Hamaker constant measurement

In an extensive SFA study of protein receptor-ligand interactions, Leckband and co-workers [114] showed the importance of electrostatic, dispersion, steric, and hydrophobic forces in mediating the strong streptavidin-biotin interaction. Israelachvili and co-workers [66, 115] have measured the Hamaker constant for the dispersion interaction between phospholipid bilayers and find A = 7.5 1.5 X 10 erg in water. 

SFA has been traditionally used to measure the forces between modified mica surfaces. Before the JKR theory was developed, Israelachvili and Tabor [57] measured the force versus distance (F vs. d) profile and pull-off force (Pf) between steric acid monolayers assembled on mica surfaces. The authors calculated the surface energy of these monolayers from the Hamaker constant determined from the F versus d data. In a later paper on the measurement of forces between surfaces immersed in a variety of electrolytic solutions, Israelachvili [93] reported that the interfacial energies in aqueous electrolytes varies over a wide range (0.01-10 mJ/m-). In this work Israelachvili found that the adhesion energies depended on pH, type of cation, and the crystallographic orientation of mica. 

Thus, the spacing of the chains relative to the neutral, free, swollen size of the buoy blocks is, for a given chemical system and temperature, a unique function of the solvent-enhanced size asymmetry of the diblock polymer and a weak function of the effective Hamaker constant for adsorption. The degree of crowding of the nonadsorbing blocks, measured by a decrease in the left-hand side of Eq. 28, increases with increasing asymmetry of the block copolymer. 

Hamaker constants can sometimes be calculated from refractive igdex data by the Lifshitz equations (8), but it now appears that Y values are closely related to refractive indices and are a direct measure of the Lifshitz attractions. In Equation 1 a correction factor f for "retardation" of dispersion forces is shown which can be determined from Figure 2, a graph of 1/f at various values of H and a as a function of Xj, the characteristic wavelength of the most energetic dispersion forces, calculable and tabulated in the literature (9). 

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 ... 

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 ... 

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). 

It is evident from Figure 10.7 that the measurements are consistent with both unretarded and retarded attractive forces at appropriate separation distances. It has also been possible to verify directly the functional dependence on radii for the attraction between dissimilar spheres (see Table 10.4), to determine the retardation of van der Waals forces (see Table 10.1), and to evaluate the Hamaker constant for several solids, including quartz. Values in the range of 6 10 20 to 7 10 20 J have been found for quartz by this method. This is remarkably close to the value listed in Table 10.5 for Si02. 

List some reasons why it is desirable to relate Hamaker constants to measurable macroscopic properties instead of relying entirely on molecular parameters. 

Throughout most of this chapter the emphasis has been on the evaluation of zeta potentials from electrokinetic measurements. This emphasis is entirely fitting in view of the important role played by the potential in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of colloidal stability. From a theoretical point of view, a fairly complete picture of the stability of dilute dispersions can be built up from a knowledge of potential, electrolyte content, Hamaker constants, and particle geometry, as we discuss in Chapter 13. From this perspective the fundamental importance of the f potential is evident. Below we present a brief list of some of the applications of electrokinetic measurements. 

Polarizability Attraction. All matter is composed of electrical charges which move in response to (become electrically polarized in) an external field. This field can be created by the distribution and motion of charges in nearby matter. The Hamaker constant for interaction eneigy, A, is a measure of this polarizability. As a first approximation it may be computed from the dielectric permittivity, S, and the refractive index, n, of the material (15), where Vg is the frequency of the principal electronic absorption... 

Equation (6.25) not only allows us to calculate the Hamaker constant, it also allows us to easily predict whether we can expect attraction or repulsion. An attractive van der Waals force corresponds to a positive sign of the Hamaker constant, repulsion corresponds to a negative Hamaker constant. Van der Waals forces between similar materials are always attractive. This can easily be deduced from the last equation for 1 = e2 and n = n2 the Hamaker constant is positive, which corresponds to an attractive force. If two different media interact across vacuum ( 3 = n3 = 1), or practically a gas, the van der Waals force is also attractive. Van der Waals forces between different materials across a condensed phase can be repulsive. Repulsive van der Waals forces occur, when medium 3 is more strongly attracted to medium 1 than medium 2. Repulsive forces were, for instance, measured for the interaction of silicon nitride with silicon oxide in diiodomethane [121]. Repulsive van der Waals forces can also occur across thin films on solid surfaces. In the case of thin liquid films on solid surfaces there is often a repulsive van der Waals force between the solid-liquid and the liquid-gas interface [122],... 

Example 6.7. For the interaction of lipid bilayers across a layer of water, a Hamaker constant of 7.5x 10 21 J is calculated. A value of only 3x 1CP21 J was measured. One reason is probably a reduction of the first term by the presence of ions [131]. 

Example 6.8. Helium [109], As an interatomic distance often a value of 1.6 A is used. The Hamaker constant of helium-vacuum-helium is 5.7 x 10" 22 J. Calculating the surface energy with Eq. (6.29) leads to 0.29 mJ/m2. This value is in good agreement with measured values for liquid helium of 0.12-0.35 mJ/m2. 

One effect of a lubricant is to reduce adhesion between the solids. Adhesion between solids is usually dominated by van der Waals forces. Hydrocarbons have a small Hamaker constant and their presence leads to a reduction in the adhesion and hence friction. Films of only a monomolecular thickness are sufficient to have a pronounced effect. This shows up when we measure friction between solids, which are coated with monomolecular layers (see example 11.1). In that case, friction can be as small as friction with plenty of lubricant. A monomolecular film affects significantly the frictional properties [495], At least with metals it can be shown that the number of microcontacts is not changed by the lubricant. Only the contact intensity is reduced. The reduced van der Waals attraction can thereby diminish the actual contact area. 

These quartz measurements, together with several other less-successful attempts by others, had been fiercely contested.20 Theories had been fitted to faulty measurements there had been no adequate theory yet available for good measurements. "Measurement" drove theory. Hamaker constants (coefficients of interaction energy) were so uncertain that they were allowed to vary by factors of 100 or 1000 in order to fit the data. The Lifshitz theory put an end to all that. Disagreement meant that either there was a bad measurement or there was something acting besides a charge-fluctuation force. 

One still hears the archaic designation "Hamaker constant" for Anam = AAm/Bm(D from the time when people did not recognize that the coefficient could itself vary with separation l. In modern usage this spatially varying coefficient, evaluated at zero separation, remains a popular and useful measure of the strength of van der Waals forces. 

While one can, in principle, explain the swelling of bilayers upon addition of salt even by assuming that the Hamaker constant does not depend on the electrolyte concentration, this was not the purpose of this paper. Note that the value -ff = 9 Debyes employed in the previous calculations is too large for surface dipoles. We fully agree with Petrache et al. that the Hamaker constant is decreased by the addition of salt, maybe not as much as 75% but nevertheless by a measurable quantity. What we tried to emphasize here is that there are so many unknown quantities in the interactions between lipid bilayers, that it is very difficult to obtain reliable information for the dependence of the Hamaker constant on electrolyte concentrations from this type of experiments. 

TABLE 1 Accessible Force Measuring Range for vdW Interactions as a Function of Spring Constant. The Calculations Assume a Tip Radius of 5 x IO-h m and a Composite Hamaker Constant of l x lO-20 J... 

FIGURE 10.4 The force measured between two curved mica surfaces in solutions of 2 1 electrolytes Ca Sr and Ba ) at pH 5.8. The solid lines are based on the theory for potentials and concentrations shown along with the van der Waals attraction corresponding to a Hamaker constant of 2.2 x 10 J. Redrawn from Pashley and Israelachvili [17]. Reprinted with permission from Academic Press. 

Part of these data on inform us about the relative preference of SL over LL Interactions, which is related to the difference between the Hamaker constants and Aj. Particularly with water as the liquid, the molar enthalpy of wetting can act as a measure of the otherwise vague notion of surface... 

Figure 2.14 Measured electrostatic double-layer and van der Waals forces between two surfaces of curved mica of radius 1 cm in (a) water and (b) dilute KNO3 and Ca(N03)2 solutions. The lines are the predictions of the DLVO theory with a Hamaker constant of 2.2 x 10 J in the limits of constant surface charge and constant surface potential here xfrQ = -(j/s, the particle surface potential. (The lines for constant surface charge are slightly higher than those for constant surface potential at small D.) The inset in (b) is the measured force in 0.1 M KNO3, which shows a force minimum at a distance of around 7 nm. Since this minimum in force occurs away from the deep minimum at the surface, it is called a secondary minimum. (From Israelachvili and Adams 1978 and Israelachvili 1992, reprinted with permission from Academic Press.)...

Zeta potentials of slun particles and wafer surfaces were measured to calculate the DLVO total interaction energy between them at various pHs. Instead of the Debye-Huckel low potential approximation, Overbeek s approximate was applied to the calculation. The repulsive energy was calculated between silica and TEOS wafers. Particle dip test also showed no deposition of particles on TEOS wafer. Due to the low cell constant of conductive W plate, it was not possible to measure the zeta potentials of W. The Hamaker constants of A1 and W were calculated and applied to the calculation of total interaction energy. The theoretical calculation was agreed well with the experimental results. The strong attractive interaction between metal surfaces and alumina particles were observed in both the calculation and experiments. 

Recall that we already derived a similar expression from the van der Waals theory under a number of restrictive simplifications, see (2.5.44 and 45]. There the geometric mean was related to the same mean of Hamaker constants. This equation can be tested experimentally for liquids like water, in which a variety of forces are operative, y can be established by measuring interfacial tensions against organic liquids in which the interaction is dominated by the dispersion forces. This analysis can be illustrated with the data of table 2.3. In (2.11.19] a is an organic liquid (like a hydrocarbon, he) for which it was assumed that only dispersion forces determined the surface tension y = Consequently, y" is the only unknown. Its value appears to be invariant at about 22 mJ m", comprising 30% of the total tension. 

Medout-Madere, V. (2000). A simple experimental way of measuring the Hamaker constant A,j of divided solids by immersion calorimetry in apolar liquids. J. Colloid Interface Sci., 228, 434-7. 

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