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Van der Waals Forces vs Casimir forces Desireé Cotto Figueroa Department of Physics & Astronomy Ohio University ABSTRACT visible phenomena. He was the first to suggest This paper is a summary of the basic concepts how the dispersion forces might arise. The and some applications of the Van der Waals Dipole-dipole forces, are also called “Keesom forces and the Casimir forces forces”. Willem Hendrik Keesom (1876 – 1956) was a Dutch physicist who, in 1926, invented a method to solidify helium. In 1921 he produced INTRODUCTION the first mathematical description of the dipole- The weak attractive forces between atoms or molecules, Van der Waals forces, were named in honor of Johannes van der Waals since he was one of the first to postulate an intermolecular force. Van der Waals (1837-1923), a Dutch physicist, received the Nobel prize in physics in 1910 for his research on the gaseous and liquid states of matter in which he developed a formula for the continuity of all gases, the van der Waals equation. There are two types of Van der Waals forces : Dispersion forces and Dipole-Dipole forces. The Dispersion forces are also known as “London Forces”. Fritz Wolfgang London (1900-1954), a German-born American physicist, in the 1930s began to explore the ways in which quantum principles could be used to explain dipole interactions. The other main force that we are interested in is the Casimir force. Have you asked yourself what would happen if you arrange two mirrors facing each other in empty space? Your first thought would probably be "nothing at all". But it results to be that the simple presence of the vacuum made the two mirrors mutually attracted to each other. This phenomenon was first predicted in 1948 by the Dutch physicists Hendrik B. G. Casimir (1909-2000) and Dirk Polder (1919-2001) while participating in a research at Philips Research Labs. The phenomenon is known as the Casimir effect, while the force between the mirrors is known as the Casimir-Polder force. VAN DER WAALS FORCES Dipole-Dipole Forces: Dispersion Forces: These exist between non-polar molecules or atoms. They are the weakest of all the intermolecular forces. On average the negative charge of the electrons in an atom or molecule is spread evenly. For brief periods of time the electrons are concentrated on one side of the atom or molecule more than the other. This gives the atom or molecule a temporary partial negative charge δ-, a temporary dipole moment. This dipole moment will induce a temporary dipole in Fig.2 The electron-rich region is colored red; the electron-poor hydrogen atom is shown in blue. The molecules align when they pass close to each other because the positive end of one molecule is attracted to the negative end of the other. The yellow glow a neighboring atom by attracting or repelling its indicates the formation of a weak intermolecular electron charge cloud (Fig. 1). A fraction of a attraction during a close encounter., the dipole-dipole second later the electron distribution changes force. causing the temporary dipole-dipole attraction to break. Dipole Dipole forces occur in polar molecules, that is, molecules that have an unequal sharing of electrons. In a system composed of polar molecules, the partial positive on one molecule will be attracted to the partial negative on a second molecule. For example, HCl comprised of the atom Hydrogen and Chlorine is polar. The Chlorine atom has an extra electron, which came Fig.1 The electrons get piled up on one side of the atom. This creates a temporary polarity in the atom. A temporary polarity in the neighboring atom is also then induced when its electrons are repelled by the first The temporarily induced polarity allows the two from the hydrogen atom. Because of this, the chlorine part of the molecule is negatively charged, and the hydrogen side of the molecule is positively charged. So in a solution where there atoms to be attracted to each other very weakly, when are thousands of these molecules around that are the partially negative end of the first atom is attracted slightly charged on each side, the molecules to the partially positive end of the neighboring atom. naturally orient themselves. The positive part of one molecule will move until it is next to the 0 The lowest energy solution of H is just the negative part of a neighboring molecule. These product of the ground states of the atoms A, B, forces between molecules tend to make them that is, 'stick' together. 0 100 A 100 B . Van der Waals interaction: Now for a large R ( R >> ao) we may expand the We can calculate the long-range interaction, or denominators in powers of rA / R, rB / R . Van der Waals force, between two hydrogen first non-vanishing terms are second order: atoms in their ground states using pertubation theory. The energy between the two hydrogen V e 2 rA rB atoms is attractive and varies as r-6 for large R1 e 2 The r r 3 rA R rB R A 3B R5 R At large separation the interaction between the separation. atoms is just a dipole-dipole interaction. Taking now the z-axis in the direction R , this interaction becomes: e2 xA xB y A yB 2 z A zB R3 Now lets assume that the atom A an d the atom B V are in the states |nA> and |nB> respectively. Then Fig 3. Two hydrogen atoms with their protons (+) the unperturbed energy of the two atoms will be separated by a fixed distance r and their electrons (-) EnA +EnB. The first-order correction to the at displacements ri (i=A,B) from them. ground state energy of the two-atom system from Consider the interaction between two hydrogen the atoms as in fig.3. H 1 V and | n 0 > = |nA nB >. To the first order Then the Hamiltonian is H H 0 V , where H0 2 2m interaction is En1 n 0 H 1 n 0 the energy will be : E = EnA +EnB. + 2A 2B e2 e2 rA rB where Now, if we look at the at the interaction term we have and the electrostatic interaction between the two atoms is : e2 e2 e2 e2 V R R rB rA R rB R rA , . Every term in the expression above is zero because the ground widely separated hydrogen in their ground states. states are spherically This potential is known as a Van der Waals symmetric. Therefore the first correction to the interaction. Even though the hydrogen atom has energy appears in second order. To second order, no permanent electric dipole moment in its the energy correction is given by: ground state, a second atom near to the first one will induce a dipole moment on the first one. CASIMIR FORCES In the classical everyday sense we think of a where m is not equal to n. vacuum as what is left after we have removed all of the stuff, molecules atoms etc. But that still We can notice from this interaction that if both leaves photons, so if we remove those as well, sums are in their ground states then the second including all the thermal energy then surely we order correction is negative. We can also notice should now have an absolute vacuum which that the interaction energy falls off as the inverse contains precisely nothing and is featureless. sixth power of the separations of the atoms. If However, in quantum mechanics , the vacuum is we define ζ as : a much more complex entity. It is far from featureless and far from empty. Every field have fluctuations, specially the electromagnetic fields. This means that at any moment their values varies around a constant, mean value. Even a ζ is a dimensionless constant with a value of 6.5 perfect vacuum at absolute zero has fluctuating from detailed calculations for the case of two fields known as "vacuum fluctuations" which hydrogen atoms in their ground state. Then the have observable consequences. An example of a total energy of the two atoms is given by: consequence of the vacuum fluctuations is an atom that is not in the ground state and return to it by emitting a photon. If you try to hold a . pencil upright on the end of your finger it will The second order energy acts as an effective stay there if your hand is stable and nothing potential energy for the interaction between two pertubs the equilibrium. Once there is a pertubation the pencil will fall into a more stable and they also exert pressure on surfaces which position. Similarly, vacuum fluctuations cause an increases with the energy of the electromagnetic excited atom to fall into its ground state and the field and hence its frequency. Casimir force is the most famous mechanical pressure inside the cavity is stronger than outside effect of this fluctuations at a cavity-resonance frequency, therefore the mirrors are pushed apart. The radiation In the other hand the mirrors are drawn towards each other when te radiation pressure inside the cavity is smaller than outside, which happens with a frequency out of resonance. It results to be that the attractive components, on balance, have a slightly stronger impact that the repulsive components. The Casimir force is attractive for two perfect, plane, parallel mirrors, therefore the mirrors are pulled together. The Casimir force can be measured for Fig.4 mirrors that are within microns of each other, for Lets consider the gap between two plane mirrors larger distance it is too small. For two plane in All parallel metallic plates of area of 1cm2 separated electromagnetic fields have a characteristic by a distance of 1 µm, the value of the attractive spectrum. And in a free vacuum there is not a force is approximately 10-7N. This force is quite frequency that is more important than the rest. small but the Casimir force becomes the strongest But the situation is different inside the cavity, force between two neutral objects at distances where the field is reflected back and forth below a micrometre. The net attractive force per between the mirrors. The field is amplified if unit area between two plates is given by: the figure above as a cavity. integer multiples of half a wavelength (which corresponds to a "cavity resonance") can fit exactly inside the cavity ad it is suppressed at other wavelengths. Another important physical quantity is the field radiation pressure. The vacuum field and every other field carries energy All electromagnetic fields can propagate in space where c is the speed of light, is Planck constant/2p, and d is the separation between the plates. When the separation between the surfaces decreases, the Casimir pressure increases rapidly, reaching about 1 atmosphere at d~ 10 nm.. Measurements of the Casimir Force Technology in Stockholm, Sweden,. He Although Casimir proposed his effect in 1948 it measured the force between two gold-coated was very difficult to measure using the cylinders and his results agreed to within 1% of equipment of that time. One of the first the theory values (see fig. 6). experiments that showed evidence for the Casimir force was carried out in 1958 by Marcus Spaarnay at Philips in Eindhoven. He investigated the Casimir force between two flat, metallic mirrors made from either aluminium, chromium or steel, but experimental uncertainties were too large for a quantitative verification of the effect. It was not until 1997 that S. K. Lamoreaux measured the force using a torsion al pendulum. He found that his experimental measurements agreed with theory to an accuracy of 5%. The following year the casimir force was confirmed by U Mohideen and Anushree Roy, using an atomic force microscope. They were able to measure the Casimir force to within 1% of the expected theoretical value (see fig. 5). In both cases, one of the surfaces was chosen to be Fig 5. (http://physicsweb.org/articles/world/15/9/6) spherical to avoid the problem of keeping two flat The experiment of U Mohideen and Anushree Roy surfaces huge measures the Casimir force between a metallized plate The and a metallized sphere fixed to the tip of the Casimir force when one plate is spherical is given cantilever of an atomic force microscope. When the parallel which cause the uncertainty in the experiment of 1958. by : sphere is brought near to the plate, an attractive Casimir force causes the cantilever to bend. This bending is monitored by bouncing a laser off the top of the cantilever and using photodiodes to record the where R is the radius of the spherical surface. reflected light. The electron micrograph shows a Another experiment using an atomic force metallized sphere attached to the triangular cantilever microscope to study the Casimir effect was made tip of an atomic force microscope. by Thomas Ederth at the Royal Institute of Carugno, and Roberto Onofrio at the University of Padova in Italy. They measured the force between a rigid chromium-coated plate and the flat surface of a cantilever made from the same material. Their result for the Casimir force agreed to within 15% of the expected theoretical value. Fig.6 (http://physicsweb.org/articles/world/15/9/6) This experiment of Thomas Ederth measures the As we can see the percent increases because of the difficulties involved in the experiment. Casimir force between two gold-coated cylinders positioned at right angles to one another. The upper cylinder can be lowered using the piezoelectric tube, VAN DER WAALS FORCES & CASIMIR which changes shape when a voltage is applied. The FORCES lower cylinder is mounted on a piezoelectric deflection sensor (known as a bimorph spring) that Initially the Casimir force was thought to be generates a charge when it is bent. When the two similar to the van der Waals force. The Van der cylinders are close together, the Casimir force causes Waals force results from the fluctuating dipole the lower cylinder to be attracted to the upper one, moment of the materials involved. thereby deflecting the spring in the process. The related to the Casimir effect for dielectric media. linearly variable displacement transducer (LVDT) monitors the nonlinear expansion of the piezotube. However, only a few number of recent experiments have measured the Casimir force using the original configuration. This configuration is more difficult because the They are The first detailed calculations of this were done in 1955 by E. M. Lifshitz. He generalized the Van der Waals force between two extended bodies as the force between fluctuating dipoles induced by the zero point electromagnetic fields The Casirmir force : mirrors have to be kept perfectly parallel during the experiment. It is more easier to bring a , sphere close up to a mirror because the separation is obtained by letting the dielectric constant € in between the mirror and the sphere is just the the Lifshitz theory approach infinity. It is also distance recent important to notice that the Van der Waals force experiment that tried to replicate Casimir's is always attractive, whereas the sign of the original set-up was carried out by Gianni Casimir force depends on the geometry. of closest approach. One For example, if a thin spherical conducting shell is http://antoine.frostburg.edu/chem/senese/ cut in half, the two hemispheres will experience a 101/liquids/faq/h-bonding-vs-london- mutual repulsive force. The Casimir force and forces.shtml the Van der Waals forces are quite different. http://www.ithacasciencezone.com/chemz one/lessons/03bonding/mleebonding/van_ der_waals_forces.htm REFERENCES Modern Quantum Mechanics, J.J. Sakurai (323-324). Lectures on Quantum Mechanics, Baym, (234-237). The Casimir effect: a force from nothing. Available at : http://physicsweb.org/articles/world/15/9/6 Van der Waals forces between atoms Available at: http://galileo.phys.virginia.edu/classes/752.mf 1i.spring03/vanderWaals.htm H. B. Chan et al. 1941 (2001); 291 Science U. Mohideen and Anushree Roy; Phys. Rev. Lett. 81, 4549 - 4552 (1998) T. H. Boyer; Phys. Rev. 174, 1764 (1968). G. L. Klimchitskaya , U. Mohideen and V. M. Mostepanenko;Phys. Rev. A 61, 062107 (2000) S. K. Lamoreaux; Phys. Rev. Lett. 78, 5 8 (1997) http://www.wikipedia.org