Th e fact that liquids and solids exist at all means that there must exist forces that bind molecules together under some conditions so that individual molecules do not simply evaporate into the gas phase. On the other hand, we know that it is extremely hard (taking considerable energy) to compress solids and liquids so as to reduce their volume. Th is implies that as we try to push atoms and mol- ecules even closer together a force acts to keep them apart. Th us, we conceive a model of intermolecular forces between two molecules that are highly repulsive at small intermolecular distances but attractive at longer distances. In this sec- tion we develop this concept to explore the origins of these forces, how they are modeled, and some other direct demonstrations of their existence.
1.4.1 What Is the Intermolecular Potential Energy?
Consider fi rst the interaction of two spherical neutral atoms a and b. Th e total energy Etot(r) of the pair of atoms at a separation r is written as
tot( ) a b ( ).
E r = E +E +φ r (1.32)
Here, Ea and Eb are the energies of the isolated atoms, and φ(r) is the contribu- tion to the total energy arising from interactions between them. We call φ(r) the intermolecular pair-potential energy function and, in the present example it depends only on the separation of the two atoms. Since this energy is equal to the work done in bringing the two atoms from infi nite separation to the separation
r, it is given in terms of the intermolecular force F(r) by φ( ) =
∫
∞ ( ) d .r
r F r r (1.33)
By convention, the force F is positive when repulsive and negative when attractive.
Th e general forms of φ(r) and F(r) are illustrated in Figure 1.1 (Maitland et al. 1981). We see as foreshadowed above that, at short range, a strong repulsion acts between the molecules while, at longer range, there is an attractive force, which decays to zero as r → ∞. Consequently, the potential energy φ(r) is large and positive at small separations but is negative at longer range. It is known that, for neutral atoms at least, there is only one minimum and no maximum in either F(r) or φ(r). Th e parameters σ, r0, and ε usually employed to characterize the intermolecular pair-potential energy are defi ned in Figure 1.1. σ is the sep- aration at which the potential energy crosses zero, r0 is the separation at which
φ(r) is minimum, and –ε is the minimum energy.
For molecules that are not spherically symmetric the situation is more complex because the force between the molecules, or equivalently the
0 0 0 r0 r –ε σ F(r) φ (r )
Figure 1.1 Th e intermolecular pair-potential energy φ(r) and force F(r) as a function
intermolecular potential energy, depends not just upon the separation of the center of the molecules but also upon the orientation of the two molecules with respect to each other. Th us, the intermolecular potential is not spheric- ally symmetric. We shall consider this in a little more detail later.
In general, the potential energy U of a cluster of molecules is a function of the intermolecular interactions, which in turn depend upon the type and num- ber of molecules under consideration, the separation between each molecule, and their mutual orientation. Th e term confi guration is used to defi ne the set of coordinates that describe the relative position and orientation of the molecules in a cluster.
To estimate the potential energy of a confi guration it is usual, and often nec- essary, to make some or all of the following simplifi cations:
1. Th e term intermolecular pair-potential energy is used to describe the potential energy involved in the interaction of an isolated pair of mol- ecules. It is very convenient to express the total potential energy U of a cluster of molecules in terms of this pair potential φ. Th is leads to a very important assumption, the pair-additivity approximation, according to which the total potential energy of a system of molecules is equal to the summation of all possible pair interaction energies. Th is implies that the interaction between a pair of molecules is unaff ected by the proximity of other molecules.
2. Th e second important assumption is that the pair-potential energy depends only on the separation of the two molecules. As we have argued, this assumption is valid only for monatomic species where, owing to the spherical symmetry, the centers of molecular interaction coincide with the centers of mass.
3. Finally, since the intermolecular potential is known accurately for only a few simple systems, model functions need to be adopted in most cases. Typically, such models give U as a function only of the separ- ation between molecules but nevertheless the main qualitative fea- tures of molecular interactions are incorporated.
For a system of N spherical molecules, the general form of the potential energy
U may be written as φ φ − = + =
∑ ∑
+ ∆ … 1 2 ij 1 i 1 j=i 1 ( , , , N) , N N NU
r r r (1.34)where φij is the potential energy of the isolated pair of molecules i and j, and ΔφN
and above the strictly pairwise additive interactions. According to the pair- additivity approximation, this reduces to
1 2 ij ij i<j 1 i 1 j=i 1 ( , , , N) . N N
U
φ φ − = + =∑ ∑ ∑
= r r … r (1.35)Th e approximation of Equation 1.35 implies that the N-body interactions (with
N > 2) are negligible compared with the pairwise interactions. In fact, many-
body forces are known to make a small but signifi cant contribution to the total potential energy when N ≥ 3 and, for systems at higher density, the pair-addi- tivity approximation can lead to signifi cant errors. However, it is often possible to employ an eff ective pair potential that gives satisfactory results for the dense fl uid while still providing a reasonable description of dilute-gas properties.
1.4.2 What Is the Origin of Intermolecular Forces?
Intermolecular forces are known to have an electromagnetic origin (Maitland et al. 1981) and the main contributions are well established. Th e strong repul- sion that arises at small separations is associated with overlap of the electron clouds. When this happens, there is a reduction in the electron density in the overlap region leaving the positively charged nuclei incompletely shielded from each other. Th e resulting electrostatic repulsion is referred to as an over-
lap force. At greater separations, where attractive forces predominate, there is
little overlap of electron clouds and the interaction arises in a diff erent man- ner. Here, the attractive forces are associated with electrostatic interactions between the essentially undistorted charge distributions that exist in the mol- ecules; for a more detailed description the reader is referred to the specialized literature (Maitland et al. 1981).
Th ere are in fact three distinct contributions to the attractive forces that will be discussed here only briefl y; for a more detailed description the reader is referred to a specialized literature (Maitland et al. 1981). For polar molecules, such as HCl, the charge distribution in each molecule gives rise to a permanent electric dipole and, when two such molecules are close, there is an electrostatic
force between them that depends upon both separation and orientation. Th e force between any two molecules may be either positive or negative, depending upon the mutual orientation of the dipoles, but the averaged net eff ect on the bulk properties of the fl uid is that of an attractive force.
Such electrostatic interactions are not associated exclusively with dipole moments. Molecules such as CO2, which have no dipole moment but a quad- rupole moment, also have electrostatic interactions of a similar nature. Th ese interactions exist in general when both molecules have one or more nonzero multipole moments.
Th ere is a second contribution to the attractive force that exists when at least one of the two molecules possesses a permanent multipole moment. Th is is known as the induction force and it arises from the fact that molecules are polarizable; so that a multipole moment is induced in a molecule when it is placed in any electric fi eld including that of another molecule. Th us, a perman- ent dipole moment in one molecule will induce a dipole moment in an adjacent molecule. Th e permanent and induced moments interact to give a force that is always attractive and, at long range, proportional to r –6.
Th e third contribution to the attractive force, and the only one present when both molecules are nonpolar, is known as the dispersion force. Th is arises from the fact that even nonpolar molecules generate fl uctuating elec- tric fi elds associated with the motion of the electrons. Th ese fl uctuating fi elds around one molecule give rise to an induced dipole moment in a second nearby molecule and a corresponding energy of interaction. Like induction forces, dispersion forces are always attractive and, at long range, vary like r –6 to leading order.
1.4.3 What Are Model Pair Potentials and Why Do We Need Them?
Th e diffi culties encountered in the evaluation of the intermolecular pair- potential energy from an ab initio basis have led to the adoption of the fol- lowing heuristic approach. We use the spherically symmetric potential as an example. Th e evaluation procedure starts with the assumption of an analyt- ical form for the relationship between the potential energy φ and the distance
r between molecules. Subsequently, macroscopic properties are calculated
using the appropriate molecular theory. Comparisons between calculated and experimental values of these macroscopic properties provide a basis for the determination of the parameters in the assumed intermolecular potential- energy function. Finally, predictions may be made of thermodynamic proper- ties of the fl uid in regions where experimental information is unavailable.
In the following sections, we present some of the most widely used model potential-energy functions. For a more comprehensive discussion the reader is referred to specialized literature (Maitland et al. 1981).
1.4.3.1 What Is a Hard-Sphere Potential?
In this model, the molecules are assumed to behave as smooth, elastic, hard spheres of diameter σ. It is apparent that the minimum possible distance between the molecules is then equal to σ and that the energy needed to bring
the molecules closer together than r = σ is infi nite as shown in Figure 1.2. For separation r > σ, there is no interaction between the molecules. Th e mathem- atical form of the potential is given by the following discontinuous function