Editing Van der Waals force (section)

==Van der Waals forces between macroscopic objects==
For [[macroscopic scale|macroscopic]] bodies with known volumes and numbers of atoms or molecules per unit volume, the total van der Waals force is often computed based on the "microscopic theory" as the sum over all interacting pairs. It is necessary to integrate over the total volume of the object, which makes the calculation dependent on the objects' shapes. For example, the van der Waals interaction energy between spherical bodies of radii R<sub>1</sub> and R<sub>2</sub> and with smooth surfaces was approximated in 1937 by [[H. C. Hamaker|Hamaker]]<ref>H. C. Hamaker, ''Physica'', 4(10), 1058–1072 (1937)</ref>{{fcn|reason=article title?|date=February 2024}} (using London's famous 1937 equation for the dispersion interaction energy between atoms/molecules<ref>London, F. ''Transactions of the Faraday Society'' 33, 8–26 (1937)</ref>{{fcn|reason=article title?|date=February 2024}} as the starting point) by:

{{NumBlk|:|<math>\begin{align}
     &U(z;R_{1},R_{2}) = -\frac{A}{6}\left(\frac{2R_{1}R_{2}}{z^2 - (R_{1} + R_{2})^2} + \frac{2R_{1}R_{2}}{z^2 - (R_{1} - R_{2})^2} + \ln\left[\frac{z^2-(R_{1}+ R_{2})^2}{z^2-(R_{1}- R_{2})^2}\right]\right)
\end{align}</math>|{{EquationRef|1}}}}

where A is the [[Hamaker constant|Hamaker coefficient]], which is a constant (~10<sup>−19</sup> − 10<sup>−20</sup> J) that depends on the material properties (it can be positive or negative in sign depending on the intervening medium), and ''z'' is the center-to-center distance; i.e., the sum of ''R''<sub>1</sub>, ''R''<sub>2</sub>, and ''r'' (the distance between the surfaces): <math>\ z = R_{1} + R_{2} + r</math>.

The van der Waals ''[[force]]'' between two spheres of constant radii (''R''<sub>1</sub> and ''R''<sub>2</sub> are treated as parameters) is then a function of separation since the force on an object is the negative of the derivative of the potential energy function,<math>\ F_{\rm VdW}(z) = -\frac{d}{dz}U(z)</math>. This yields:

{{NumBlk|:|<math>\ F_{\rm VdW}(z)= -\frac{A}{6}\frac{64R_{1}^3R_{2}^3z}{[z^2-(R_{1}+R_{2})^2]^2[z^2-(R_{1}-R_{2})^2]^2}</math>|{{EquationRef|2}}}}

In the limit of close-approach, the spheres are sufficiently large compared to the distance between them; i.e., <math>\ r \ll R_{1}</math> or <math>R_{2}</math>, so that equation (1) for the potential energy function simplifies to:

{{NumBlk|:|<math>\ U(r;R_{1},R_{2})= -\frac{AR_{1}R_{2}}{(R_{1}+R_{2})6r}</math>|{{EquationRef|3}}}}

with the force:

{{NumBlk|:|<math>\ F_{\rm VdW}(r)= -\frac{AR_{1}R_{2}}{(R_{1}+R_{2})6r^2}</math>|{{EquationRef|4}}}}

The van der Waals forces between objects with other geometries using the Hamaker model have been published in the literature.<ref>{{cite journal |first=R. |last=Tadmor|title=The London–Van der Waals interaction energy between objects of various geometries |journal=[[Journal of Physics: Condensed Matter]]|volume=13|issue=9 |date=March 2001|pages=L195–L202|doi=10.1088/0953-8984/13/9/101|bibcode=2001JPCM...13L.195T|s2cid=250790137 }}</ref><ref>{{cite book|author=Israelachvili J.|title=Intermolecular and Surface Forces|publisher=[[Academic Press]] |date=1985–2004|isbn=978-0-12-375181-2}}</ref><ref>{{cite book|first=V. A. |last=Parsegian |title=Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists |publisher=[[Cambridge University Press]] |date=2006 |isbn=978-0-521-83906-8}}</ref>

From the expression above, it is seen that the van der Waals force decreases with decreasing size of bodies (R). Nevertheless, the strength of inertial forces, such as gravity and drag/lift, decrease to a greater extent. Consequently, the van der Waals forces become dominant for collections of very small particles such as very fine-grained dry powders (where there are no capillary forces present) even though the force of attraction is smaller in magnitude than it is for larger particles of the same substance. Such powders are said to be cohesive, meaning they are not as easily fluidized or pneumatically conveyed as their more coarse-grained counterparts. Generally, free-flow occurs with particles greater than about 250 μm.

The van der Waals force of adhesion is also dependent on the surface topography. If there are surface asperities, or protuberances, that result in a greater total area of contact between two particles or between a particle and a wall, this increases the van der Waals force of attraction as well as the tendency for mechanical interlocking.

The microscopic theory assumes pairwise additivity. It neglects [[Many-body problem|many-body interactions]] and [[Retarded potential|retardation]]. A more rigorous approach accounting for these effects, called the "[[Lifshitz Theory of van der Waals Force|macroscopic theory]]", was developed by [[Evgeny Lifshitz|Lifshitz]] in 1956.<ref>E. M. Lifshitz, ''Soviet Physics—JETP'', 2, 73 (1956)</ref>{{fcn|reason=article title?|date=February 2024}} [[D. Langbein|Langbein]] derived a much more cumbersome "exact" expression in 1970 for spherical bodies within the framework of the Lifshitz theory<ref>D. Langbein, ''Physical Review B'', 2, 3371 (1970)</ref>{{fcn|reason=article title?|date=February 2024}} while a simpler macroscopic model approximation had been made by [[Boris Derjaguin|Derjaguin]] as early as 1934.<ref>B. V. Derjaguin, ''Kolloid-Zeitschrift'', 69, 155–164 (1934)</ref>{{fcn|reason=article title?|date=February 2024}} Expressions for the van der Waals forces for many different geometries using the Lifshitz theory have likewise been published.