Force between macroscopic objects

In this section we consider electromagnetic dispersion forces between macroscopic objects. There are two approaches to this problem in the first, microscopic model, one assumes pairwise additivity of the dispersion attraction between molecules from Eq. VI-15. This is best for surfaces that are near one another. The macroscopic approach considers the objects as continuous media having a dielectric response to electromagnetic radiation that can be measured through spectroscopic evaluation of the material. In this analysis, the retardation of the electromagnetic response from surfaces that are not in close proximity can be addressed. A more detailed derivation of these expressions is given in references such as the treatise by Russel et al. [3] here we limit ourselves to a brief physical description of the phenomenon. 

One of the most important things to bear in mind in studying van der Waals forces is that this topic has ramifications that extend far beyond our discussion here. Van der Waals interactions, for example, contribute to the nonideality of gases and, closer to home, gas adsorption. We also see how these forces are related to surface tension, thereby connecting this material with the contents of Chapter 6 (see Vignette X below). These connections also imply that certain macroscopic properties and measurements can be used to determine the strength of van der Waals forces between macroscopic objects. We elaborate on these ideas through illustrative examples in this chapter. 

More often than not one deals with colloidal objects immersed in a liquid or other such media, and therefore interactions between similar or dissimilar materials in an arbitrary medium are of importance in colloid science. Moreover, it is very useful to relate such dissimilar interactions to those between identical particles in vacuum. In the last section (Section 10.8) we present what are known as combining relations for accomplishing this. The van der Waals forces between macroscopic objects are usually attractive, but under certain circumstances they (and, as a consequence, the Hamaker constant) can be negative, as noted in Vignette X. A brief discussion of this completes Section 10.8. 

Forces between macroscopic objects result from a complex interplay of the interaction between molecules in the two objects and the medium separating them. The basis for an understanding of intermolecular forces is the Coulomb1 force. The Coulomb force is the electrostatic force between two charges Qi and Q-2-... 

Attractive forces between macroscopic objects have been measured directly by a number of investigators. In the first experiment of this kind, Deryagin and Abricossova107,202 203 used a sensitive electronic feed-back balance to measure the attraction for a planoconvex polished quartz system from which all residual electric charge had been removed. Relatively large separations were involved and the... 

Theories of interparticle forces play a fundamental part in many theoretical aspects of colloidal behaviour. It is therefore of great importance to have experimental evidence for the validity of these theories. One approach to this is to study the forces between macroscopic objects, to which the same theoretical equations should apply. Since these forces arc exceedingly small until the bodies come into very close proximity, work in this area has faced considerable experimental difficulties. Experiments on the force between two plates and between a plate and a lens have been of limited validity because of the difficulty in achieving adequate surface smoothness and in completely eliminating dust. The... 

Overbeek, J. and Sparnaay, M., Experiments on long-range attractive forces between macroscopic objects, J. Colloid Interface Sci., 343-345 (1952). 

Forces between macroscopic objects result from a complex interplay of the interaction between molecules in the two objects and the medium separating... 

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. 

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). 

Owing to their fundamental interest and their practical importance in issues such as colloid stability, much experimental effort has been devoted to the measurement of electric double layer and van der Waals interactions between macroscopic objects at close separations. Such measurements involve balancing the force(s) to be measured with an externally applied force. 

As it was already written above, we would like to study structural changes in the charge distribution between macroscopic objects, that is caused by the image forces, and depends on the wall-to-wall distance. To obtain direct structural information about the system, we will introduce a configurational analogue of the phase-space distribution function. At equilibrium, the definition of an fth order distribution function given by Eq. (12) can be applied to the equilibrium probability density [Eq. (13)], and the integration with respect to impulses can easily be carried out. We write for the rth order local density... 

These forces are always present and always attractive between particles of the. same nature. They are the result of fluctuations in the dipolar interactions at the molecular level [2,3,13]. The potential energy of this interaction is a function of the separation distance r between dipoles, and has an r dependence. The sum of the interactions between macroscopic objects (as far as molecular dimensions are concerned) yields an interaction energy that is a function of... 

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 ... 

For macroscopic objects the adhesion force is often small compared to the load. For microscopic bodies this can be different. The reason is simple the weight of an object sliding over a surface usually decreases with the third power of its diameter (or another length characterizing its size). The decrease of the actual contact area and hence the adhesion force follows a weaker dependence. For this reason, friction between microbodies is often dominated by adhesion while in the macroscopic world we can often neglect adhesion. 

The force of gravity dominates our macroscopic world. Gravity can be described as the universal attraction between all objects. Even though gravity is the weakest of the four fundamental forces, it is ultimately responsible for perhaps the most violent of all objects in the universe, black holes. Newton s Law of Universal Gravitation gives us the mathematical description of the attractive gravitational force between two point objects of mass mx and m2 ... 

Image forces may also alter the interaction between uncharged macroscopic objects. It is known that the van der Waals interaction between particles is remarkably screened if an electrolyte is present [3,31,34]. For xh> 1 the... 

Casimir and Polder also showed that retardation effects weaken the dispersion force at separations of the order of the wavelength of the electronic absorption bands of the interacting molecules, which is typically 10 m. The retarded dispersion energy varies as R at large R and is determined by the static polarizabilities of the interacting molecules. At very large separations the forces between molecules are weak but for colloidal particles and macroscopic objects they may add and their effects are measurable. Fluctuations in particle position occur more slowly for nuclei than for electrons, so the intermolecular forces that are due to nuclear motion are effectively unretarded. A general theory of the interaction of macroscopic bodies in terms of the bulk static and dynamic dielectric properties... 

The two system-specific parameters in the LJ equation encompass a and s. If their values, the number density of species within the interacting bodies, and the form/shape of the bodies are known, the mesoscopic/macroscopic interaction forces between two bodies can be calculated. The usual treatment of calculating net forces between objects includes a pairwise summation of the interaction forces between the species. Here, we neglect multibody interactions, which can also be considered at the expense of mathematical simplicity. Additivity of forces is assumed during summation of the pairwise interactions, and retardation effects are neglected. The corresponding so-called Hamaker summation method is well described in standard texts and references [5,6]. Below we summarize a few results relevant for AFM. 

Molecules do not roam around being lonely. They exist in populous societies, wherein they form aggregates that are held hy forces between molecules (called by chemists intermolecularforces/interactions), which endow these aggregates with 3D architecture, which in turn determines the architecture of the macroscopic matter (macroscopic objects are ones we can see with the naked eye). Figure 7.11 shows some of these architectures. 

For large times t, the exponential factor in Eq. (231), which takes the damping of the particle trajectories into account, restricts the time that each particle spends between collisions to be on the order of t the mean free time. Upon comparing (231) with (137), the expansion of the force on a sphere in a rarefied gas in powers of the inverse Knudsen number, one can see that these two expansions have a remarkably similar structure. This similarity has its source in the fact that the coefficients in the density expansion of 17/170 and in the expansion of F/Fq are determined by dynamical events of the same basic types, as may be seen by comparing Fig. 15 with Figs. 22 and 24. The dynamical events that contribute to F/Fq differ from those which contribute to 17/170 only in that in F/Fq one of the gas particles is replaced by a macroscopic object, in this case a sphere. 

Recently, capillary action has been used to self-assemble macroscopic objects. Objects of various shapes were cut from polydimethylsiloxane, a polymer that is not wettable by water but is wetted by fluorinated hydrocarbons. Designated surfaces were then made wettable by water by using controlled oxidation. These objects were then floated at an interface between perfluorodecalin (CioFig) and water. When two non-oxidized surfaces (wettable by CioFis) approached each other within a distance of approximately 5 mm, they moved into contact, which with time created an ordered, self-assembled pattern of the objects. The movement and self-assembly was driven by the solvent adhesive forces that produce the capillary action, thereby leading to an elimination of the curved menisci between non-oxidized surfaces. One such pattern is shown to the right. 

Particle behavior in colloidal suspensions is dominated by forces and mechanisms different from that relevant for macroscopic objects. Their behavior will be an interplay of sedimentation, random Brownian motion, viscous drag, and interparticle interactions. Thus, in order to use colloidal particles as building blocks for patterning in the micrometer and sub-micrometer range, the balance between these forces must be controlled to provide a method for the robust and reprodudble placement of the particles. In the following a short description of the forces will be given. 

The main purpose of the SFA is to measure the forces exerted by a thin fluid film on a solid substrate with nearly molecular precision (143). In the SFA, a thin film is confined between the surfaces of two macroscopic cylinders arranged sucli that their axes are at a right angle (143). In an alternative setup, the fluid is confined between the surface of a macroscopic sphere and a planar substrate [144]. However, crosscd-cylinder and sphere-plane configurations can be mapped onto one another by differential-geometrical arguments [145]. The surface of each macroscopic object is covered by a thin mica sheet with a silver backing, which permits one to measure the separation h between the surfaces by optical interferometry [143]. 

One curious physical phenomenon associated with gases is the fact that when there is a mixture of gases in a given volume they behave independentiy so that their pressures are additive. In fact this raises the issue of what we mean by pressure. Common sense may lead us to expect that volumes are additive as indeed they are for macroscopic objects such as bricks. Thus, it is somewhat thought provoking that several gases can be easily confined in the same volume. This same sort of question also arises for mixtures of liquids to a much less extent as discussed later in Chapter 6. These considerations go to the very heart of the concept of the size of atoms and molecules and how much space is between them in a liquid or gas. As we will soon see, the space between gas molecules is about 100 times their size at 1 atm so there is plenty of space for other molecules. In addition, it will soon become evident that pressure is (force/area) caused by many collisions of gas molecules with the wall of the container. Cavendish in 1781 and Dalton in 1810 contributed to the concept now known as Dalton s law. ... 

Dispersion forces are universal because they attract all molecules together, regardless of their specific chemical nature. The potential energy of dispersion attraction between two isolated molecules decays with the sixth power of the separation distance. Based on the so-called Hamaker theory (i.e., the method of pair-wise summation of intermolecular forces) or the more modern Lifshitz macroscopic treatment of strictly additive London forces, it is possible to develop the so-called Lifshitz-Van der Waals expression for the macroscopic interactions between macroscopic-in-size objects (i.e., macrobodies) [19, 21], Such an expression strongly depends on the shapes of the interacting macrobodies as well as on the separation distance (non-retarded or retarded interaction). For two portions of the same phase of infinite extent bounded by parallel flat surfaces, at a distance h apart, the potential energy of macroscopic attraction is ... 

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. 

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