Intermolecular Forces

1. Intermolecular Forces At the level of molecular mechanism, biological structure and function are determined byintermolecular forces that are electrostatic in origin and which can be adequately described byclassical methods1. Electrostatic forces also lie at the heart of the barrier function of membranesas well as the mechanisms whereby this barrier is overcome. The principles involved can beunderstood using a few examples based on simple or symmetric charge configurations. Forthese, the electrostatic interaction energies can be derived quite easily, and it is found that avariety of distance dependences arise. 1Hydrogen bonding requires some quantum mechanical corrections for precise representation, and Londondispersion forces are deeply quantum mechanical in origin, but both are very adequately described by classicalmodels for chemical and biological application.

2. Interactions between discrete charges are given by Coulomb’s Law, which describes the 1/r2dependence for the force between point charges. The energy (U(r) = - ∫F(r)·dr)of a point chargein 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 repulsiveconfigurations, changing both the strength of interaction, which decreases, and the distancedependence, 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 fieldof another dipole, the interaction falls off even more steeply:

3. Pairwise electrostatic interaction energies

5. The dipole forms of electrostatic interaction are especially important in determining the structure and properties of macromolecules. Although the short range interactions listed here areoften described as being "weak", in many circumstances they are the only ones present and canbe very effective. Furthermore, the characterization as weak and short-range comes from theirbehavior 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. Allhave 1/r6 distance dependence and are known collectively as van der Waals forces. The finalone (#6), the London dispersion force, is of particular interest and importance in understandingthe behavior of non-ionic compounds in the liquid phase, and it is the dominant force in apolarand weakly polar materials. We know, forexample, that even totally symmetrical, unchargedmolecules 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 notsolidify 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.

6. 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 electrondistribution can give rise to a small, transient - or instantaneous - dipole. The fluctuations are ofquantum mechanical origin, and are independent of temperature (note that kBT is lacking in allexpressions 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 nearbyatoms, in proportion to their polarizability. The induced dipoles are always complementary tothe instantaneous dipole, regardless of what orientation the latter might have momentarily, andthe dipole fluctuations are strongly correlated. Thus, the interaction between them -instantaneous dipole-induced dipole - is always attractive, in vacuo:

7. This is the London dispersion force. Since the actual interaction is between rapidly fluctuatingdipoles, the distance dependence of the energy is 1/r6, as for two freely rotating, permanentdipoles (but without the kBT factor). The modern theory of dispersion forces, based on the workof Lifshitz, is complex, but it reveals very explicitly the role of electronic fluctuations over thewhole range of frequencies. In principle, therefore, the force is derivable directly andquantitatively 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 ofas very short range, on account of their elementary 1/r6 dependence. The dispersion force is byfar the most important, in a general sense, because it is the only one that is universal and, exceptfor very polar molecules, such as water, it is actually stronger than the rotating dipole-dipoleinteraction (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 dispersionforce is generally small compared to these other electrostatic forces. Although the dispersion force is always attractive between similar molecules, in solution itcan actually be repulsive between dissimilar solutes. Also, for large structures in condensedmedium, 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 macromoleculesand membranes in solution. 2 The other two are the Keesom force - the orientation force between two freely mobile dipoles - and the Debyeforce - the induction force between a freely rotating dipole and an apolar (but polarizable) species.

8. The effect of the medium on electrostatic interactions A dielectric medium between two charges or charge distributions will decrease the pairwiseinteractions by a factor ε or ε2, depending on the nature of the interaction. However, when weconsider the effective force of attraction (or repulsion) between neutral, polar molecules, wemust recognize that two particles will only "notice" each other if they are distinguishable fromthe solvent. Thus, for non-ionic interactions, the determining property is the excesspolarizability of the solutes over the solvent. This makes the effective pairwise interactionseven 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 stillrecognizable. In a medium of dielectric constant ε, with an applied electric field, E, such asphere will be polarized to acquire a dipole moment:

9. 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 dielectricconstant of the sphere, i.e., εi > 1. Indeed, for a sphere of high dielectric (εi» 1), in vacuo, thepolarizability is αi≈ 4πε0ai3 = 3ε0vi. This is readilyderivable for a simple one-electron Bohr atom. However, if ε > εi, as might occur in acondensed medium (readily in water, where ε = 80), the polarizability is negative, implying thatthe direction of the induced dipole is opposite to that produced in free space! This can beunderstood in terms of the highly polarizable medium responding to the electric field to such anextent as to produce an opposite local field around the molecule of interest. Such unexpected effects of the medium are encountered in all interactions of electrostaticorigin, 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 isproportional to α1.α2 = α2, and even two microscopic air bubbles will attract each other in aliquid. 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 isan unlikely situation in aqueous medium, and in any case it is easily over-ridden by stronger, oriented net dipoleeffects.

10. 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 asthe 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 knownthat an attractive force arises as if there were an equal but opposite charge situated equidistantbehind the plane (so the separation is 2d). This is the "image charge", and the net force isproportional to -Q2/(2d)2. Because a conductor supports no potential differences, the electric field lines must be normal to the conductingplane. Consequently, the charge Q feels a force, apparently from an image charge behind the conducting surface.

11. For a charge, Q, in a medium of dielectric constant ε1, at a distance d from the planar surfaceof 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 theion is effectively pulled out of low dielectric medium by the attraction of the image charge ofopposite sign. This is consistent with our expectation that an ion will be less happy in a mediumof 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 theimage charge is the same as that of the charge itself! The effect is to repel the charge away fromthe interface. The interaction energy is equivalent to the energetic barrier for an ion to enter aregion of low dielectric.

12. Although the final result in the latter case (ε1 > ε2) is consistent with our intuition about charges anddielectrics, the idea that a charge can induce a charge of the same sign (or that a dipole caninduce 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 replacedby medium of lower dielectric - a net force acts on the charge in a direction normal to and away from the cuttingplane (F > F’). If the piece removed is substituted by medium of greater polarizability (higher ε), the force will acttoward the border (F < F’). (From B. L. Silver, 1985.)

13. Forces and interaction energies between two atoms, in vacuo

14. Repulsive contact forces It is well known that liquids are not very compressible, which implies that a repulsive forcequickly comes into play when atoms approach closer than their normal separation in the liquidphase. 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), inthe gas phase, has an attractive term, proportional to 1/r6, which is well-founded in the Londondispersion force. However, the repulsive term is not well developed theoretically. It is oftenmodeled 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 particularchoice 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 ofpotential function are still widely used, even though an exponential dependence is known to fit the repulsive termmuch better, because they can be factored.

15. Van der Waals interactions between large structures The familiar characterization of van der Waals forces as weak and short range, derived fromthe pairwise energy of interaction in vacuo, breaks down completely when one considers theinteraction between large structures (such as macromolecules and membranes) in a dielectricmedium. A complete, modern theoretical treatment is based on classical continuum phaseproperties rather than quantum mechanics, but is mathematically complex, and a very adequatefirst order approach can be made by assuming pairwise additivity for all the interactions in thetwo bodies. This is fundamentally incorrect, as van der Waals forces are not additive6, but theerror 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 directlyand indirectly, after reflection from other atoms, which also see it. It is therefore impossible for all dipoles to bealigned in a mutually energy-minimized way and the ensemble of dipoles is inherently “frustrated”. Second, theinteraction is propagated by an electromagnetic wave traveling at the finite speed of light. In order for theinstantaneous dipole and the induced dipole to be positively correlated, the time for a round-trip between them, atspeed c, must be short compared to the frequency of the dipole fluctuations. The electronic fluctuations occur overthe whole spectrum of frequencies, from vibrational to electronic, and the relationship between the force and theoptical frequency spectrum gives rise to the term "dispersion". At long distances, the "cause and effect" fall out ofphase. The interaction is then said to be retarded and the interaction in vacuo exhibits a 1/r7 dependence, ratherthan 1/r6.

16. Starting with a pairwise interaction energy of the form, U(r) = -C/r6, we can integrate theenergies 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 weakerdistance 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 aproportionality constant called the Hamaker coefficient7: A = π2Cρ1ρ2 where C is the coefficient in the pairwise atom-atom potential. For a variety of interactionsinvolving large bodies, the distance dependence is remarkably small - often in the range of 1/r to1/r2. For two plane surfaces (membranes), the dependence is 1/r2 for distances up to a few tensof Å. At greater separation it falls off faster than this, but it remains a potent force, and can havelong range effects to beyond 100 Å. Typical values of the Hamaker coefficient between large structures are on the order of 10-19 Jfor interactions across a vacuum, and this is not very sensitive to the molecular identity of thematerial, only its (electron) density. In a dielectric medium, the absolute magnitude of theconstant is smaller, but the same considerations apply as discussed above, and the net interactioncan be attractive, neutral or repulsive, depending on the dielectric constants of the medium andthe two bodies. However, as noted above, the more bizarre behavior requires uncommoncircumstances. 7The term Hamaker coefficient is better than Hamaker constant as it varies with distance - the non-additivity andretardation are incorporated into it.

17. An important attribute of the London dispersion force, in solution, is that it is not effectivelyscreened by salts8. This is because the electronic fluctuations that underly it are at much higherfrequencies than can be followed by ionic redistributions. Thus, although charge-charge effects are intrinsically much stronger than van der Waals interactions, under physiological conditionswith ionic strengths of 0.1-0.5 M, the former are often almost obliterated, leaving the van derWaals forces relatively strong, and sometimes even dominant. 8See Gouy-Chapman theory for surface charges and potentials.

18. Van der Waals forces between surfacesNon-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 ρ1and 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.