MATERIALES Y MÉTODOS
PROGRAMAS INFORMÁTICOS
Intermolecular interactions are primarily the result of charge interactions between atoms or small molecules. At the atomic and molecular scale, intermolecular bonds are weak, generally on the order of 4-400 kJ/mol (see Table 2.1). The emphasis in this section is placed on van der Waals and electrostatic interactions. Simple theoretical potentials known for these interactions have been validated extensively by experiment. The resulting forces are long-range and play an important role in room temperature adhesion of microscopic and macroscopic bodies. Repulsive exclusion forces that occur between atoms and molecules at close separations are studied by introducing the Lennard-Jones pair-potential, which adds a repulsive term to the attractive van der Waals potential.
2.2.1. Van der Waals forces
Van der Waals interactions generically refer to dipole interactions that result between both polar and nonpolar molecules. In the later case, the interactions are generally referred to as induced dipole-dipole interactions, which result from polarization by the electrons and nuclei of neighboring molecules. These interactions are long-range and are always present. Attempts to quantify van der Waals interactions dates back to the early 1920’s with the Keesom and Debye Theories (Debye, 1921; Keesom, 1921; Israelachvilli, 1985; Maugis, 2000). Keesom treated the polar molecules as dipoles and calculates the dipole-dipole interaction energy. Debye took the derivation one step further by considering the interaction between a dipole and a non-polar molecule that is polarized by the neighboring dipole. With the introduction of quantum mechanics came an understanding of electron fluctuations, which prompted the London theory (Kallmann and London, 1929; London, 1930; Maugis, 2000). In London’s theory, nonpolar molecules become polarized by the fluctuating electron clouds of neighboring atoms or molecules and, therefore, no permanent dipoles are needed to explain the resulting interactions. All three models predict a potential energy:
6 C U
r
= − , (2.1)
where the constant C>0 depends on the model. The energy (2.1) describes dipole-dipole
(Keesom energy), dipole-nonpolar (Debye energy), and nonpolar-nonpolar (London dispersion energy) interactions (Israelachvilli, 1985).
Only the London dispersion forces act between nonpolar molecules. For London’s theory, the constant C (London constant) is given in terms of the permittivity of free
space
ε
o, the absorption frequency of the molecules ν, the polarizability of the moleculesα, and Plank’s constant h=6.625 10× −34 Js:
(
)
A B A B 2 A B o 3 1 2 (4 ) h Cα α
ν ν
ν
ν
πε
= + , (2.2)where subscripts A and B distinguish between the two interacting molecules. The
magnitude of the London constant is typically on the order of 10−79J m⋅ 6. In 1948 Casimir and Polder modified London’s expression to account for the time delay caused by the electrostatic interactions traveling between the molecules (Casimir and Polder, 1948). London’s equation was found to hold only for r50 nm and correction factor of
1r is required for large distances (r>500 nm
) due to retardation of the force (Maugis,
2000).
2.2.2. The Lennard-Jones potential
Quantum mechanics explains the repulsive force between two atoms at very small separation and predicts repulsion has an exponential dependence on the separation, proportional to exp
(
r ro)
. For mathematical convenience, the repulsive interactions areoften approximated as 1 n
r , with n>10. The Lennard-Jones potential combines the attractive van der Waals interactions with a repulsive term that dominates at close separations: 12 6 LJ 12 6 4 o B C a a U U r r r r = − = − , (2.3)
where C is London’s constant from (2.1) and B is a proportionality constant. The
potential energy minimum −Uo at the equilibrium separation r ao =1.12. The interaction force is given from differentiation of (2.3) with respect to its’ thermodynamic conjugate r: 13 7 LJ LJ m d 10.01 2 d U a a F F r r r = = − − , (2.4)
where the maximum force Fmax =2.39Uo rooccurs at a separation rm =1.11ro. The potential energy (2.3) and force (2.4) between two Lennard-Jones molecules is plotted as a function of their separation in Fig. 2.1a. In this figure, and throughout the discussion, negative force values indicate repulsive interactions. Although Lennard-Jones interactions only describe interactions between noble gasses, it is feverously applied to other systems because of its simplicity and because it captures the correct generic behavior between single molecule pairs. As demonstrated in Fig. 2.1a, the interactions are repulsive at close separations, attractive at moderate separations, and quickly decay to zero at far separations. The maximum force coincides with the inflection point of the adhesive energy curve, whereas the equilibrium separation corresponds to the potential energy minimum.
2.2.3. Electrostatic Coulomb interactions
Forces due to static point charges are described by Coulomb’s law. The interaction force Fq and potential energy Uq for the two charges qA and qB are expressed in terms
of the separation r, the dielectric constant
ε
of the surroundings, and the permittivity ofA B q 2 o 4 q q F r
πεε
= − , (2.5) A B q o 4 q q U r πεε = . (2.6)The tractions are attractive if charges qA and qB are of opposite sign, which is indicated
by a positive force values. The potential energy (2.6) and force (2.5) between two identical point charges are plotted as a function of their separation in Fig. 2.1b. The force becomes strongly repulsive at close separations and decays to zero at far separations.
2.2.4. Hydrogen bonds
Hydrogen bonds are the result of interactions between polar molecules that contain hydrogen covalently bonded to other electronegative atoms such as oxygen, nitrogen, or fluorine (Israelachvilli, 1985). Unequal sharing of electrons results in a positively polarized hydrogen, which easily interacts with neighboring molecules due to its small size. These interactions are responsible for the unique properties of water. Despite their complexity, the interaction energy of hydrogen bonds is approximately given by (Israelachvilli, 1985):
2
U = −k r , (2.7)
where k is a constant for proportionality in units J m⋅ 2.