Hamaker coefficients

Let us first assume that we have a spherical particle with a radius of 5 p.m similar to an idealized toner particle, which is comprised of polystyrene, in contact with an electrically conducting substrate. A typical electric charge on a toner particle of that size is of the order of 10" " C. The Hamaker coefficient (Eq. 15) for such as system would be about 1.5 eV. 

The Hamaker coefficient. A, is a measure of the interaction and is dependent on the material of the particle as well as on the surrounding medium. Heavy atoms, which are generally more polarizable (i.e., the electron distribution can be more... 

The VDW interactions seem to have little effect on the rate of aggregation of small vesicles in a primary minimum. However, this statement may be made only because the magnitudes of Hamaker coefficients are less than 10 13 erg (5 X 10 14 erg), in contrast to much higher values frequently used in treatments in colloid science (3). Our estimates of VDW parameters for phospholipid vesicles are based on the analysis of a significant amount of recent data (33,42). 

The VDW interaction between two different substances, 1 and 2, immersed or dissolved in Liquid 3 will be repulsive when the Hamaker coefficients of the two substances An and A22 stand in either of the following relations to that of the liquid, A33 (4)... 

The situation is somewhat different for a solution, particularly if a surface-active component is present. The measured surface tension is strongly influenced by adsorption at the liquid-vapor surface (16). Yet adsorption does not influence the values of the Hamaker coefficients that must be used in Equations 3 and 10, which are related to those of the pure substances by the volume fraction weighted averages. Thus, for solution /... 

A = effective Hamaker coefficient d =3 separation distance d0 = equilibrium separation distance AF = free energy difference mol wt = molecular weight... 

Detailed HRTEM characterization of the specimens showed that all metal-ceramic interfaces in the two different nanocomposites had thin ( 1 nm thick) amorphous films (see Fig. 11.5). In addition, occluded particles were found inside the alumina grains which also had thin amorphous films at their interfaces with alumina. Analytical microscopy showed the films to contain Ca, Si, and Al.41 Hamaker coefficients were calculated for metal-ceramic interfaces in the presence of a Si02-based film, which indicated that a stronger attractive force is expected for intergranular films at metal-alumina interfaces,... 

Avishai, A. and Kaplan, W.D., Intergranular films at metal-ceramic interfaces Part II - Calculation of Hamaker coefficients , Acta. Mater., 2005 53(5) 1571-1581. 

Sampling frequencies 25 If I take that formula seriously, then two bubbles or even two pockets of vacuum will attract across a material body. How can two nothings do anything 26 Does charge-fluctuation resonance translate into specificity of interaction 26 How does retardation come in 27 Can there ever be a negative Hamaker coefficient and a positive charge-fluctuation energy 28... 

Table Pr.2. Typical estimates of Hamaker coefficients in the limit of small separation...

PROBLEM PR.2 Calculate the effective Hamaker coefficient between the spherical atom... 

For the real answer to this question, read past this Prelude. Here, to keep to familiar notation, write energies in terms of a Hamaker coefficient (but never Hamaker constant). In this language the interaction energy per unit area between parallel, infinitely thick walls A and B across a medium m of thickness l looks like -[AHam/(12jr/2)]. 

In a weak sense, yes, as long as we are careful to define "specificity." Imagine that A and B are two different materials. At each sampling frequency in the summation, compare the terms in the Hamaker coefficients for... 

Because the van der Waals interactions go as the negative of the Hamaker coefficients, the sum of the (negative) A — A and B — B interactions at a given separation is more negative than two A — B interactions at that same separation. 

Can there ever be a negative Hamaker coefficient and a positive charge-fluctuation energy ... 

For historical reasons, we write the interaction free energy G(Z) between the plane-parallel half-spaces of Fig. Ll.l in terms of a "Hamaker coefficient" AHam so as to put the interaction per unit area in a form... 

To first approximation, the Hamaker coefficient AAm/Bm(Z) for interactions between interfaces Am and Bm is... 

The integrated consequence of this retardation screening shows up as a change in the contribution to the Hamaker coefficient AAm/Bm(0- This diminution in AAm/BmfZ) looks different when plotted versus log(Z) (see Fig. LI.16) or plotted versus Z by itself (see Fig. LI. 17). 

Here the Hamaker coefficient is constant only over a very small range of separations. With an absorption wavelength of 500 A, there is a diminution in (/) by 50%... 

Table LI.3. Typical Hamaker coefficients, symmetric systems, retardation screening neglected...

From these summations, neglecting any retardation or ionic screening, the unscreened Hamaker coefficients AAm/Bm(/ = 0) (3Z T/2) 0 AAmABm emerge as the... 

How strong are these long-range van der Waals forces between semi-infinite bodies A and B across a medium m or across a vacuum For the four materials whose dielectric responses are plotted in the preceding section, the corresponding Hamaker coefficients (with the neglect of retardation) make an instructive table. See Table LI. 3. 

Rather than the one interaction GAm/Bim(Z) = — [AAm/Bjm /)]/12nl2 between one pair of interfaces across a distance Z, there are now two pairs of interacting surfaces, each pair with its own coefficient and its own distance of separation. Because there are three materials involved, the subscripts on the free energies and on the Hamaker coefficients are written to show which surfaces are correspondingly involved. For clarity, we use an outside-inside subscripting for the materials at the different interfaces. 

The evaluation of the two Hamaker coefficients is much as before with the following two exceptions ... 

Looking at the summations that make the Hamaker coefficients, we see that this Aeff has the succinct form... 

The generalization from these last two examples is straightforward. Between every pair of interfaces there is a separate term with its own Hamaker coefficient and inverse-... 

In the language of the Hamaker coefficient /l m/2m) which is restricted to situations in which e1( s2, and em are nearly equal, Gpp(Z) looks like the same expression already used several times for planes ... 

These simplified expressions depend on the assumption of a constant Hamaker coefficient with dielectric responses e1, e2, and em of similar magnitudes. 

This formula can also be written in Hamaker form with the Hamaker coefficient AAmBQt T) (see Table P.l.a.2) ... 

In fact, the graft is exceedingly helpful for geometries in which field equations of the modern theory are too difficult to solve but pairwise summation (actually integration) can be effected. The distance dependence of the interaction is taken from summation whereas the Hamaker coefficient is estimated with modern theory. To see how to connect old and new, consider the formal procedure for summation, then see its equivalence to a much-reduced version of the general theory. 

In this small-difference limit then, at which it is accurate to compute the Hamaker coefficient as... 

To show features at higher frequencies, the function Re[/cv0 R) multiplies s"( r) by whereas e(i ) depends on e"(r. These s(/ ) are then used to compute Hamaker coefficients (in 1 zj = 10-21 J) in the limit of no retardation for attraction across a vacuum13 or across water14 as in this table ... 

Even exhaustive full-spectral computations have their frustrating uncertainties. Compare these tabulated Hamaker coefficients with those in parentheses,15 quoted in the Prelude, which used earlier, slightly different, data and slightly different procedures16 to create e(/ ). The comparison reminds us to continue to search for the best data and to be aware of the unavoidable ambiguities due to limited data and to computational procedure. 

Temperature comes into computation two ways. First, there is the way temperature affects electromagnetic fluctuations, how variable T is handled in formulae. Second, changes in temperature actually affect spectral response. By measuring response at different temperatures, we can determine both these consequences of varied temperature. Figure L2.32 shows the response of AI2O3 at different temperatures.17 The nonretarded Hamaker coefficient for AI2O3 across vacuum goes from 145 zj at 300 K to 152 zj at 800 K and then down to 125 zj at T = 1925 K.14... 

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