AB interface disappears. Thus, if there is an interface, DW must be negative (and gAB
must be positive).
A, B dispersed
AA and BB associated
Solvent
ΔW −(A − C)2 − (B − C)2
ΔW −(A − C)(B − C)
ΔW −(A − B)2
A,B associated
(a) (b)
(c)
C
C
C
C
C
C
C
C
C
C
C
C
A
A
A
A
A
A
B
B
B
B
B
B
FIGURE 10.4 (a) and (b) Two unlike molecules or particles A and B may attract or repel each other in a third
medium C. Repulsion will occur (DW positive) if the properties of C are intermediate between those of A and
B—for example, for gravitational forces, if the density of C (e.g., water) is between that of A (e.g., iron) and B
(e.g., wood). In such cases the dispersed state (a) is energetically favored. (c) The associated state of like
molecules has a lower energy than either (a) or (b).
198 INTERMOLECULAR AND SURFACE FORCES
energetically unfavorable—in other words, the energy lost in water going up is not
recovered by the energy gained inwood coming down. This displacement principle applies
to all interactions in a medium, including intermolecular interactions (cf. Problem 10.4).
Equation (10.13) tells us that if C is intermediate between A and B, two particles (or
surfaces) will repel each other, an effect that was previously noted for van der Waals
forces (Section 6.7) where the operative properties of media A, B, and C are their dielectric
constants and refractive indices. However, the association or dissociation of A and B in
medium C is not the end of the story. As shown in Figure 10.4, whatever the relation
between A, B and C, the most favored final state will be that of particles A associating with
particles A, B with B, and C with C (Figure 10.4c).
This procedure may be extended to mixtures with more species. We may therefore
generalize our earlier conclusion: there is always an effective attraction between like
molecules or particles in a multicomponent mixture (again with the proviso that the
interactions are not dominated by Coulombic or H-bonding forces). And in addition,
unlike particles may attract or repel each other in a solvent.
10.4 Particle-Surface and Particle-Interface
Interactions
The preceding analysis can be extended to the case of a particle C near an interface
dividing two immiscible liquid media A and B (Figure 10.5). Four situations may arise:
Adsorption
Adsorption
Engulfing (A B)
 (A B)
Engulfing
Negative adsorption
from both sides
or ejection
(A solid)
Medium A Medium B
Medium A Medium B
Interface
C C C
C C
CC
C
C
(a)
Engulfin
g
Medium B
C
Adsorption
Negative adsorption from
both sides (impossible)
Medium ATOTW
(b)
FIGURE 10.5 (a) Different possible modes of interaction of a particle C with a liquid-liquid interface. (b)
Corresponding schematic energy versus distance profiles (assumed monotonic) for DWtot < 0. In the case of
adsorbed particles, their configuration at the interface depends on their shape and whether the two media
A and B are liquid or solid. If both media are liquid and the particle is spherical, it will pass through the interface
or subtend a finite contact angle q at the interface. If medium B is solid, particle C could adsorb on, but not
penetrate into or pass through, the interface.
Chapter 10 • Unifying Concepts in Intermolecular and Interparticle Forces 199
200 INTERMOLECULAR AND SURFACE FORCES
• Adsorption: The particle is attracted to the interface from either side.
• Desorption: The particle is repelled from the interface on either side of it (also
known as “negative adsorption”).
• Engulfing: The particle is (1) attracted from one side (left or right) but (2) repelled
from the other side (right or left).
Applying Eq. (10.13) we may write for the energy change of a particle coming up to the
interface shown in Figure 10.5:
from the left:
!
DWf� ðC � AÞðB � AÞ; (10.14)
from the right:
 
DWf� ðC � BÞðA� BÞ: (10.15)
The preceding equations predict the following possibilities, illustrated in Figure 10.5a:
First, if the particle’s properties C are intermediate between those of media A and B (i.e., if
A > C > B or A < C < B), both
!
DW and
 
DW are negative. The particle will therefore be
attracted to the interface from either side, leading to adsorption at the interface. The
adsorption of amphiphilic molecules at hydrocarbon-water interfaces is an example of
such a phenomenon. Amphiphilic molecules such as detergents and surfactants (derived
from the words ‘surface-active’) are partly hydrophilic and partly hydrophobic and so
have properties intermediate between the two liquids. Second, if A > B > C or A < B < C
(B intermediate),
!
DW will be negative, but
 
DW will be positive. The particle will therefore
be attracted to the interface from the left but repelled from the right (engulfing by
medium B). Finally, if B > A > C or B < A < C (A intermediate),
!
DW will be positive, but 
DW will be negative, and the particle will now be attracted from the right but repelled
from the left (reverse engulfing or ejection frommedium B). Since these six combinations
exhaust all the possibilities, we see that repulsion from both sides of an interface (i.e.,
negative adsorption from both sides) cannot occur and that either adsorption or
engulfing will be the rule. It is for this reason that surfaces are so prone to adsorbing
molecules or particles from vapor or solution.
In the following two sections we discuss the cases of engulfing and adsorption in turn.
10.5 Engulfing and Ejection
The special case of engulfing or ejection involves the complete transfer of a particle from
the interior of one bulk medium into another. The total energy of transfer
!
DWtot from
medium A into medium B can be determined by combining Eqs (10.14) and (10.15) to
give
!
DWtot ¼ DWA/B ¼
!
DW � DWf� ðC � AÞðB � AÞ þ ðC � BÞðA� BÞ
fðB � CÞ2 � ðA� CÞ2 (10.16a)
fgBC � gAC (10.16b)
Chapter 10 • Unifying Concepts in Intermolecular and Interparticle Forces 201
where gBC and gAC are the interfacial energies of the respective particle-media interfaces.
Thus, a particle will always move into a medium where its interfacial energy is lowest
(
!
DWtot < 0), which in the case of engulfing by medium B implies that gBC < gAC. For
example, for a spherical particle of surface area 4pr2, its change in energy on going from
bulk medium B to bulk medium A is
DWA/B ¼ Surface area� ðgBC � gACÞ ¼ 4pr2ðgBC � gACÞ: (10.17)
This energy determines the partitioning of particles and molecules between different
phases according to the Boltzmann factor e�DW=kT , where the free energy of transfer DW
can be seen to be the same as the change in self-energy mi0 of the particle or molecule,
discussed in Chapter 2. Since DW is proportional to the area ofa molecule or particle, the
partitioning becomes increasingly more pronounced for larger molecules—for example,
polymers of higher MW—even when the type of interaction (e.g., van der Waals) remains
the same.
The preceding two equations for the energy of engulfing or transfer should not be
taken to imply that engulfing will occur whenever gBC and gAC are different, which they
usually are. We also have to exclude the possibility of adsorption and desorption, and
when this is done (see Worked Example 10.2), one finds that the necessary condition for
engulfing may be expressed in terms of the interfacial energies as
gBC þ gAC > gAB: (10.18)
n n n
Worked Example 10.2
Question: Obtain Eq. (10.18) from first principles—that is, from a consideration of the condi-
tions necessary for engulfing based only on the elementary pair potentials AB, AC, and BC.
Answer: From the discussion following Eqs. (10.14) and (10.15), it is clear that for engulfing to
occur from either side of the interface,
!
DWand
 
DWmust have opposite signs. This means that!
DW � DW < 0. In terms of the elementary potentials, this may be expressed as (C� A)(B� A)�
(C � B)(A � B) < 0, and since (B � A)(A � B) ¼ �(A � B)2 must be negative, this equation
immediately simplifies to (C � A)(C � B) > 0. In terms of the elementary interactions as
defined by Eq. (10.8), Eq. (10.18) is equivalent to 1
2n(B � C)2 þ 1
2n(A � C)2 > 1
2n(A � B)2. When
this equation is expanded, the 1
2nA
2 and 1
2nB
2 terms cancel and one is left with (C� A)(C� B)> 0,
which is the necessary condition for engulfing.
n n n
10.6 Adsorbed Surface Films: Wetting
and Nonwetting
The previous examples and Figure 10.5 apply only to isolated particles or molecules of C;
in other words, they apply only to dilute concentrations of C below its solubility limit in
media A and B. At higher concentrations the molecules or particles of C may associate
into a separate phase either in media A or B, or at the A–B interface. Which of these
Ads
orb
ed
film
 of
 C
Negative
adsorption of C
Intermediate
case
Medium B
Medium B
Medium B
Medium B
Solid A
Solid A
c
c
c
c c
c
c
cc
cc c
c c
c
c
c
c
c
AC
BC
AB
(a)
(b) (c)
(d)
FIGURE 10.6 (a) Low concentration of solute molecules C in medium B (i.e., below saturation). (b) Wetting film: an
adsorbed film of C develops and grows in thickness as the concentration of C in B approaches saturation. This
corresponds to cos q > 1 in Eq. (10.19). (c) Unwetting: resulting from repulsion between C and A in medium B above
saturation. This corresponds to cos q<�1 in Eq. (10.19). (d) Partial wetting: intermediate case between the two above,
corresponds to 1 > cos q > �1.
202 INTERMOLECULAR AND SURFACE FORCES
happens depends on the relative magnitudes of A, B, and C. In this final section we shall
consider the factors favoring the formation of thick adsorbed liquid films on a solid
surface (the case of a liquid film or droplet adsorbed at a liquid-liquid or solid-liquid
interface is further considered in Chapter 17).
In Figure 10.6a we have, initially, a solid surface of A in contact with a binary liquid
mixture of solute molecules C in solvent B. Again there are three possibilities:
• For C intermediate between A and B, molecules C will be attracted to A, while mole-
cules B will be repelled from it, and an adsorbed monolayer or film of C will be
energetically favorable (Figure 10.6b). In this case, C is said to completely wet the
surface.
• For B intermediate between A and C (Figure 10.6c), the roles of B and C are reversed,
favoring adsorption of B and negative adsorption of C. In this case, C does not adsorb
or wet the surface (whereas B will wet the surface if C were the solvent). This is known
as nonwetting, unwetting or dewetting.
Chapter 10 • Unifying Concepts in Intermolecular and Interparticle Forces 203
• Finally, when A is intermediate, both molecules B and C are attracted to the solid
surface. Under such circumstances no uniformly adsorbed film of B or C will form, but
different regions of the interface will collect macroscopic droplets of the B or C phase
(Figure 10.6d). This is known as partial-wetting or incomplete wetting.
It is left as an exercise for the interested reader (see problem 10.5) to ascertain that when
the total surface energies of the whole system is minimized, the contact angle q formed by
these droplets (Figure 10.6d) is given by
cosq ¼ ðB þ C � 2AÞ=ðB � CÞ; (10.19)
which leads to values of cos q between 1 and �1 (q between 0� and 180�) only when A is
intermediate between B and C. Equation (10.19) may also be written in the forms
gAC þ gBC cos q ¼ gAB ðYoung equationÞ (10.20)
gBCð1þ cos qÞ ¼ DWABC ðYoung�Dupr�e equationÞ; (10.21)
where DWABC is the adhesion energy per unit areas of surfaces A and C adhering in
medium B. These important fundamental equations will be derived using a different
approach and discussed further in Chapter 17.
The purpose of the phenomenological discussion of this chapter is to illustrate how
a few basic notions concerning two-particle interaction energies can be applied to
progressively more complex situations, and vice versa—that is, how fairly complex
situations can arise from, and be understood in terms of, the simplest possible pair
potential, Eq. (10.2), as was illustrated in the Worked Example 10.2. However, this type of
nonspecific approach, while conceptually useful, has its limits in that it does not take into
account the way interaction energies vary with distance. Two particles or surfaces may
have an adhesive energy minimum at contact, but if the force law is not monotonic—it
may be repulsive before it becomes attractive closer in—the particles will remain sepa-
rated—that is, they effectively repel each other. In addition, as we have seen, certain
important interactions do not follow these simple rules. Only a quantitative analysis in
terms of the magnitudes of the operative forces and their distance dependence can
provide a full understanding of interparticle and interfacial phenomena.
PROBLEMS AND DISCUSSION TOPICS
10.1 Figures 10.1a and b show that in two dimensions the associated state (b) is ener-
getically more favorable than the dispersed state (a) by a factor of �9(A � B)2.
Calculate this energy in the case of three dimensions—that is, for a spherical cluster
of 12 molecules surrounding the central molecule. [Answer: DW ¼ �22(A � B)2.]
10.2 In the examples of Figure 10.1 a–c, all the molecules or particles have the same
radius, r or a. In Sections 5.1 and 6.1, it was shown that the polarizability of
204 INTERMOLECULAR AND SURFACE FORCES
a molecule is proportional to its volume—that is, to r3, which appears in the
numerator of the London equation for the van der Waals pair potential. Show that
for the more realistic case where W ¼ �AB/(rAþ rB)
6 and where A f r3A and Bf r3B,
the associated state is still the energetically favored one.
10.3 Show that for the geometries of Figures 10.1 and 10.2, n z R/a, which leads to the
following important approximate relation:
Adhesion or binding energy of particles z Binding energy of atoms� Radius of particles
Radius of atoms
:
(10.23)
For which systems or conditions does the preceding relation not hold?
10.4) An incompressible spherical particle has the same density and optical properties as
water at 30�C, and all its properties may be assumed to be independent of
temperature and pressure. It is suspended in a vertical column of water across
which there is a uniform temperature gradient. In one case the top is maintained at
50�C, the bottom at 10�C. In another, the temperatures are reversed. In which
directions do the following four forces act on the particle in the above two cases:
gravitational (buoyancy) forces, van der Waals forces, forces due to viscous flow,
and the forces due to Brownian motion—that is, osmotic pressure, thermal pres-
sure or the molecular collisions of the water molecules. If your answer is “zeroforce,” state whether the “equilibrium” is stable or unstable—that is, whether the
particle will return to the center if it is displaced from it by a small amount up or
down, or whether it will continue to move away from the center.
10.5) Derive Eq. (10.19) by finding the condition that minimizes the total surface energy
of the system. [Hint: Consider all the surface energies of a truncated sphere of
constant volume on a surface, including the curved and flat liquid areas, Ac and Af,
and the flat area As of the solid surface. You should find that the minimum energy
condition is dAc/dAf ¼ cos q , which leads to Eq. (10.19).]
10.6 Two immiscible liquids B and C are in contact with a solid surface A, where there is
a finite contact angle q between the B�C interface and A, as shown in Figure 10.6d.
Would you expect q to change as one approaches the critical point of the B�C
system, and if so how would this be seen visually?
10.7 What is the relative surface density of B and C molecules at the solid surface of
A in Figure 10.6(a) if B and C are miscible and present in equal amounts (a 50/50
mixture)? Under what conditions will the densities at the surface be the same
as in the bulk?
	Chapter 10 -
Unifying Concepts in Intermolecular and Interparticle Forces
	The Association of Like Molecules or Particles in a Medium
	Two Like Surfaces Coming Together in a Medium: Surface and Interfacial Energy
	The Association of Unlike Molecules, Particles, or Surfaces in a Third Medium
	Particle-Surface and Particle-Interface Interactions
	Engulfing and Ejection
	Adsorbed Surface Films: Wetting and Nonwetting
	Problems and Discussion Topics