Van-der-Waals force

Van der Waals forces are weaker than ionic interactions and include three types of interactions dipole-dipole, dipole-induced dipole and induced dipole-induced dipole. Dipolar forces can be compared to magnetic forces where a 

Van der Waals forces are only relevant if the surface of the primary particles in contact with each other is significantly deformed and can be the forces responsible for initiating the contact prior to sintering. Van der Waals forces are based on temporary load shifts in neighboring surfaces. The van der Waals forces between particles can be estimated by calculating the electrostatic forces between two parallel circular plates with a diameter x according to Lifshitz (1956) or Hamaker (1937) using Eq. 7.4  

A is a constant valid for parallel circular contact areas and a is the distance between the contacting surfaces. A can be calculated using the Lifshitz-van der Waals constant has = 10 ° — 10 J) or the Hamaker constant H = 10 — 10 J). 

Since the van der Waals forces depend strongly on the distance a between the particles and on the interparticle contact area, they can be increased by decreasing the former and/or increasing the latter (Eq. 7.4). This can be achieved by plastic or viscoelastic deformation of particle surfaces which depends on the mechanical material properties. 

The adhesion force between two plastically deformed spheres can be calculated according to Rumpf et al. (1976)  

Ft is the force with which the particles are pressed together and Ppi is the plastic yield pressure. The distance a is assumed to be approximately 0.4 nm when the particles are in contact (Rumpf et al, 1976). 

The van der Waals forces in the strict sense, also called dispersive forces, are the attractive forces between two neutral, nonpolar molecules, for example anthracene molecules, which thus have no static dipole moments. Were the charge dishibution within the molecules rigid, then there would indeed be no interactions between them. However, due to their temporally fluctuating charge distributions, they also have fluctuating dipole moments and these can induce dipoles in other molecules, compare Fig. 2.3. This results in an attractive force, as we aheady calculated in the section on inductive forces. To distinguish the two cases (of a permanent dipole and fluctuating dipoles), these forces due to fluctuations are also termed dispersive forces. 

From Eq. (2.9) we can find the dispersive force between two molecules with po-larisability a 

The factor A = A76 has here of course the same meaning as in Eq. (2.11). A specific model for the differential calculation of the dispersion energy Vdisp will be treated in Sect. 2.1.4. 

4 shows the different distance dependences of dipolar, inductive, and dispersive energies of attraction, computed for the NH3 molecule, which has a permanent dipole moment. In this case, not only inductive and dispersive forces, but also dipole forces play a role. 

In colloidal systems, van der Waals forces play a prominent role. When any two particles (neutral or with charges) come very close to each other, the van der Waals forces will be strongly dependent on the surrounding medium. In a vacuum, two identical particles will always exhibit an attractive force. On the other hand, if two different particles are present in a medium (in water), then there may be repulsion forces. This can be due to one particle adsorbing with the medium more strongly than with the other particle. One example will be silica particles in water medium and plastics (as in wastewater treatment). It is critical to understand under what conditions it is possible that colloidal particles remain suspended. For example, if paint aggregates in the container, then it is obviously useless for its intended purpose. 

The positive-positive particles will show repulsion. On the other hand, the positive-negative particles will attract each other. The ion distribution will also depend on the concentration of any counterions or coions in the solution. Even glass, when dipped in water, exchanges ions with its surroundings. Such phenomena can be easily investigated by measuring the change in the conductivity of the water. 

The force F12, acting between these oppositely charges, is given by Coulomb s law, with charges q and q2 separated at a distance R12 in a dielectric medium De  

Another very important physical parameter one must consider is the size distribution of the colloids. A system consisting of particles of the same size is called a monodis-perse. A system with different sizes is called polydisperse. It is also obvious that systems with monodisperse will exhibit different properties from those of polydispersed systems. In many industrial application (such as coating on tapes used for recording music and coatings on CDs or DVDs), latter kind of quality of coatings is needed. 

The methods used to prepare monodisperse colloids aim to achieve a large number of critical nuclei in a short interval of time. This induces all equally sized nuclei to grow simultaneously, thus producing a monodisperse colloidal product. 

The Gibbs free energy of the universal van der Waals interactions (AGv) between two identical spheres with radius a separated by a distance // in a vacuum HIa 1) can be calculated by the following equation  

For the colloidal particles dispersed in some medium other than a vacuum, the attractive van der Waals forces between two particles will be reduced due to the effect of the molecules present in the medium. To derive an expression for the effective Ffamaker constant for the particles (component 2, ) 

With the assumption that the Gibbs free energy terms are linearly proportional to the corresponding Hamaker constants, then Eq. (2.10) can be further reduced to the following relationships  

In a similar manner, the effective Hamaker constant for the different particles (components 2 and 3) dispersed in the medium (component 1) can be estimated by the following relationships  

Inside an atom, the nucleus carries a positive charge. Negatively charged electrons are located at off-centre and fluctuating positions relative to the nucleus. This asymmetry gives rise to attractive interatomic electromagnetic forces. In some cases, the asymmetry extends to the molecular scale (polar molecules), but the effect is globally the same. All forces of this kind are known as London-van der Waals forces. 

The corresponding energy of attraction between two molecules decreases as the sixth power of the distance between them. However, when the electron clouds of the atoms begin to interpenetrate, strong repulsive forces come into play (Born forces), so that an equilibrium position exists for intermolecular distances of order 0.2-0.4 nm, known as the van der Waals radius. 

If we now consider the forces exerted between two particles (or more generally, between several particles), we can assume to a first approximation that the van der Waals forces are additive. Within a given particle, each atom or molecule is thus subject to attractive forces from all the atoms of the neighbouring particles. The global attraction can be calculated for each geometrical arrangement of the particles. 

two identical spherical objects of radius r are coupled with energy 

When the spheres are immersed in a fluid M, the same formula applies but the constant changes, and is denoted Am 7 A. It is, in fact, a complicated function depending on the intrinsic physical characteristics of both the dispersed material and the medium M. Table 3.2 gives the values of Hamaker s constant A for various materials dispersed in air, and the values of Am for the aqueous dispersions. 

If a second dipole of moment ps is placed in this field at position r, an interaction energy U results which reads 

Molecules that do not possess permanent dipole posses a non-zero instantaneous dipole moment because of fluctuations caused for instance by electromagnetic radiation fields. Then instantaneous values of dipolar moment must be taken into 

Let us go back to the definition of the electric dipolar moment p, ,( in a molecule. The instantaneous configuration of a molecule consisting of a set of nuelei and electrons with charges at positions r . The net charge is given by 

When two molecules A and B are brought from an infinite separation to a distance R, the charges on each particle will interaet. The interaction energy is 

If the intermolecular distance R is greater than r4j and r j the last equation can be expanded in powers of l/R. 

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Much of chemistry is concerned with the short-range wave-mechanical force responsible for the chemical bond. Our emphasis here is on the less chemically specific attractions, often called van der Waals forces, that cause condensation of a vapor to a liquid. An important component of such forces is the dispersion force, another wave-mechanical force acting between both polar and nonpolar materials. Recent developments in this area include the ability to measure... 

In 1930, London [1,2] showed the existence of an additional type of electromagnetic force between atoms having the required characteristics. This is known as the dispersion or London-van der Waals force. It is always attractive and arises from the fluctuating electron clouds in all atoms that appear as oscillating dipoles created by the positive nucleus and negative electrons. The derivation is described in detail in several books [1,3] and we will outline it briefly here. 

Fig. VI-6. The force between two crossed cylinders coated with mica and carrying adsorbed bilayers of phosphatidylcholine lipids at 22°C. The solid symbols are for 1.2 mM salt while the open circles are for 10.9 roM salt. The solid curves are the DLVO theoretical calculations. The inset shows the effect of the van der Waals force at small separations the Hamaker constant is estimated from this to be 7 1 x 10 erg. In the absence of salt there is no double-layer force and the adhesive force is -1.0 mN/m. (From Ref. 66.)...

There is always some degree of adsorption of a gas or vapor at the solid-gas interface for vapors at pressures approaching the saturation pressure, the amount of adsorption can be quite large and may approach or exceed the point of monolayer formation. This type of adsorption, that of vapors near their saturation pressure, is called physical adsorption-, the forces responsible for it are similar in nature to those acting in condensation processes in general and may be somewhat loosely termed van der Waals forces, discussed in Chapter VII. The very large volume of literature associated with this subject is covered in some detail in Chapter XVII. 

The adhesion between two solid particles has been treated. In addition to van der Waals forces, there can be an important electrostatic contribution due to charging of the particles on separation [76]. The adhesion of hematite particles to stainless steel in aqueous media increased with increasing ionic strength, contrary to intuition for like-charged surfaces, but explainable in terms of electrical double-layer theory [77,78]. Hematite particles appear to form physical bonds with glass surfaces and chemical bonds when adhering to gelatin [79]. 

The second general cause of a variable heat of adsorption is that of adsorbate-adsorbate interaction. In physical adsorption, the effect usually appears as a lateral attraction, ascribable to van der Waals forces acting between adsorbate molecules. A simple treatment led to Eq. XVII-53. 

Dalgarno A and Kingston A E 1961 van der Waals forces for hydrogen and the inert gases Proc. Phys. Soc. London 78 607... 

Perez-Jordy J M and Becke A D 1995 A density functional study of van der Waals forces rare gas diatomics Chem. Phys. Lett. 233 134... 

Pekeris C L 1934 The rotation-vibration coupling in diatomic molecules Phys. Rev. 45 98 Slater J C and Kirkwood J G 1931 The van der Waals forces in gases Phys. Rev. 37 682... 

Adsorbates can physisorb onto a surface into a shallow potential well, typically 0.25 eV or less [25]. In physisorption, or physical adsorption, the electronic structure of the system is barely perturbed by the interaction, and the physisorbed species are held onto a surface by weak van der Waals forces. This attractive force is due to charge fiuctuations in the surface and adsorbed molecules, such as mutually induced dipole moments. Because of the weak nature of this interaction, the equilibrium distance at which physisorbed molecules reside above a surface is relatively large, of the order of 3 A or so. Physisorbed species can be induced to remain adsorbed for a long period of time if the sample temperature is held sufficiently low. Thus, most studies of physisorption are carried out with the sample cooled by liquid nitrogen or helium. 

Note that the van der Waals forces tliat hold a physisorbed molecule to a surface exist for all atoms and molecules interacting with a surface. The physisorption energy is usually insignificant if the particle is attached to the surface by a much stronger chemisorption bond, as discussed below. Often, however, just before a molecule fonus a strong chemical bond to a surface, it exists in a physisorbed precursor state for a short period of time, as discussed below in section AL7.3.3. 

As the tip is brought towards the surface, there are several forces acting on it. Firstly, there is the spring force due to die cantilever, F, which is given by = -Icz. Secondly, there are the sample forces, which, in the case of AFM, may comprise any number of interactions including (generally attractive) van der Waals forces, chemical bonding interactions, meniscus forces or Bom ( hard-sphere ) repulsion forces. The total force... 

Mutter J L and Bechhoefer J 1994 Measurement and manipulation of van der Waals forces in atomic-force microscopy J. Vac. Sc/. Technol. B 12 2251... 

In accordance with equation (Bl.20.1). one can plot the so-called surface force parameter, P = F(D) / 2 i R, versus D. This allows comparison of different direct force measurements in temis of intemiolecular potentials fV(D), i.e. independent of a particular contact geometry. Figure B 1.20.2 shows an example of the attractive van der Waals force measured between two curved mica surfaces of radius i 10 nun. 

Tabor D and Winterton R H S 1969 The direct measurement of normal and retarded van der Waals forces Proc. R. Soc. London A 312 435-50... 

Similarly, van der Waals forces operate between any two colloidal particles in suspension. In the 1930s, predictions for these interactions were obtained from the pairwise addition of molecular interactions between two particles [38]. The interaction between two identical spheres is given by... 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

For other compounds, the agreement is not always so good. The assumption that the lattice is always wholly ionic is not always true there may be some degree of covalent bonding or (where the ions are very large and easily distorted) some appreciable van der Waals forces between the ions (p.47). 

To separate the non-bonded forces into near, medium, and far zones, pair distance separations are used for the van der Waals forces, and box separations are used for the electrostatic forces in the Fast Multipole Method,[24] since the box separation is a more convenient breakup in the Fast Multipole Method (FMM). Using these subdivisions of the force, the propagator can be factorized according to the different intrinsic time scales of the various components of the force. This approach can be used for other complex systems involving long range forces. 

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