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Friday, October 30, 2015

Kinetic theory of gases


From Wikipedia, the free encyclopedia


The temperature of an ideal monatomic gas is proportional to the average kinetic energy of its atoms. The size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have a certain, average speed, slowed down here two trillion fold from room temperature.
Kinetic theory defines temperature in its own way, not identical with the thermodynamic definition.[1]

Under a microscope, the molecules making up a liquid are too small to be visible, but the jittering motion of pollen grains or dust particles can be seen. Known as Brownian motion, it results directly from collisions between the grains or particles and liquid molecules. As analyzed by Albert Einstein in 1905, this experimental evidence for kinetic theory is generally seen as having confirmed the concrete material existence of atoms and molecules.

Assumptions

The theory for ideal gases makes the following assumptions:
  • The gas consists of very small particles known as molecules. This smallness of their size is such that the total volume of the individual gas molecules added up is negligible compared to the volume of the smallest open ball containing all the molecules. This is equivalent to stating that the average distance separating the gas particles is large compared to their size.
  • These particles have the same mass.
  • The number of molecules is so large that statistical treatment can be applied.
  • These molecules are in constant, random, and rapid motion.
  • The rapidly moving particles constantly collide among themselves and with the walls of the container. All these collisions are perfectly elastic. This means, the molecules are considered to be perfectly spherical in shape, and elastic in nature.
  • Except during collisions, the interactions among molecules are negligible. (That is, they exert no forces on one another.)
This implies:
1. Relativistic effects are negligible.
2. Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than the thermal de Broglie wavelength and the molecules are treated as classical objects.
3. Because of the above two, their dynamics can be treated classically. This means, the equations of motion of the molecules are time-reversible.
  • The average kinetic energy of the gas particles depends only on the absolute temperature of the system. The kinetic theory has its own definition of temperature, not identical with the thermodynamic definition.
  • The time during collision of molecule with the container's wall is negligible as compared to the time between successive collisions.
  • Because they have mass, the gas molecules will be affected by gravity.
More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions.

An important book on kinetic theory is that by Chapman and Cowling.[1] An important approach to the subject is called Chapman–Enskog theory.[2] There have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.[3] In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number.

Properties

Pressure and kinetic energy

Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V=L3. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is:
\Delta p = p_{i,x} - p_{f,x} = p_{i,x} - (-p_{i,x}) = 2 p_{i,x} = 2 m v_x\,
where vx is the x-component of the initial velocity of the particle.

The particle impacts one specific side wall once every
\Delta t = \frac{2L}{v_x}
(where L is the distance between opposite walls).

The force due to this particle is:
F = \frac{\Delta p}{\Delta t} = \frac{m v_x^2}{L}.
The total force on the wall is
F = \frac{Nm\overline{v_x^2}}{L}
where the bar denotes an average over the N particles. Since the assumption is that the particles move in random directions, we will have to conclude that if we divide the velocity vectors of all particles in three mutually perpendicular directions, the average value along each direction must be equal (though their proportions are arbitrary for individual particles). That is,
\overline{v_x^2} = \overline{v^2}/3 .
We can rewrite the force as
F = \frac{Nm\overline{v^2}}{3L}.
This force is exerted on an area L2. Therefore the pressure of the gas is
P = \frac{F}{L^2} = \frac{Nm\overline{v^2}}{3V}
where V=L3 is the volume of the box. The ratio n=N/V is the number density of the gas (the mass density ρ=nm is less convenient for theoretical derivations on atomic level). Using n, we can rewrite the pressure as
P = \frac{n m \overline{v^2}}{3}.
This is a first non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule {1 \over 2} m\overline{v^2} which is a microscopic property.

Temperature and kinetic energy

Rewriting the above result for the pressure as PV = {Nm\overline{v^2}\over 3} , we may combine it with the ideal gas law
\displaystyle PV = N k_B T ,




(1)
where \displaystyle k_B is the Boltzmann constant and \displaystyle T the absolute temperature defined by the ideal gas law, to obtain
k_B T = {m\overline{v^2}\over 3} ,
which leads to the expression of the average kinetic energy of a molecule,
\displaystyle \frac {1} {2} m\overline{v^2} = \frac {3} {2} k_B T.
The kinetic energy of the system is N times that of a molecule, namely K= \frac {1} {2} N m \overline{v^2} . Then the temperature \displaystyle T takes the form
\displaystyle T = {m\overline{v^2}\over 3 k_B}




(2)
which becomes
\displaystyle T = \frac {2} {3} \frac {K} {N k_B}.




(3)
Eq.(3) is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature. From Eq.(1) and Eq.(3), we have
\displaystyle PV = \frac {2} {3} K.




(4)
Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy.

Eq.(1) and Eq.(4) are called the "classical results", which could also be derived from statistical mechanics; for more details, see .[4]

Since there are \displaystyle 3N degrees of freedom in a monatomic-gas system with \displaystyle N particles, the kinetic energy per degree of freedom per molecule is
\displaystyle \frac {K} {3 N} = \frac {k_B T} {2}




(5)
In the kinetic energy per degree of freedom, the constant of proportionality of temperature is 1/2 times Boltzmann constant. In addition to this, the temperature will decrease when the pressure drops to a certain point.[why?] This result is related to the equipartition theorem.

As noted in the article on heat capacity, diatomic gases should have 7 degrees of freedom, but the lighter gases act as if they have only 5.

Thus the kinetic energy per kelvin (monatomic ideal gas) is:
  • per mole: 12.47 J
  • per molecule: 20.7 yJ = 129 μeV.
At standard temperature (273.15 K), we get:
  • per mole: 3406 J
  • per molecule: 5.65 zJ = 35.2 meV.....

Collisions with container

One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time.

Assuming an ideal gas, a derivation[5] results in an equation for total number of collisions per unit time per area:
A = \frac{1}{4}\frac{N}{V} v_{avg} = \frac{n}{4} \sqrt{\frac{8 k_{B} T}{\pi m}} . \,
This quantity is also known as the "impingement rate" in vacuum physics.

Speed of molecules

From the kinetic energy formula it can be shown that
v_\mathrm{rms} = \sqrt {{3 k_{B} T}\over{m}}
with v in m/s, T in kelvins, and m is the molecular mass (kg). The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (isotropic distribution of speeds).

Transport properties

The kinetic theory of gases deals not only with gases in thermodynamic equilibrium, but also very importantly with gases not in thermodynamic equilibrium. This means considering what are known as 'transport properties', such a viscosity and thermal conductivity.

History

In approximately 50 BCE, the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other.[6] This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant.


Hydrodynamica front cover

In 1738 Daniel Bernoulli published Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.[7]:36–37

Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),[8] Georges-Louis Le Sage (ca. 1780, published 1818),[9] John Herapath (1816)[10] and John James Waterston (1843),[11] which connected their research with the development of mechanical explanations of gravitation. In 1856 August Krönig (probably after reading a paper of Waterston) created a simple gas-kinetic model, which only considered the translational motion of the particles.[12]

In 1857 Rudolf Clausius, according to his own words independently of Krönig, developed a similar, but much more sophisticated version of the theory which included translational and contrary to Krönig also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. [13] In 1859, after reading a paper by Clausius, James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics.[14] In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases."[15] In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. Also the logarithmic connection between entropy and probability was first stated by him.

In the beginning of the twentieth century, however, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905)[16] and Marian Smoluchowski's (1906)[17] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.
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Thursday, October 29, 2015

Green Climate Fund


From Wikipedia, the free encyclopedia
Green Climate Fund
Emblem of the United Nations.svg
Green Climate Fund Logo.png
Abbreviation GCF
Formation 2010
Legal status Active
Headquarters Songdo, Incheon, South Korea
Website www.gcfund.org

The Green Climate Fund (GCF) is a fund within the framework of the UNFCCC founded as a mechanism to assist developing countries in adaptation and mitigation practices to counter climate change. The GCF is based in the new Songdo district of Incheon, South Korea. It is governed by a Board of 24 members and initially supported by a Secretariat.

‘The Green Climate Fund will support projects, programmes, policies and other activities in developing country Parties using thematic funding windows’.[1] It is intended to be the centrepiece of efforts to raise Climate Finance of $100 billion a year by 2020. This is not an official figure for the size of the Fund itself, however. Disputes also remain as to whether the funding target will be based on public sources, or whether "leveraged" private finance will be counted towards the total.[2] Only a fraction of this sum had been pledged as of July 2013, mostly to cover start-up costs.

According to the Climate & Development Knowledge Network, at the third meeting of the Board in Berlin, Germany, in March 2013, members agreed on how to move forward with the fund’s Business Model Framework (BMF). They identified the need to assess various options for how nations could access the fund, approaches for involving the private sector, plus ways to measure results and ensure requests for monies are country-driven.[3] At the fourth Board meeting in Songdo, South Korea, in June 2013, Hela Cheikhrouhou, a Tunisian national, was selected to become the Fund's first Executive Director.[4] "Resource mobilisation" (establishing a process for funding pledges) is expected to be the most contentious issue for the fifth Board meeting in Paris, France, in October 2013.[5]

History

The Copenhagen Accord, established during the 15th Conference Of the Parties (COP-15) in Copenhagen in 2009 mentioned the "Copenhagen Green Climate Fund". The fund was formally established during the 2010 United Nations Climate Change Conference in Cancun and is a fund within the UNFCCC framework.[6] Its governing instrument was adopted at the 2011 UN Climate Change Conference (COP 17) in Durban, South Africa.[7]

Organization

During COP-16 in Cancun, the matter of governing the GCF was entrusted to the newly founded Green Climate Fund Board, and the World Bank was chosen as the temporary trustee.[6] To develop a design for the functioning of the GCF, the ‘Transitional Committee for the Green Climate Fund’ was established in Cancun too. The committee met four times throughout the year 2011, and submitted a report to the 17th COP in Durban, South Africa. Based on this report, the COP decided that the ‘GCF would become an operating entity of the financial mechanism’ of the UNFCCC,[8] and that on COP-18 in 2012, the necessary rules should be adopted to ensure that the GCF ‘is accountable to and functions under the guidance of the COP’.[8] Researchers at the Overseas Development Institute state that without this last minute agreement on a governing instrument for the GCF, the "African COP" would have been considered a failure.[9] Furthermore, the GCF Board was tasked with developing rules and procedures for the disbursement of funds, ensuring that these should be consistent with the national objectives of the countries where projects and programmes will be taking place. The GCF Board was also charged with establishing an independent secretariat and the permanent trustee of the GCF.[8]

Sources of finance

The Green Climate Fund is intended to be the centrepiece of Long Term Financing under the UNFCCC, which has set itself a goal of raising $100 billion per year by 2020. Uncertainty over where this money would come from led to the creation of a High Level Advisory Group on Climate Change Financing (AGF) was founded by UN Secretary-General Ban Ki-Moon in February 2010.
There is no formal connection between this Panel and the GCF, although its report is one source for debates on "resource mobilisation" for the GCF, an item that will be discussed at the Fund's October 2013 Board meeting.[10]

The lack of pledged funds and potential reliance on the private sector is controversial and has been criticized by developing countries.[11]

Pledges to the fund reached $10.2 billion on May 28, 2015.[12]

Issues

The process of designing the GCF has raised several issues. These include ongoing questions on how funds will be raised,[13] the role of the private sector,[14] the level of "country ownership" of resources,[15] and the transparency of the Board itself.[16] In addition, questions have been raised about the need for yet another new international climate institution which may further fragment public dollars that are put toward mitigation and adaptation annually.[17]

The Fund is also pledged to offer "balanced" support to adaptation and mitigation, although there is some concern amongst developing countries that inadequate adaptation financing will be offered, in particular if the fund is reliant on "leveraging" private sector finance.[18]

Role of the private sector

One of the most controversial aspects of the GCF concerns the creation of the Fund's Private Sector Facility (PSF). Many of the developed countries represented on the GCF board advocate a PSF that appeals to capital markets, in particular the pension funds and other institutional investors that control trillions of dollars that pass through Wall Street and other financial centers. They hope that the Fund will ultimately use a broad range of financial instruments.[19]

However, several developing countries and non-governmental organizations have suggested that the PSF should focus on "pro-poor climate finance" that addresses the difficulties faced by micro-, small-, and medium-sized enterprises in developing countries. This emphasis on encouraging the domestic private sector is also written into the GCF’s Governing Instrument, its founding document.[20]

Additionality of funds

The Cancun agreements clearly specify that the funds provided to the developing countries as climate finance, including through the GCF, should be ‘new’ and ‘additional’ to existing development aid.[6]
The condition of funds having to be new means that pledges should come on top of those made in previous years. As far as additionality is concerned, there is no strict definition of this term, which has already led to serious problems in evaluating the additionality of emission reductions through CDM-projects, leading to counterproductivity, and even fraud.[21][22]

A lack of stakeholder involvement

Using the money in the right way in order to enforce actual change on the ground is one of the biggest challenges ahead. Many academics argue that, in order to do this in an efficient way, all stakeholders should be involved in the process, instead of using a top-down approach. They point to that fact that, without their input, it is harder to achieve targets set. Moreover, projects often even miss out on their actual purpose.[18][23][24][25][26] A group of researchers associated with the Australian National University,[27] call for the foundation of so-called ‘National Implementing Entities’ (NIE) in each country, that would become responsible for ‘the implementation of sub-national projects’.[27] This would avoid national governments getting too involved, because in the past, they ‘often hindered the flow of international support to subnational scale reform for sustainable development’.[27] Overall, this view on the need for more stakeholder involvement can be framed within the movement in environmental governance calling for a shift from traditional ways of government to governance.[28] The Climate & Development Knowledge Network is funding a research project that aims to help the GCF Board, by analysing how best to allocate resources among countries. The project will research and present four case studies of how federal or central government money is presently distributed to sub-national entities. Chosen for the diversity in their underlying political systems, these are: China, India, Switzerland and the USA.[29]

Failure to ban fossil fuel funding under climate finance

At its board meeting in South Korea held in March 2015, the GCF refused an explicit ban on fossil fuel projects. Japan, China, and Saudi Arabia were opposing the ban. “It’s like a torture convention that doesn’t forbid torture,” Karen Orenstein, a campaigner for Friends of the Earth US who attended the meeting told the Guardian. “Honestly it should be a no-brainer at this point.”[30][31]

Accredited entities

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Wednesday, October 28, 2015

van der Waals force


From Wikipedia, the free encyclopedia


Geckos can stick to walls and ceilings because of Van der Waals forces; see the section below.

In physical chemistry, the van der Waals forces (or van der Waals' interaction), named after Dutch scientist Johannes Diderik van der Waals, is the sum of the attractive or repulsive forces between molecules (or between parts of the same molecule) other than those due to covalent bonds, or the electrostatic interaction of ions with one another, with neutral molecules, or with charged molecules.[1] The resulting van der Waals forces can be attractive or repulsive.[2]

The term includes:
It is also sometimes used loosely as a synonym for the totality of intermolecular forces. Van der Waals forces are relatively weak compared to covalent bonds, but 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 define many properties of organic compounds, including their solubility in polar and non-polar media.

In low molecular weight alcohols, the hydrogen-bonding properties of the 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 define the solubility. Van der Waals forces quickly vanish at longer distances between interacting molecules.

In 2012, the first direct measurements of the strength of the van der Waals force for a single organic molecule bound to a metal surface was made via atomic force microscopy and corroborated with density functional calculations.[3]

Definition


Attractive interactions resulting from dipole-dipole interaction of two hydrogen chloride molecules

Van der Waals forces include attractions 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]).

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 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 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 nearest need to be considered instead of all the particles. The greater is the attraction if the molecules are closer due to Van der Waals forces.
  • Van der Waals forces are independent of temperature except dipole - dipole interactions.

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:
\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}




(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): \ 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., \ r \ll R_{1} or R_{2}, so that equation (1) for the potential energy function simplifies to:
\ 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,\ F_{VW}(r) = -\frac{d}{dr}U(r). This yields:
\ 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 easily 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 spiders


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 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] There were efforts in 2008 to create a dry glue that exploits the effect,[23] and success was achieved in 2011 to create an adhesive tape on similar grounds.[24] In 2011, a paper was published relating the effect to both velcro-like hairs and the presence of lipids in gecko footprints.[25]

Some spiders have convergently evolved 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.[26]

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 8m tall glass wall using only two climbing paddles. Tests are ongoing, but DARPA hopes one day to make the technology available for military use.
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Dipole -- More on what causes a greenhouse gas


From Wikipedia, the free encyclopedia


The Earth's magnetic field, approximated as a magnetic dipole. However, the "N" and "S" (north and south) poles are labeled here geographically, which is the opposite of the convention for labeling the poles of a magnetic dipole moment.

In physics, there are several kinds of dipole:
  • An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some (usually small) distance. A permanent electric dipole is called an electret.
  • A magnetic dipole is a closed circulation of electric current. A simple example of this is a single loop of wire with some constant current through it.[1][2]
  • A current dipole is a current from a sink of current to a source of current within a (usually conducting) medium. Current dipoles are often used to model neuronal sources of electromagnetic fields that can be measured using Magnetoencephalography or Electroencephalography.
Dipoles can be characterized by their dipole moment, a vector quantity. For the simple electric dipole given above, the electric dipole moment points from the negative charge towards the positive charge, and has a magnitude equal to the strength of each charge times the separation between the charges. (To be precise: for the definition of the dipole moment, one should always consider the "dipole limit", where e.g. the distance of the generating charges should converge to 0, while simultaneously the charge strength should diverge to infinity in such a way that the product remains a positive constant.)

For the current loop, the magnetic dipole moment points through the loop (according to the right hand grip rule), with a magnitude equal to the current in the loop times the area of the loop.

In addition to current loops, the electron, among other fundamental particles, has a magnetic dipole moment. This is because it generates a magnetic field that is identical to that generated by a very small current loop. However, the electron's magnetic moment is not due to a current loop, but is instead an intrinsic property of the electron.[3] It is also possible that the electron has an electric dipole moment, although this has not yet been observed (see electron electric dipole moment for more information).


Contour plot of the electrostatic potential of a horizontally oriented electrical dipole of finite size. Strong colors indicate highest and lowest potential (where the opposing charges of the dipole are located).

A permanent magnet, such as a bar magnet, owes its magnetism to the intrinsic magnetic dipole moment of the electron. The two ends of a bar magnet are referred to as poles (not to be confused with monopoles), and may be labeled "north" and "south". In terms of the Earth's magnetic field, these are respectively "north-seeking" and "south-seeking" poles, that is if the magnet were freely suspended in the Earth's magnetic field, the north-seeking pole would point towards the north and the south-seeking pole would point twards the south. The dipole moment of the bar magnet points from its magnetic south to its magnetic north pole. The north pole of a bar magnet in a compass points north. However, this means that Earth's geomagnetic north pole is the south pole (south-seeking pole) of its dipole moment, and vice versa.

The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanical spin since the existence of magnetic monopoles has never been experimentally demonstrated.

The term comes from the Greek δίς (dis), "twice"[4] and πόλος (pòlos), "axis".[5][6]

Classification


Electric field lines of two opposing charges separated by a finite distance.

Magnetic field lines of a ring current of finite diameter.

Field lines of a point dipole of any type, electric, magnetic, acoustic, …

A physical dipole consists of two equal and opposite point charges: in the literal sense, two poles. Its field at large distances (i.e., distances large in comparison to the separation of the poles) depends almost entirely on the dipole moment as defined above. A point (electric) dipole is the limit obtained by letting the separation tend to 0 while keeping the dipole moment fixed. The field of a point dipole has a particularly simple form, and the order-1 term in the multipole expansion is precisely the point dipole field.

Although there are no known magnetic monopoles in nature, there are magnetic dipoles in the form of the quantum-mechanical spin associated with particles such as electrons (although the accurate description of such effects falls outside of classical electromagnetism). A theoretical magnetic point dipole has a magnetic field of exactly the same form as the electric field of an electric point dipole. A very small current-carrying loop is approximately a magnetic point dipole; the magnetic dipole moment of such a loop is the product of the current flowing in the loop and the (vector) area of the loop.

Any configuration of charges or currents has a 'dipole moment', which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration. This is simply one term in the multipole expansion when the total charge ("monopole moment") is 0 — as it always is for the magnetic case, since there are no magnetic monopoles. The dipole term is the dominant one at large distances: Its field falls off in proportion to 1/r3, as compared to 1/r4 for the next (quadrupole) term and higher powers of 1/r for higher terms, or 1/r2 for the monopole term.

Molecular dipoles

Many molecules have such dipole moments due to non-uniform distributions of positive and negative charges on the various atoms. Such is the case with polar compounds like hydrogen fluoride (HF), where electron density is shared unequally between atoms. Therefore, a molecule's dipole is an electric dipole with an inherent electric field which should not be confused with a magnetic dipole which generates a magnetic field.

The physical chemist Peter J. W. Debye was the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in units named debye in his honor.

For molecules there are three types of dipoles:
  • Permanent dipoles: These occur when two atoms in a molecule have substantially different electronegativity: One atom attracts electrons more than another, becoming more negative, while the other atom becomes more positive. A molecule with a permanent dipole moment is called a polar molecule. See dipole-dipole attractions.
  • Instantaneous dipoles: These occur due to chance when electrons happen to be more concentrated in one place than another in a molecule, creating a temporary dipole. See instantaneous dipole.
  • Induced dipoles: These can occur when one molecule with a permanent dipole repels another molecule's electrons, inducing a dipole moment in that molecule. A molecule is polarized when it carries an induced dipole. See induced-dipole attraction.
More generally, an induced dipole of any polarizable charge distribution ρ (remember that a molecule has a charge distribution) is caused by an electric field external to ρ. This field may, for instance, originate from an ion or polar molecule in the vicinity of ρ or may be macroscopic (e.g., a molecule between the plates of a charged capacitor). The size of the induced dipole is equal to the product of the strength of the external field and the dipole polarizability of ρ.

Dipole moment values can be obtained from measurement of the dielectric constant. Some typical gas phase values in debye units are:[7]

The linear molecule CO2 has a zero dipole as the two bond dipoles cancel.

KBr has one of the highest dipole moments because it is a very ionic molecule (which only exists as a molecule in the gas phase).


The bent molecule H2O has a net dipole. The two bond dipoles do not cancel.

The overall dipole moment of a molecule may be approximated as a vector sum of bond dipole moments. As a vector sum it depends on the relative orientation of the bonds, so that from the dipole moment information can be deduced about the molecular geometry. For example the zero dipole of CO2 implies that the two C=O bond dipole moments cancel so that the molecule must be linear. For H2O the O-H bond moments do not cancel because the molecule is bent. For ozone (O3) which is also a bent molecule, the bond dipole moments are not zero even though the O-O bonds are between similar atoms. This agrees with the Lewis structures for the resonance forms of ozone which show a positive charge on the central oxygen atom.
Resonance Lewis structures of the ozone molecule
An example in organic chemistry of the role of geometry in determining dipole moment is the cis and trans isomers of 1,2-dichloroethene. In the cis isomer the two polar C-Cl bonds are on the same side of the C=C double bond and the molecular dipole moment is 1.90 D. In the trans isomer, the dipole moment is zero because the two C-Cl bond are on opposite sides of the C=C and cancel (and the two bond moments for the much less polar C-H bonds also cancel).


Cis isomer, dipole moment 1.90 D

Trans isomer, dipole moment zero

Another example of the role of molecular geometry is boron trifluoride, which has three polar bonds with a difference in electronegativity greater than the traditionally cited threshold of 1.7 for ionic bonding. However, due to the equilateral triangular distribution of the fluoride ions about the boron cation center, the molecule as a whole does not exhibit any identifiable pole: one cannot construct a plane that divides the molecule into a net negative part and a net positive part.

Quantum mechanical dipole operator

Consider a collection of N particles with charges qi and position vectors ri. For instance, this collection may be a molecule consisting of electrons, all with charge −e, and nuclei with charge eZi, where Zi is the atomic number of the i th nucleus. The dipole observable (physical quantity) has the quantum mechanical dipole operator:[citation needed]
\mathfrak{p} = \sum_{i=1}^N \, q_i \, \mathbf{r}_i \, .
Notice that this definition is valid only for non-charged dipoles, i.e. total charge equal to zero. To a charged dipole we have the next equation:
\mathfrak{p} = \sum_{i=1}^N \, q_i \, (\mathbf{r}_i - \mathbf{r}_c) \, .
where \mathbf{r}_c is the center of mass of the molecule/group of particles.[8]

Atomic dipoles

A non-degenerate (S-state) atom can have only a zero permanent dipole. This fact follows quantum mechanically from the inversion symmetry of atoms. All 3 components of the dipole operator are antisymmetric under inversion with respect to the nucleus,
\mathfrak{I} \;\mathfrak{p}\; \mathfrak{I}^{-1} = - \mathfrak{p},
where \stackrel{\mathfrak{p}}{} is the dipole operator and \stackrel{\mathfrak{I}}{}\, is the inversion operator. The permanent dipole moment of an atom in a non-degenerate state (see degenerate energy level) is given as the expectation (average) value of the dipole operator,
\langle \mathfrak{p} \rangle = \langle\, S\, | \mathfrak{p} |\, S \,\rangle,
where |\, S\, \rangle is an S-state, non-degenerate, wavefunction, which is symmetric or antisymmetric under inversion: \mathfrak{I}\,|\, S\, \rangle= \pm |\, S\, \rangle. Since the product of the wavefunction (in the ket) and its complex conjugate (in the bra) is always symmetric under inversion and its inverse,
\langle \mathfrak{p} \rangle = \langle\, \mathfrak{I}^{-1}\, S\, | \mathfrak{p} |\, \mathfrak{I}^{-1}\, S \,\rangle = \langle\, S\, | \mathfrak{I}\, \mathfrak{p} \, \mathfrak{I}^{-1}| \, S \,\rangle = -\langle \mathfrak{p} \rangle
it follows that the expectation value changes sign under inversion. We used here the fact that \mathfrak{I}\,, being a symmetry operator, is unitary: \mathfrak{I}^{-1} = \mathfrak{I}^{*}\, and by definition the Hermitian adjoint \mathfrak{I}^*\, may be moved from bra to ket and then becomes \mathfrak{I}^{**} = \mathfrak{I}\,. Since the only quantity that is equal to minus itself is the zero, the expectation value vanishes,
\langle \mathfrak{p}\rangle = 0.
In the case of open-shell atoms with degenerate energy levels, one could define a dipole moment by the aid of the first-order Stark effect. This gives a non-vanishing dipole (by definition proportional to a non-vanishing first-order Stark shift) only if some of the wavefunctions belonging to the degenerate energies have opposite parity; i.e., have different behavior under inversion. This is a rare occurrence, but happens for the excited H-atom, where 2s and 2p states are "accidentally" degenerate (see article Laplace–Runge–Lenz vector for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).

Field of a static magnetic dipole

Magnitude

The far-field strength, B, of a dipole magnetic field is given by

B(m, r, \lambda) = \frac {\mu_0} {4\pi} \frac {m} {r^3} \sqrt {1+3\sin^2\lambda} \, ,
where
B is the strength of the field, measured in teslas
r is the distance from the center, measured in metres
λ is the magnetic latitude (equal to 90° − θ) where θ is the magnetic colatitude, measured in radians or degrees from the dipole axis[note 1]
m is the dipole moment (VADM=virtual axial dipole moment), measured in ampere square-metres (A·m2), which equals joules per tesla
μ0 is the permeability of free space, measured in henries per metre.
Conversion to cylindrical coordinates is achieved using r2 = z2 + ρ2 and
\lambda = \arcsin\left(\frac{z}{\sqrt{z^2+\rho^2}}\right)
where ρ is the perpendicular distance from the z-axis. Then,
B(\rho,z) = \frac{\mu_0 m}{4 \pi (z^2+\rho^2)^{3/2}} \sqrt{1+\frac{3 z^2}{z^2 + \rho^2}}

Vector form

The field itself is a vector quantity:
\mathbf{B}(\mathbf{m}, \mathbf{r}) = \frac {\mu_0} {4\pi} \left(\frac{3(\mathbf{m}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}}{r^3}\right) + \frac{2\mu_0}{3}\mathbf{m}\delta^3(\mathbf{r})
where
B is the field
r is the vector from the position of the dipole to the position where the field is being measured
r is the absolute value of r: the distance from the dipole
\hat{\mathbf{r}} = \mathbf{r}/r is the unit vector parallel to r;
m is the (vector) dipole moment
μ0 is the permeability of free space
δ3 is the three-dimensional delta function.[note 2]
This is exactly the field of a point dipole, exactly the dipole term in the multipole expansion of an arbitrary field, and approximately the field of any dipole-like configuration at large distances.

Magnetic vector potential

The vector potential A of a magnetic dipole is
\mathbf{A}(\mathbf{r}) = \frac {\mu_0} {4\pi} \frac{\mathbf{m}\times\hat{\mathbf{r}}}{r^2}
with the same definitions as above.

Field from an electric dipole

The electrostatic potential at position r due to an electric dipole at the origin is given by:
\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\,\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^2}
where
\hat{\mathbf{r}} is a unit vector in the direction of r, p is the (vector) dipole moment, and ε0 is the permittivity of free space.
This term appears as the second term in the multipole expansion of an arbitrary electrostatic potential Φ(r). If the source of Φ(r) is a dipole, as it is assumed here, this term is the only non-vanishing term in the multipole expansion of Φ(r). The electric field from a dipole can be found from the gradient of this potential:
\mathbf{E} = - \nabla \Phi =\frac {1} {4\pi\epsilon_0} \left(\frac{3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{r^3}\right) - \frac{1}{3\epsilon_0}\mathbf{p}\delta^3(\mathbf{r})
where E is the electric field and δ3 is the 3-dimensional delta function.[note 2] This is formally identical to the magnetic H field of a point magnetic dipole with only a few names changed.

Torque on a dipole

Since the direction of an electric field is defined as the direction of the force on a positive charge, electric field lines point away from a positive charge and toward a negative charge.

When placed in an electric or magnetic field, equal but opposite forces arise on each side of the dipole creating a torque τ:
\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}
for an electric dipole moment p (in coulomb-meters), or
\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}
for a magnetic dipole moment m (in ampere-square meters).

The resulting torque will tend to align the dipole with the applied field, which in the case of an electric dipole, yields a potential energy of
U = -\mathbf{p} \cdot \mathbf{E}.
The energy of a magnetic dipole is similarly
U = -\mathbf{m} \cdot \mathbf{B}.

Dipole radiation


Evolution of the magnetic field of an oscillating electric dipole. The field lines, which are horizontal rings around the axis of the vertically oriented dipole, are perpendicularly crossing the x-y-plane of the image. Shown as a colored contour plot is the z-component of the field. Cyan is zero magnitude, green–yellow–red and blue–pink–red are increasing strengths in opposing directions.

In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to spherical wave radiation.
In particular, consider a harmonically oscillating electric dipole, with angular frequency ω and a dipole moment p_0 along the \hat{z} direction of the form
\mathbf{p}(\mathbf{r},t)=\mathbf{p}(\mathbf{r})e^{-i\omega t} = p_0\hat{\mathbf{z}}e^{-i\omega t} .
In vacuum, the exact field produced by this oscillating dipole can be derived using the retarded potential formulation as:

\mathbf{E} = \frac{1}{4\pi\varepsilon_0} \left\{ \frac{\omega^2}{c^2 r} ( \hat{\mathbf{r}} \times \mathbf{p} ) \times \hat{\mathbf{r}} + \left( \frac{1}{r^3} - \frac{i\omega}{cr^2} \right) \left[ 3 \hat{\mathbf{r}} (\hat{\mathbf{r}} \cdot \mathbf{p}) - \mathbf{p} \right] \right\} e^{i\omega r/c} e^{-i\omega t}
\mathbf{B} = \frac{\omega^2}{4\pi\varepsilon_0 c^3} \hat{\mathbf{r}} \times \mathbf{p} \left( 1 - \frac{c}{i\omega r} \right) \frac{e^{i\omega r/c}}{r} e^{-i\omega t}.

For \scriptstyle r \omega /c \gg 1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:[9]
\mathbf{B} = \frac{\omega^2}{4\pi\varepsilon_0 c^3} (\hat{\mathbf{r}} \times \mathbf{p}) \frac{e^{i\omega (r/c-t)}}{r} = \frac{\omega^2 \mu_0 p_0 }{4\pi c} (\hat{\mathbf{r}} \times \hat{\mathbf{z}}) \frac{e^{i\omega (r/c-t)}}{r} = -\frac{\omega^2 \mu_0 p_0 }{4\pi c} \sin\theta \frac{e^{i\omega (r/c-t)}}{r} \mathbf{\hat{\phi} }
\mathbf{E} = c \mathbf{B} \times \hat{\mathbf{r}} = -\frac{\omega^2 \mu_0 p_0 }{4\pi} \sin\theta (\hat{\phi} \times \mathbf{\hat{r} } )\frac{e^{i\omega (r/c-t)}}{r} = -\frac{\omega^2 \mu_0 p_0 }{4\pi} \sin\theta \frac{e^{i\omega (r/c-t)}}{r} \hat{\theta}.
The time-averaged Poynting vector

\langle \mathbf{S} \rangle = \bigg(\frac{\mu_0p_0^2\omega^4}{32\pi^2 c}\bigg) \frac{\sin^2\theta}{r^2} \mathbf{\hat{r}}

is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the spherical harmonic function ( \sin\theta ) responsible for such "donut-shaped" angular distribution is precisely the l=1 "p" wave.

The total time-average power radiated by the field can then be derived from the Poynting vector as
P = \frac{\mu_0 \omega^4 p_0^2}{12\pi c}.
Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the Rayleigh scattering, and the underlying effects why the sky consists of mainly blue colour.

A circular polarized dipole is described as a superposition of two linear dipoles.

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