Van der Waals forces between macroscopic bodies

Hamaker (1932) and de Boer (1936) calculated van der Waals forces between macroscopic bodies using the summation method. 

CALCULATING VAN DER WAALS FORCES BETWEEN MACROSCOPIC BODIES... 

One such approximation was proposed by Good and Girilalco assuming that mainly van der Waals forces act between all molecules 11081. Based on the theory of van der Waals forces between macroscopic bodies they suggested... 

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.

Note the difference between Hq.(20) and (25). The attraction van der Waals force between macroscopic bodies is clearly dependent on the geometries of two units. 

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... 

The universal van der Waals attraction which occurs in all disperse systems is described in Vol. 1. The dipole-dipole, dipole-induced dipole and London dispersion forces for atoms and molecules are described. This is followed by the microscopic theory of Hamaker for colloidal particles and definition of the Hamaker constant. This microscopic theory is based on the assumption of additivity of all atom or molecular attractions in each particle or droplet. The variation of van der Waals attraction with separation distance h between the particles is schematically represented. This shows a sharp increase in attraction at small separation distances (of the order of a few nanometers). In the absence of any repulsion, this strong attraction causes particle or droplet coagulation which is irreversible. The effect of the medium on the overall van der Waals attraction is described in terms of the effective Hamaker constant which is now determined by the difference in Hamaker constant between the particles and the medium. The macroscopic theory of van der Waals attraction is briefly described, with reference to the retardation effect at long sepeiration distances. The methods that can be applied for determination of the van der Waals attraction between macroscopic bodies are briefly described. 

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

These van der Waals forces between colloid particles or surfaces are very strong, as shown in Figure 6.10, where some numerical examples are provided. We can see that the van der Waals forces between macroscopic particles are large and not only when the bodies arc in contact ... 

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. 

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 ... 

The strategy for scaling up the van der Waals attraction to macroscopic bodies requires that all pairwise combinations of intermolecular attraction between the two bodies be summed. This has been done for several different geometries by Hamaker. We consider only one example of the calculations involved, namely, the case of blocks of material with planar surfaces. This example serves to illustrate the method and also provides a foundation for connecting van der Waals forces with surface tension, the subject of the next section. 

Figure 6.1 Calculating the van der Waals force between a macroscopic body and a molecule.

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]. 

Attractive surface forces, generally referred to as van der Waals forces, exist between all atoms and molecules regardless of what other forces may be involved. They have been discussed in detail by Isrealachvili (4). We shall first examine the origins of the van der Waals forces between atoms and molecules and later consider the attractive forces between macroscopic bodies such as particles. 

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 ... 

The van der Waals forces at the atomic level as well as those between macroscopic bodies (Chapter 10)... 

Our objectives in this chapter are to look into the origin of van der Waals forces, see how they affect macroscopic behavior and properties of materials, and establish relations for scaling up the molecular-level forces to forces between macroscopic bodies. 

The van der Waals forces are always attractive (although, as we see in Section 10.8b, the London forces between two macroscopic bodies immersed in a medium can be repulsive, depending on the material properties). 

It is extremely difficult to measure the Hamaker constant directly, although this has been the object of considerable research efforts. Direct evaluation, however, is complicated either by experimental difficulties or by uncertainties in the values of other variables that affect the observations. The direct measurement of van der Waals forces has been undertaken by literally measuring the force between macroscopic bodies as a function of their separation. The distances, of course, must be very small, so optical interference methods may be used to evaluate the separation. The force has been measured from the displacement of a sensitive spring (or from capacitance-type measurements). 

Here we already see a remarkable property of the van der Waals forces The energy of a molecule and a macroscopic body decreases less steeply than the energy between two molecules. Instead of the I) 6 dependence the energy decreases proportional to I) 3. 

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]... 

The van der Waals force is ubiquitous in colloidal dispersions and between like materials, always attractive and therefore the most common cause of dispersion destabilization. In its most common form, intermolecular van der Waals attraction originates from the correlation, which arises between the instantaneous dipole moment of any atom and the dipole moment induced in neighbouring atoms. On this macroscopic scale, the interaction becomes a many-body problem where allowed modes of the electromagnetic field are limited to specific frequencies by geometry and the dielectric properties of the system. 

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... 

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. 

J. N. Israelachvili, The calculation of van der Waals dispersion forces between macroscopic bodies, Proc. Royal Soc. (London) 331A 39 (1972). See also Chap. 7 in J. Mahanty and B. W. Ninham, Dispersion Forces. Academic Press, London, 1976. 

The van der Waals forces represent an averaged dipole-dipole interaction, which is a superposition of orientation interactions (between two permanent dipoles, Keesom 1913), induction interaction (between one permanent dipole and one induced dipole, Debye 1920) and dispersion interaction (between two induced dipoles, London 1930). The interaction between two macroscopic bodies depends on the geometry of the system (see Fig. 3). For a plane-parallel film with uniform thickness, h, from component 3 located between two semi-infinite... 

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. 

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. 

---