Waals Interactions Between Macroscopic Bodies

As we saw in Chapter 2, van der Waals forces consist only of long-range forces the interaction pair potential, V(r), decreases with the inverse sixth power of the distance between molecules, r-6 and the corresponding interaction force, F(r), decreases as r 7. When particle-particle or particle-surface attractions are considered, polar Keesom and Debye 

---

Lifshitz (1955-60) developed a complete quantum electrodynamic (continuum) theory for the van der Waals interaction between macroscopic bodies. 

In colloid and surface science we are interested in calculating the van der Waals interaction between macroscopic bodies, such as spherical particles and planar surfaces. If the dispersion interaction, for example, were additive we could sum the total interaction between every molecule in a body with that in another. Thus, if the separation distance between any two molecules i and f in a system is... 

Though the original work is difficult to understand very good reviews about the van der Waals interaction between macroscopic bodies have appeared [114,120], In the macroscopic treatment the molecular polarizability and the ionization frequency are replaced by the static and frequency dependent dielectric permittivity. The Hamaker constant turns out to be the sum over many frequencies. The sum can be converted into an integral. For a material 1 interacting with material 2 across a medium 3, the non-retarded5 Hamaker constant is... 

Source Equation numbers are for formulae derived in V. A. Parsegian and G. H. Weiss, "On van der Waals interactions between macroscopic bodies having inhomogeneous dielectric susceptibilities," J. Colloid Interface Sci., 40, 35-41 (1972). 

Van der Waals Interactions Between Macroscopic Bodies and Adhesion Forces... 

The Derjaguin approximation (also called proximity force approximation) allows the calculation of the van der Waals interaction between macroscopic bodies with complex geometries from the knowledge of the interaction potential between planar surfaces, as long as the radii of curvature of the objects are large compared to the separation between them. 

Once we have established reasonable values for the Hamaker constants we shonld be able to calculate, for example, adhesion and surface energies, as well as the interaction between macroscopic bodies and colloidal particles. Clearly, this is possible if the only forces involved are van der Waals forces. That this is the case for non-polar liquids such as hydrocarbons can be illustrated by calculating the surface energy of these liqnids, which can be directly measured. When we separate a liquid in air we mnst do work Wc (per unit area) to create new surface, thus ... 

This approach enables the interactions between macroscopic bodies to be calculated, making assumptions about their molecular structure. The van der Waals attraction energy between individual molecules at separation r is given by... 

Lifshitz avoided this problem of additivity by developing a continuum theory of van der Waals forces that used quantum field theory. Simple accounts of the theory are given by Israelachvili and Adamson. Being a continuum theory, it does not involve the distinctions associated with the names of London, Debye and Keesom, which follow from considerations of molecular structure. The expressions for interaction between macroscopic bodies (e.g. Eqns. 2-4) remain valid, except that the Hamaker constant has to be calculated in an entirely different way. 

The van-der-Waals interaction between atomic dipoles leads to corresponding interactions between macroscopic bodies. The effective, macroscopic van-der-Waals forces can be approximated from the pairwise interaction between all atoms (Hamaker-de-Boer theory). Alternatively, one can relate the macroscopic van-der-Waals forces to the fluctuating electromagnetic fields that originate from the macrobodies (Lifshitz theory). These fields arise from the local fluctuations of electron density and can, therefore, be related to the dielectric properties of the material, which are accessible by experiment. 

The above short description of the theory of van der Waals forces is based upon interparticle interactions. The extension of this approach to many-body systems is a difficult task, as discussed in refs. 55 and 56. Since the van der Waals force between macroscopic bodies is measurable and since it originates in the fluctuations of electrons, Lifschitz sought to develop a theory based upon macroscopic electromagnetic concepts. The derivations are rather lengthy and the reader is referred to Lifschitz and coworkers articles and to a very readable review by Grimley. A somewhat simplified result is given in Eq. (93),... 

To determine the van der Waals forces between macroscopic bodies (e.g., two particles), we assume that the interaction between one molecule and a macroscopic body is simply the sum of the interactions with all the molecules in the body. We are therefore assuming simple additivity of the van da- Waals forces and ignore how the induced fields are affected by the intervening molecules. 

Figure 4.2 Additivity of the van der Waals force between macroscopic bodies. The total interaction is taken as the sum of the interactions between infinitesimally small elements in the two bodies.

When we consider the long-range interactions between macroscopic bodies (such as colloidal particles) in liquids, we find that the two most important forces are the van der Waals forces and electrostatic forces, although in the shorter distance, solvation forces often dominate over both. In this section, some important types of surface forces are discussed, and it should be helpful for calculation of surface forces. 

We move from the interaction between two molecules to the interaction between two macroscopic solids. It was recognized soon after London had published his explanation of the dispersion forces that dispersion interaction could be responsible for the attractive forces acting between macroscopic objects. This idea led to the development of a theoretical description of van der Waals forces between macroscopic bodies based on the pairwise summation of the forces between all molecules in the objects. This concept was developed by Hamaker [9] based on earlier work by Bradley [10] and de Boer [11]. This microscopic approach of Hamaker of pairwise summation of the dipole interactions makes the simplifying assumption that the... 

We have already seen from Example 10.1 that van der Waals forces play a major role in the heat of vaporization of liquids, and it is not surprising, in view of our discussion in Section 10.2 about colloid stability, that they also play a significant part in (or at least influence) a number of macroscopic phenomena such as adhesion, cohesion, self-assembly of surfactants, conformation of biological macromolecules, and formation of biological cells. We see below in this chapter (Section 10.7) some additional examples of the relation between van der Waals forces and macroscopic properties of materials and investigate how, as a consequence, measurements of macroscopic properties could be used to determine the Hamaker constant, a material property that represents the strength of van der Waals attraction (or repulsion see Section 10.8b) between macroscopic bodies. In this section, we present one illustration of the macroscopic implications of van der Waals forces in thermodynamics, namely, the relation between the interaction forces discussed in the previous section and the van der Waals equation of state. In particular, our objective is to relate the molecular van der Waals parameter (e.g., 0n in Equation (33)) to the parameter a that appears in the van der Waals equation of state ... 

In order to determine the interaction between macroscopic solids, in the first step we calculate the van der Waals energy between a molecule A and an infinitely extended body with a planar surface made of molecules B. This is also of direct relevance in understanding the adsorption of gas molecules to surfaces. We sum up the van der Waals energy between molecule A and all molecules in the solid B. Practically this is done via an integration of the molecular density pB over the entire volume of the solid ... 

Van der Waals interactions are noncovalent and nonelectrostatic forces that result from three separate phenomena permanent dipole-dipole (orientation) interactions, dipole-induced dipole (induction) interactions, and induced dipole-induced dipole (dispersion) interactions [46]. The dispersive interactions are universal, occurring between individual atoms and predominant in clay-water systems [23]. The dispersive van der Waals interactions between individual molecules were extended to macroscopic bodies by Hamaker [46]. Hamaker s work showed that the dispersive (or London) van der Waals forces were significant over larger separation distances for macroscopic bodies than they were for singled molecules. Through a pairwise summation of interacting molecules it can be shown that the potential energy of interaction between flat plates is [7, 23]... 

According to the microscopic theory by flamaker, the van der Waals interaction between two macroscopic bodies can be found by integration of Equation 5.166 over aU couples of molecules, followed by subtraction of the interaction energy at infinite separation between the bodies. The result depends on the geometry of the system. For a plane-parallel film from component 3 located between two semi-infinite phases composed from components 1 and 2, the van der Waals interaction energy per unit area and the respective disjoining pressure, stemming from Equation 5.166, are ... 

Van der Waals forces are briefly discussed in Section 3.1 for interaction between atoms or small molecules, their strength decays with intermolecular distance to the power —6. In the Hamaker-de Boer treatment, two macroscopic bodies are considered, and the van der Waals interaction between each atom in one of the bodies with all of the atoms in the other body are summed (actually a double integration procedure is applied). The result is that the total interaction energy V can be given by the product of a material property, called the Hamaker constant A (expressed in J or units of kBT), and a term depending on the geometry of the system. These relations are relatively simple. 

The attractive component of the DLVO force balance arises from van der Waals or dispersion interactions. In summary, all molecular species (apart from the proton) have, via the interaction of the electron cloud with the background electromagnetic field, the ability to induce dipoles in nearby molecules, and to be so affected themselves. This dipole-dipole interaction leads to an attractive force between molecules, and indeed between macroscopic bodies (although it is important to point out that dispersion forces are not, in general, pair-wise additive). 

Historically, van der Waals forces between condensed media have been calculated by summing over pairwise interactions between the molecular constituents of each body [e.g., the first term in (5.23)]. This method, called Hamaker theory, first showed that the sum of the intermolecular interactions, as calculated by elementary second-order perturbation theory in the Schrbdinger equation, between macroscopic bodies gives realistic attractive forces in many cases. Its description can be found in any reference on modern dispersion or van der Waals forces between condensed media including LANGBEIN [5.34] and MAHANTY and NINHAM [5.35] at the advanced level or the excellent elementary introduction by PARSE6IAN [5.36]. 

According to the microscopic theory by Hamaker [387], the van der Waals interaction between two macroscopic bodies can be found by integration of Equation 4.177 over all couples of molecules. 

To calculate the contribution of van der Waals forces in the interaction energy between macroscopic bodies one may sum the energies of aU the atoms in one body with all the... 

Between macroscopic bodies, the distance dependence of the van der Waals interaction depends on the geometry of the objects. Generally, it decays less steeply than between single molecules, for example, proportional to D for two planar surfaces. 

That is, the potential energy of attraction is identical in the two cases. This is an important result as far as the extension of molecular interactions to macroscopic spherical bodies is concerned. What it says is that two molecules, say, 0.3 nm in diameter and 1.0 nm apart, interact with exactly the same energy as two spheres of the same material that are 30.0 nm in diameter and 100 nm apart. Furthermore, an inspection of Equation (49) reveals that this is a direct consequence of the inverse sixth-power dependence of the energy on the separation. Therefore the conclusion applies equally to all three contributions to the van der Waals attraction. Precisely the same forces that are responsible for the association of individual gas molecules to form a condensed phase operate —over a suitably enlarged range —between colloidal particles and are responsible for their coagulation. 

Between molecules, the van der Waals energy decreases with /D6. For macroscopic bodies the decay is less steep and it depends on the specific shape of the interacting bodies. For example, for two infinitely extended bodies separated by a gap of thickness x the van der Waals energy per unit area is... 

The major disadvantage of this microscopic approach theory was the fact that Hamaker knowingly neglected the interaction between atoms within the same solid, which is not correct, since the motion of electrons in a solid can be influenced by other electrons in the same solid. So a modification to the Hamaker theory came from Lifshitz in 1956 and is known as the Lifshitz or macroscopic theory." Lifshitz ignored the atoms completely he assumed continuum bodies with specific dielectric properties. Since both van der Waals forces and the dielectric properties are related with the dipoles in the solids, he correlated those two quantities and derived expressions for the Hamaker constant based on the dielectric response of the material. The detailed derivations are beyond the scope of this book and readers are referred to other publications. The final expression that Lifshitz derived is... 

Hamaker [4] suggested that the London dispersion interactions between atoms or molecules in macroscopic bodies (such as emulsion droplets) can be added, resulting in strong van der Waals attractions, particularly at close distances of separation between the droplets. For two droplets with equal radii, R, at a separation distance h, the van der Waals attraction G is given by the following equation (due to Hamaker) ... 

The theory for the van der Waals interactions presented in the previous section applies to macroscopic media only in a qualitative sense. This is because (i) the additivity of the interactions is assumed — i.e., the energies are written as sums of the separate interactions between every pair of molecules (ii) the relationship of the Hamaker constant to the dielectric constant is based on a very oversimplified quantum-mechanical model of a two-level system (iii) finite temperature effects on the interaction are not taken into account since it is a zero-temperature description. Here, we present a simplified derivation of the van der Waals interaction in continuous media, based upon arguments first presented by Ninham et al a more rigorous treatment can be found in Ref. 4. The van der Waals interactions arise from the free energy of the fluctuating electromagnetic field in the system. For bodies whose separations... 

---