# Van der Waals force

van der Waals forces help geckos walk effortlessly along walls and ceilings, but this ability is mainly due to electrostatic interaction according to a recent study.
Rain water flux from a canopy. Among the forces that govern drop formation: van der Waals force, surface tension, cohesion, Plateau–Rayleigh instability.

In molecular physics, the van der Waals forces, named after Dutch scientist Johannes Diderik van der Waals, are distance-dependent interactions between atoms or molecules. Unlike ionic or covalent bonds, these attractions are not a result of any chemical electronic bond, and they are comparatively weak and more susceptible to being perturbed. Van der Waals forces quickly vanish at longer distances between interacting molecules.

Van der Waals forces play a fundamental role in fields as diverse as supramolecular chemistry, structural biology, polymer science, nanotechnology, surface science, and condensed matter physics. Van der Waals forces also define many properties of organic compounds and molecular solids, including their solubility in polar and non-polar media.

If no other forces are present, the point at which the force becomes repulsive rather than attractive as two atoms near one another is called the van der Waals contact distance. This results from the electron clouds of two atoms unfavorably coming into contact.[1] It can be shown that van der Waals forces are of the same origin as that of the Casimir effect, arising from quantum interactions with the zero-point field.[2] The term "van der Waals forces" is also sometimes used loosely as a synonym for the totality of intermolecular forces.[3] The term always includes the force between instantaneously induced dipoles (London dispersion force), sometimes includes the force between a permanent dipole and a corresponding induced dipole (Debye force), and – less frequently – includes the force between permanent dipoles (Keesom force).

## Definition

Van der Waals forces include attraction and repulsions between atoms, molecules, and surfaces, as well as other intermolecular forces. They differ from covalent and ionic bonding in that they are caused by correlations in the fluctuating polarizations of nearby particles (a consequence of quantum dynamics[4]).

Being the weakest of the weak chemical forces, with a strength between 0.4 and 4kJ/mol, they may still support an integral structural load when multitudes of such interactions are present. Such a force results from a transient shift in electron density. Specifically, the electron density may temporarily shift more greatly to one side of the nucleus. This generates a transient charge to which a nearby atom can be either attracted or repelled. When the interatomic distance of two atoms is greater than 0.6 nm the force is not strong enough to be observed. In the same vein, when the interatomic distance is below 0.4 nm the force becomes repulsive.

Intermolecular forces have four major contributions:

1. A repulsive component resulting from the Pauli exclusion principle that prevents the collapse of molecules.
2. Attractive or repulsive electrostatic interactions between permanent charges (in the case of molecular ions), dipoles (in the case of molecules without inversion center), quadrupoles (all molecules with symmetry lower than cubic), and in general between permanent multipoles. The electrostatic interaction is sometimes called the Keesom interaction or Keesom force after Willem Hendrik Keesom.
3. Induction (also known as polarization), which is the attractive interaction between a permanent multipole on one molecule with an induced multipole on another. This interaction is sometimes called Debye force after Peter J.W. Debye.
4. Dispersion (usually named London dispersion interactions after Fritz London), which is the attractive interaction between any pair of molecules, including non-polar atoms, arising from the interactions of instantaneous multipoles.

Returning to nomenclature, different texts refer to different things using the term "van der Waals force". Some texts describe the van der Waals force as the totality of forces (including repulsion); others mean all the attractive forces (and then sometimes distinguish van der Waals-Keesom, van der Waals-Debye, and van der Waals-London).

All intermolecular/van der Waals forces are anisotropic (except those between two noble gas atoms), which means that they depend on the relative orientation of the molecules. The induction and dispersion interactions are always attractive, irrespective of orientation, but the electrostatic interaction changes sign upon rotation of the molecules. That is, the electrostatic force can be attractive or repulsive, depending on the mutual orientation of the molecules. When molecules are in thermal motion, as they are in the gas and liquid phase, the electrostatic force is averaged out to a large extent, because the molecules thermally rotate and thus probe both repulsive and attractive parts of the electrostatic force. Sometimes this effect is expressed by the statement that "random thermal motion around room temperature can usually overcome or disrupt them" (which refers to the electrostatic component of the van der Waals force). Clearly, the thermal averaging effect is much less pronounced for the attractive induction and dispersion forces.

The Lennard-Jones potential is often used as an approximate model for the isotropic part of a total (repulsion plus attraction) van der Waals force as a function of distance.

Van der Waals forces are responsible for certain cases of pressure broadening (van der Waals broadening) of spectral lines and the formation of van der Waals molecules. The London-van der Waals forces are related to the Casimir effect for dielectric media, the former being the microscopic description of the latter bulk property. The first detailed calculations of this were done in 1955 by E. M. Lifshitz.[5] A more general theory of van der Waals forces has also been developed.[6][7]

The main characteristics of van der Waals forces are:[8]

• They are weaker than normal covalent and ionic bonds.
• Van der Waals forces are additive and cannot be saturated.
• They have no directional characteristic.
• They are all short-range forces and hence only interactions between the nearest particles need to be considered (instead of all the particles). Van der Waals attraction is greater if the molecules are closer.
• Van der Waals forces are independent of temperature except dipole – dipole interactions.

In low molecular weight alcohols, the hydrogen-bonding properties of their polar hydroxyl group dominate other weaker van der Waals interactions. In higher molecular weight alcohols, the properties of the nonpolar hydrocarbon chain(s) dominate and determine their solubility.

## London dispersion force

London dispersion forces, named after the German-American physicist Fritz London, are weak intermolecular forces that arise from the interactive forces between instantaneous multipoles in molecules without permanent multipole moments. These forces dominate the interaction of non-polar molecules, and are often more significant than Keesom and Debye forces in polar molecules. London dispersion forces are also known as 'dispersion forces', 'London forces', or 'instantaneous dipole–induced dipole forces'. The strength of London dispersion forces is proportional to the polarizability of the molecule, which in turn depends on the total number of electrons and the area over which they are spread. Any connection between the strength of London dispersion forces and mass is coincidental.

## Van der Waals forces between macroscopic objects

For 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 R1 and R2 and with smooth surfaces was approximated in 1937 by Hamaker[9] (using London's famous 1937 equation for the dispersion interaction energy between atoms/molecules[10] as the starting point) by:

{\displaystyle {\begin{aligned}&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{aligned}}}

(1)

where A is the Hamaker coefficient, which is a constant (~10−19 − 10−20 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 R1, R2, and r (the distance between the surfaces): ${\displaystyle \ z=R_{1}+R_{2}+r}$.

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

${\displaystyle \ U(r;R_{1},R_{2})=-{\frac {AR_{1}R_{2}}{(R_{1}+R_{2})6r}}}$

(2)

The van der Waals force between two spheres of constant radii (R1 and R2 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,${\displaystyle \ F_{VW}(r)=-{\frac {d}{dr}}U(r)}$. This yields:

${\displaystyle \ F_{VW}(r)=-{\frac {AR_{1}R_{2}}{(R_{1}+R_{2})6r^{2}}}}$

(3)

The van der Waals forces between objects with other geometries using the Hamaker model have been published in the literature.[11][12][13]

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 interactions and retardation. A more rigorous approach accounting for these effects, called the "macroscopic theory" was developed by Lifshitz in 1956.[14] Langbein derived a much more cumbersome "exact" expression in 1970 for spherical bodies within the framework of the Lifshitz theory[15] while a simpler macroscopic model approximation had been made by Derjaguin as early as 1934.[16] Expressions for the van der Waals forces for many different geometries using the Lifshitz theory have likewise been published.

## Use by geckos and arthropods

Gecko climbing a glass surface

The ability of geckos – which can hang on a glass surface using only one toe – to climb on sheer surfaces has been for many years mainly attributed to the van der Waals forces between these surfaces and the spatulae, or microscopic projections, which cover the hair-like setae found on their footpads.[17][18] A later study suggested that capillary adhesion might play a role,[19] but that hypothesis has been rejected by more recent studies.[20][21][22]

A recent study has however shown that gecko adhesion is in fact mainly determined by electrostatic interaction (caused by contact electrification), not van der Waals or capillary forces.[23]

There were efforts in 2008 to create a dry glue that exploits the effect,[24] and success was achieved in 2011 to create an adhesive tape on similar grounds.[25] In 2011, a paper was published relating the effect to both velcro-like hairs and the presence of lipids in gecko footprints.[26]

Among the arthropods, some spiders have similar setae on their scopulae or scopula pads, enabling them to climb or hang upside-down from extremely smooth surfaces such as glass or porcelain.[27][28]

## In modern technology

In May 2014, DARPA demonstrated the latest iteration of its Geckskin by having a 100 kg researcher (saddled with 20 kg of recording gear) scale an 8-metre-tall (26 ft) glass wall using only two climbing paddles. Tests are ongoing, but DARPA hopes one day to make the technology available for military use, giving soldiers Spider-Man-like abilities in urban combat.[29]

## References

1. ^ Garrett, Reginald H.; Grisham, Charles M. (2016). Biochemistry (sixth ed.). University of Virginia. pp. 12–13.
2. ^ G. L. Klimchitskaya, V. M. Mostepanenko. 2015. Casimir and van der Waals forces: Advances and problems. https://arxiv.org/ftp/arxiv/papers/1507/1507.02393.pdf
3. ^ Tschumper, Gregory S. (2008-01-01). Lipkowitz, Kenny B.; Cundari, Thomas R., eds. Reviews in Computational Chemistry. John Wiley & Sons. pp. 39–90. doi:10.1002/9780470399545.ch2. ISBN 9780470399545.
4. ^ A.A. Abrikosov; L.P. Gorkov; I.E. Dzyaloshinsky (1963–1975). Methods of Quantum Field Theory in Statistical Physics. Dover Publications. ISBN 0-486-63228-8. Chapter 6 Electromagnetic Radiation in an Absorbing Medium
5. ^ For further investigation, one may consult the University of St. Andrews' levitation work in a popular article: Science Journal: New way to levitate objects discovered, and in a more scholarly version: New Journal of Physics: Quantum levitation by left-handed metamaterials, which relate the Casimir effect to the gecko and how the reversal of the Casimir effect can result in physical levitation of tiny objects.
6. ^ Dzyaloshinskii, I E; Lifshitz, E M; Pitaevskii, Lev P (1961). "General theory of van der waals' forces". Soviet Physics Uspekhi. 4 (2): 153. Bibcode:1961SvPhU...4..153D. doi:10.1070/PU1961v004n02ABEH003330.
7. ^ Zheng, Y.; Narayanaswamy, A. (2011). "Lifshitz Theory of van der Waals Pressure in Dissipative Media". Phys. Rev. A. 83: 042504. arXiv:. Bibcode:2011PhRvA..83d2504Z. doi:10.1103/PhysRevA.83.042504.
8. ^ Chemical Bonding Ms Shethi and M. Satake
9. ^ H. C. Hamaker, Physica, 4(10), 1058–1072 (1937)
10. ^ F. London, Transactions of the Faraday Society, 33, 8–26 (1937)
11. ^ R. Tadmor (March 2001). "The London-van der Waals interaction energy between objects of various geometries". Journal of Physics: Condensed Matter. 13 (9): L195–L202. Bibcode:2001JPCM...13L.195T. doi:10.1088/0953-8984/13/9/101.
12. ^ Israelachvili J. (1985–2004). Intermolecular and Surface Forces. Academic Press. ISBN 0-12-375181-0.
13. ^ V. A. Parsegian (2006). van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists. Cambridge University Press. ISBN 978-0-521-83906-8.
14. ^ E. M. Lifshitz, Soviet Phys. JETP, 2, 73 (1956)
15. ^ D. Langbein, Phys. Rev. B, 2, 3371 (1970)
16. ^ B. V. Derjaguin, Kolloid-Z., 69, 155–64 (1934)
17. ^ Russell, Anthony P.; Higham, Timothy. E. (2009). "A new angle on clinging in geckos: incline, not substrate, triggers the deployment of the adhesive system". Proceedings of the Royal Society B: Biological Sciences. 276 (1673): 3705–3709. doi:10.1098/rspb.2009.0946. ISSN 0962-8452. PMC . PMID 19656797.
18. ^ Autumn, Kellar; Sitti, Metin; Liang, Yiching A.; Peattie, Anne M.; Hansen, Wendy R.; Sponberg, Simon; Kenny, Thomas W.; Fearing, Ronald; Israelachvili, Jacob N.; Full, Robert J. (2002). "Evidence for van der Waals adhesion in gecko setae". Proceedings of the National Academy of Sciences. 99 (19): 12252–6. Bibcode:2002PNAS...9912252A. doi:10.1073/pnas.192252799. PMC . PMID 12198184.
19. ^ Huber, Gerrit; Mantz, Hubert; Spolenak, Ralph; Mecke, Klaus; Jacobs, Karin; Gorb, Stanislav N.; Arzt, Eduard (2005). "Evidence for capillarity contributions to gecko adhesion from single spatula nanomechanical measurements". Proceedings of the National Academy of Sciences. 102 (45): 16293–6. Bibcode:2005PNAS..10216293H. doi:10.1073/pnas.0506328102. PMC . PMID 16260737.
20. ^ Chen, Bin; Gao, Huajian (2010). "An alternative explanation of the effect of humidity in gecko adhesion: stiffness reduction enhances adhesion on a rough surface". Int JAppl Mech. 2: 1–9. Bibcode:2010IJAM...02....1C. doi:10.1142/s1758825110000433.
21. ^ Puthoff, Jonathan B.; Prowse, Michael S.; Wilkinson, Matt; Autumn, Kellar (2010). "Changes in materials properties explain the effects of humidity on gecko adhesion". J Exp Biol. 213 (21): 3699–3704. doi:10.1242/jeb.047654. PMID 20952618.
22. ^ Prowse, Michael S.; Wilkinson, Matt; Puthoff, Michael; Mayer, George; Autumn, Kellar (February 2011). "Effects of humidity on the mechanical properties of gecko setae". Acta Biomaterialia. 7 (2): 733–738. doi:10.1016/j.actbio.2010.09.036. PMID 20920615.
23. ^ Izadi, H.; Stewart, K. M. E.; Penlidis, A. (9 July 2014). "Role of contact electrification and electrostatic interactions in gecko adhesion". Journal of The Royal Society Interface. 11 (98): 20140371. doi:10.1098/rsif.2014.0371. PMC . We have demonstrated that it is the CE-driven electrostatic interactions which dictate the strength of gecko adhesion, and not the van der Waals or capillary forces which are conventionally considered as the main source of gecko adhesion.
24. ^ Steenhuysen, Julie (8 October 2008). "Gecko-like glue is said to be stickiest yet". Reuters. Retrieved 5 October 2016.
25. ^ Quick, Darren (6 November 2011). "Biologically inspired adhesive tape can be reused thousands of times". New Atlas. Retrieved 5 October 2016.
26. ^ Hsu, Ping Yuan; Ge, Liehui; Li, Xiaopeng; Stark, Alyssa Y.; Wesdemiotis, Chrys; Niewiarowski, Peter H.; Dhinojwala, Ali (24 August 2011). "Direct evidence of phospholipids in gecko footprints and spatula-substrate contact interface detected using surface-sensitive spectroscopy". Journal of the Royal Society Interface. 9 (69): 657–664. doi:10.1098/rsif.2011.0370. ISSN 1742-5689. PMC .
27. ^ Kesel, Antonia B.; Martin, Andrew; Seidl, Tobias (19 April 2004). "Getting a grip on spider attachment: an AFM approach to microstructure adhesion in arthropods". Smart Materials and Structures. 13 (3): 512–518. Bibcode:2004SMaS...13..512K. doi:10.1088/0964-1726/13/3/009. ISSN 0964-1726.
28. ^ Wolff, Jonas O.; Gorb, Stanislav N. (7 January 2012). "The influence of humidity on the attachment ability of the spider Philodromus dispar (Araneae, Philodromidae)". Proceedings of the Royal Society B. 279 (1726): 139–143. doi:10.1098/rspb.2011.0505. PMC  – via rspb.royalsocietypublishing.org.
29. ^ Andrew Tarantola (June 2014). "DARPA's Gecko-Inspired Gloves Let Anyone Climb Up Flat Walls". Gizmodo. Retrieved 5 October 2016.