Download Intermolecular Forces

Intermolecular Forces
At the level of molecular mechanism, biological structure and function
are determined by intermolecular forces that are electrostatic in origin
and which can be adequately described by classical methods1.
Electrostatic forces also lie at the heart of the barrier function of
membranes as well as the mechanisms whereby this barrier is
overcome. The principles involved can be understood using a few
examples based on simple or symmetric charge configurations. For
these, the electrostatic interaction energies can be derived quite easily,
and it is found that a variety of distance dependences arise.
1
Hydrogen bonding requires some quantum mechanical corrections for precise
representation, and London dispersion forces are deeply quantum mechanical in origin, but
both are very adequately described by classical models for chemical and biological
application.
Interactions between discrete charges are given by Coulomb’s Law,
which describes the 1/r2 dependence for the force between point charges.
The energy (U(r) = - ∫F(r)·dr) of a point charge in the electric field of
another point charge is the Coulomb energy, and is proportional to 1/r.
The energy of a dipole in the field of a point charge varies as 1/r2 (and
the force goes as 1/r3).
When free rotation of the dipole is allowed, the interaction samples both
attractive and repulsive configurations, changing both the strength of
interaction, which decreases, and the distance dependence, which now
goes as 1/r4. Dipole energies fall off more quickly than for a net charge,
especially when the dipole is free to rotate, making it a more short-range
interaction. In the field of another dipole, the interaction falls off even
more steeply:
Pairwise electrostatic interaction energies
The dipole forms of electrostatic interaction are especially important in determining the
structure and properties of macromolecules. Although the short range interactions listed
here are often described as being "weak", in many circumstances they are the only ones
present and can be very effective. Furthermore, the characterization as weak and shortrange comes from their behavior in vacuo, and in condensed media they can be
strikingly different.
The last three listed above (#4-6) describe the interactions of mobile, uncharged
species. All have 1/r6 distance dependence and are known collectively as van der Waals
forces. The final one (#6), the London dispersion force, is of particular interest and
importance in understanding the behavior of non-ionic compounds in the liquid phase,
and it is the dominant force in apolar and weakly polar materials. We know, for
example, that even totally symmetrical, uncharged molecules and atoms can condense
to form liquids and, usually, solids (helium is an exception). The forces holding them
together may be weak but they cannot be negligible or they would not solidify except at
0 K. Furthermore, they cannot be based on charge or fixed dipole interactions
because these species do not have net charge or fixed dipole moments.
What they do have, however - in common with all atoms - is a dynamic electron
distribution. On average they have no dipole, but at any instant in time a fluctuation in
the electron distribution can give rise to a small, transient - or instantaneous - dipole.
The fluctuations are of quantum mechanical origin, and are independent of temperature
(note that kBT is lacking in all expressions involving a “non-polar” species ). A dipole
generates an electric field around itself (propagated at the speed of light) and this will
induce complementary dipoles in any nearby atoms, in proportion to their polarizability.
The induced dipoles are always complementary to the instantaneous dipole, regardless
of what orientation the latter might have momentarily, and the dipole fluctuations are
strongly correlated. Thus, the interaction between them - instantaneous dipole-induced
dipole - is always attractive, in vacuo:
This is the London dispersion force. Since the actual interaction is between rapidly
fluctuating dipoles, the distance dependence of the energy is 1/r6, as for two freely
rotating, permanent dipoles (but without the kBT factor). The modern theory of
dispersion forces, based on the work of Lifshitz, is complex, but it reveals very
explicitly the role of electronic fluctuations over the whole range of frequencies. In
principle, therefore, the force is derivable directly and quantitatively from the
electromagnetic absorption spectrum of the atomic/molecular system.
In vacuo, all three contributions2 to the van der Waals force are attractive and are
thought of as very short range, on account of their elementary 1/r6 dependence. The
dispersion force is by far the most important, in a general sense, because it is the only
one that is universal and, except for very polar molecules, such as water, it is actually
stronger than the rotating dipole-dipole interaction (especially in liquid media), because
of the inherent correlation of the fluctuations. However, in situations involving net
charge or fixed dipoles, the magnitude of the dispersion force is generally small
compared to these other electrostatic forces.
Although the dispersion force is always attractive between similar molecules, in
solution it can actually be repulsive between dissimilar solutes. Also, for large
structures in condensed medium, it is not necessarily short range and can be significant
at distances up to 100 Å or more. It is of particular importance - and of non-intuitive
behavior - in the context of macromolecules and membranes in solution.
2
The other two are the Keesom force - the orientation force between two freely mobile dipoles - and the
Debye force - the induction force between a freely rotating dipole and an apolar (but polarizable) species.
The effect of the medium on
electrostatic interactions
A dielectric medium between two charges or charge distributions will decrease the
pairwise interactions by a factor ε or ε2, depending on the nature of the interaction.
However, when we consider the effective force of attraction (or repulsion) between
neutral, polar molecules, we must recognize that two particles will only "notice" each
other if they are distinguishable from the solvent. Thus, for non-ionic interactions, the
determining property is the excess polarizability of the solutes over the solvent. This
makes the effective pairwise interactions even smaller. What remains, however, is
sometimes counter-intuitive.
Analysis of medium effects is extremely difficult at the microscopic level, but can be
approached by continuum methods in which a molecule, i, is modeled as a sphere of
radius, ai. In a vacuum, the polarizability of a spherical atom is given by 4πε0ai3. For a
spherical molecule, one must assign a dielectric constant, εi, which complicates things
considerably. However, the form of the expression is still recognizable. In a medium of
dielectric constant ε, with an applied electric field, E, such a sphere will be polarized to
acquire a dipole moment:
Thus, in a dielectric medium, the effective or excess polarizability is:
where vi = 4/3πai3 is the volume of the sphere or molecule. In vacuo (ε = 1), this is the
Clausius-Mossotti equation and it yields a normal polarizability for any physically reasonable
dielectric constant of the sphere, i.e., εi > 1. Indeed, for a sphere of high dielectric (εi » 1), in
vacuo, the polarizability is αi ≈ 4πε0ai3 = 3ε0vi. This is readily derivable for a simple oneelectron Bohr atom. However, if ε > εi, as might occur in a condensed medium (readily in
water, where ε = 80), the polarizability is negative, implying that the direction of the induced
dipole is opposite to that produced in free space! This can be understood in terms of the highly
polarizable medium responding to the electric field to such an extent as to produce an opposite
local field around the molecule of interest.
Such unexpected effects of the medium are encountered in all interactions of electrostatic origin,
except the charge-charge Coulomb interaction. For two identical molecules (spheres), with
ε1 = ε2, in a dielectric medium, the net interaction is always attractive, since it is proportional to
α1.α2 = α2, and even two microscopic air bubbles will attract each other in a liquid. However, for
two different solute species, with ε1 > ε > ε2, the net force is repulsive!4
4This counter-intuitive behavior is true for any of the van der Waals interactions, and is
interestingly weird, but it is an unlikely situation in aqueous medium, and in any case it is
easily over-ridden by stronger, oriented net dipole effects.
Image forces and image charges
A closely related effect is evident in the approach of an ion or permanent dipole to the
interface between two phases of different polarizabilities (different dielectric constant),
such as the water-membrane interface. This can be described in terms of image
charges. For a charge, Q, in vacuum, at a distance d from an infinite, plane (flat),
conducting surface, it is well known that an attractive force arises as if there were an
equal but opposite charge situated equidistant behind the plane (so the separation is 2d).
This is the "image charge", and the net force is proportional to -Q2/(2d)2.
Because a conductor supports no potential
differences, the electric field lines must be
normal to the conducting plane.
Consequently, the charge Q feels a force,
apparently from an image charge behind
the conducting surface.
For a charge, Q, in a medium of dielectric constant ε1, at a distance d
from the planar surface of a medium of dielectric constant ε2, the image
charge has the value:
r can be + or For ε1 < ε2 (i.e., the charge is in the medium of lower dielectric), the r
factor is negative, and the ion is effectively pulled out of low dielectric
medium by the attraction of the image charge of opposite sign. This is
consistent with our expectation that an ion will be less happy in a
medium of low dielectric than in one of high dielectric (see Born energy,
later).
For ε1 > ε2 (the charge is in the medium of higher dielectric), r is
positive, and the sign of the image charge is the same as that of the
charge itself! The effect is to repel the charge away from the interface.
The interaction energy is equivalent to the energetic barrier for an ion to
enter a region of low dielectric.
Although the final result in the latter case (ε1 > ε2 ) is consistent with our
intuition about charges and dielectrics, the idea that a charge can induce a
charge of the same sign (or that a dipole can induce a repulsive dipole) is
not so intuitive, and might even be thought to violate some physical
law! However, it can be understood, at a microscopic level, in terms of
the polarization (reaction field) around an ion:
A charge polarizes the surrounding medium by polarizing and orienting neighboring
molecules (the reaction field). In an infinite isotropic medium, there is no net force on
the charge. If part of the medium is removed – or replaced by medium of lower
dielectric - a net force acts on the charge in a direction normal to and away from the
cutting plane (F > F’). If the piece removed is substituted by medium of greater
polarizability (higher ε), the force will act toward the border (F < F’). (From B. L.
Silver, 1985.)
Forces and interaction energies
between two atoms, in vacuo
Repulsive contact forces
It is well known that liquids are not very compressible, which implies
that a repulsive force quickly comes into play when atoms approach
closer than their normal separation in the liquid phase. At these close
distances, electron-electron repulsion (which is both electrostatic and
quantum mechanical in nature) becomes dominant.
The distance dependence of the energy of a pair of atoms, such as argon
(Ar, see Figure), in the gas phase, has an attractive term, proportional to
1/r6, which is well-founded in the London dispersion force. However, the
repulsive term is not well developed theoretically. It is often modeled by
an empirical expression with a very steep distance dependence such as
1/r12, although it is well known that an exponential form fits the data
much better. This particular choice yields the so-called Lennard-Jones, or
6-12, potential5:
5
This is a particular case of the more general Morse potential function, with terms in 1/rm and
1/rn. These types of potential function are still widely used, even though an exponential
dependence is known to fit the repulsive term much better, because they can be factored.
Van der Waals interactions between large structures
The familiar characterization of van der Waals forces as weak and short range, derived
from the pairwise energy of interaction in vacuo, breaks down completely when one
considers the interaction between large structures (such as macromolecules and
membranes) in a dielectric medium. A complete, modern theoretical treatment is based
on classical continuum phase properties rather than quantum mechanics, but is
mathematically complex, and a very adequate first order approach can be made by
assuming pairwise additivity for all the interactions in the two bodies. This is
fundamentally incorrect, as van der Waals forces are not additive6, but the error is
usually not more than a few percent.
6
The non-additivity arises from two sources. First, an instantaneous dipole is "seen" by a second
atom both directly and indirectly, after reflection from other atoms, which also see it. It is
therefore impossible for all dipoles to be aligned in a mutually energy-minimized way and the
ensemble of dipoles is inherently “frustrated”. Second, the interaction is propagated by an
electromagnetic wave traveling at the finite speed of light. In order for the instantaneous dipole
and the induced dipole to be positively correlated, the time for a round-trip between them, at
speed c, must be short compared to the frequency of the dipole fluctuations. The electronic
fluctuations occur over the whole spectrum of frequencies, from vibrational to electronic, and the
relationship between the force and the optical frequency spectrum gives rise to the term
"dispersion". At long distances, the "cause and effect" fall out of phase. The interaction is then
said to be retarded and the interaction in vacuo exhibits a 1/r7 dependence, rather than 1/r6.
Starting with a pairwise interaction energy of the form, U(r) = -C/r6, we can integrate
the energies of interaction for all the atoms in one body with all those in the other, and
arrive at a 'two body' potential for several simple systems. All resulting forms exhibit
much weaker distance dependence (and therefore are more long range), and are
proportional to the densities, ρ1 and ρ2 (atoms per unit volume), of the two surfaces or
bodies. These are collected into a proportionality constant called the Hamaker
coefficient7:
A = π2Cρ1ρ2
where C is the coefficient in the pairwise atom-atom potential. For a variety of
interactions involving large bodies, the distance dependence is remarkably small - often
in the range of 1/r to 1/r2. For two plane surfaces (membranes), the dependence is 1/r2
for distances up to a few tens of Å. At greater separation it falls off faster than this, but
it remains a potent force, and can have long range effects to beyond 100 Å.
Typical values of the Hamaker coefficient between large structures are on the order of
10-19 J for interactions across a vacuum, and this is not very sensitive to the molecular
identity of the material, only its (electron) density. In a dielectric medium, the absolute
magnitude of the constant is smaller, but the same considerations apply as discussed
above, and the net interaction can be attractive, neutral or repulsive, depending on the
dielectric constants of the medium and the two bodies. However, as noted above, the
more bizarre behavior requires uncommon circumstances.
7 The term Hamaker coefficient is better than Hamaker constant as it varies with
distance - the non-additivity and retardation are incorporated into it.
An important attribute of the London dispersion force, in solution, is that
it is not effectively screened by salts8. This is because the electronic
fluctuations that underly it are at much higher frequencies than can be
followed by ionic redistributions. Thus, although charge-charge effects
are intrinsically much stronger than van der Waals interactions, under
physiological conditions with ionic strengths of 0.1-0.5 M, the former are
often almost obliterated, leaving the van der Waals forces relatively
strong, and sometimes even dominant.
8
See Gouy-Chapman theory for surface charges and potentials.
Van der Waals forces
between surfaces
Non-retarded van der Waals
interaction free energies between
bodies of different geometries
calculated on the basis of pairwise
additivity (Hamaker summation
method).
The Hamaker coefficient is defined
as A = π2Cρ1ρ2 where ρ1 and ρ2 are
the number of atoms per unit
volume in the two bodies and C is
the coefficient in the atom-atom
pair potential (see above).
The forces are obtained by
differentiating the energies with
respect to distance.