Group ring - Revision history

134.99.156.36: /* Group rings over an infinite group */

2014-07-18T08:11:48Z

Group rings over an infinite group

← Older revision		Revision as of 10:11, 18 July 2014
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en>LokiClock: /* Center of a group algebra */ Uninterpreted delimiters in LaTeX

2014-02-23T05:57:49Z

Center of a group algebra: Uninterpreted delimiters in LaTeX

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en>Williamdemeo: /* Representations of a group algebra */

2014-01-11T09:42:46Z

Representations of a group algebra

← Older revision		Revision as of 11:42, 11 January 2014
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	~~Catrina Le is what~~'~~s designed on her birth diploma though she doesn~~'~~t significantly like being called prefer~~ that. ~~Her job~~ is ~~probably~~ a ~~cashier but speedily her husband~~ and ~~your will start their own home~~ [http://~~browse~~.~~deviantart~~.com/~~?qh~~=~~&section~~=~~&global~~=~~1&q~~=~~based+business based business~~]. ~~To drive~~ is ~~something~~ which will ~~she'~~s ~~been doing best~~. ~~For years she~~'~~s been living~~ in ~~Vermont~~. Go to ~~your sweetheart website to find accessible more: http~~://~~prometeu~~.~~net~~<br><br>~~Also visit my weblog~~ ... [~~http~~:~~//prometeu.net Hack Clash Of Clans 2014~~]		'''Electroosmotic flow''' (or '''electro-osmotic flow''', often abbreviated EOF; synonymous with '''electroosmosis''' or '''electroendosmosis''') is the motion of liquid induced by an applied potential across a porous material, capillary tube, membrane, microchannel, or any other fluid conduit. Because electroosmotic velocities are independent of conduit size, as long as the double layer is much smaller than the characteristic length scale of the channel, electroosmotic flow is most significant when in small channels. Electroosmotic flow is an essential component in chemical separation techniques, notably capillary electrophoresis. Electroosmotic flow can occur in natural unfiltered water, as well as [[buffer solution\|buffer]]ed solutions.

			[[File:Electroosmotic Flow Depiction.tiff\|thumb\|Electroosmotic Flow Schematic]]

			==History==

			Electroosmotic flow was first reported in 1809 by [[F.F. Reuss]] in the Proceedings of the [[Imperial Society of Naturalists of Moscow]]. He showed that water could be made to flow through a plug of [[clay]] by applying an electric voltage. Clay is composed of closely packed particles of silica and other minerals, and water flows through the narrow spaces between these particles just as it would through a narrow glass tube. Any combination of an [[electrolyte]] (a fluid containing dissolved ions) and an insulating solid would generate electro-osmotic flow, though for water/[[silica]] the effect is particularly large. Even so, flow speeds are typically only a few millimeters per second.

			==Cause==

			Electroosmotic flow is caused by the [[Coulomb force]] induced by an electric field on net mobile [[electric charge]] in a solution. Because the chemical equilibrium between a solid surface and an electrolyte solution typically leads to the interface acquiring a net fixed electrical charge, a layer of mobile ions, known as an [[electrical double layer]] or Debye layer, forms in the region near the interface. When an electric field is applied to the fluid (usually via electrodes placed at inlets and outlets), the net charge in the electrical double layer is induced to move by the resulting Coulomb force. The resulting flow is termed electroosmotic flow.

			==Description==
			The resulting flow from applying a voltage is a [[plug flow]]. Unlike a parabolic profile flow generated from a pressure differential, a plug flow’s velocity profile is approximately planar, with slight variation near the electric double layer. This offers significantly less deleterious dispersive effects and can be controlled without valves, offering a high performance method for fluid separation, although many complex factors prove this control to be difficult. Because of difficulties measuring and monitoring flow in micro fluidic channels, primarily disrupting the flow pattern, most analysis is done through numerical methods and simulation.<ref name=Yao>{{cite book \| author=G.F. Yao, H. \| title=A Computational Model for Simulation of Electroosmotic Flow in Microsystems}}</ref>
			Electroosmotic flow through micro channels can be modeled after the Navier-Stokes equation with the driving force deriving from the electric field and not the pressure differential. Thus it is governed by the [[continuity equation]]

			:<math>\nabla \cdot\mathbf{U} = 0</math>

			and [[momentum]]

			:<math>\rho \frac{D \mathbf{U}}{D t} = -\nabla p + \mu \nabla^2 \mathbf{U} + \rho_e \nabla \left( \psi + \phi \right)</math>

			where {{math\|'''U'''}} is the velocity vector, {{math\|''ρ''}} is the density of the fluid, <math>D/Dt</math> is the [[material derivative]], {{math\|''μ''}} is the viscosity of the fluid, {{math\|''ρ''<sub>''e''</sub>}} is the electric charge density, {{math\|''Φ''}} is the applied electric field, and {{math\|''ψ''}} is the electric field due to the [[zeta potential]] at the walls.

			[[Laplace’s equation]] can describe the external electric field

			:<math> \nabla^2 \phi = 0</math>

			while the potential within the electric double layer is governed by

			:<math> \nabla^2 \psi = \frac{-\rho_e}{\epsilon\epsilon_0}</math>

			where {{math\|''ε''}} is the dielectric constant of the electrolyte solution and {{math\|''ε''<sub>0</sub>}} is the [[vacuum permittivity]]. This equation can be further simplified using the [[Debye–Hückel theory\|Debye-Hückel approximation]]

			:<math> \nabla^2 \psi = k^2\psi </math>

			where {{math\|1 / ''k''}} is the [[Debye Length]], used to describe the characteristic thickness of the electric double layer. The equations for potential field within the double layer can be combined as:

			:<math>\rho_e = -\epsilon \epsilon_0 k^2 \psi \, </math>

			==Applications==

			Electroosmotic flow is commonly used in [[microfluidic]] devices,<ref name=Bruus>{{cite book \| author=Bruus, H. \| title=Theoretical Microfluidics \| year= 2007 \| isbn=0-19-923509-0}}</ref><ref name=Kirby>{{cite book \| year=2010 \| publisher=Cambridge University Press \| author=Kirby, B.J. \| title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 6: Electroosmosis \| url=http://www.kirbyresearch.com/index.cfm/wrap/textbook/microfluidicsnanofluidicsch6.html}}</ref> soil analysis and processing,<ref name=Wise>{{cite book \| author=Wise, D.L. and Trantolo, D.J., eds. \| title=Remediation of Hazardous Waste Contaminated Soils}}</ref> and chemical analysis,<ref name=Skoog>{{cite book \| author=Skoog \| title=Principles of Instrumental Analysis \| isbn=0-495-12570-9 }}</ref> all of which routinely involve systems with highly charged surfaces, often of [[oxide]]s. One example is [[capillary electrophoresis]],<ref name=Kirby/><ref name=Skoog/> in which electric fields are used to separate chemicals according to their electrophoretic mobility by applying an electric field to a narrow capillary, usually made of [[silica]]. In electrophoretic separations, the electroosmotic flow affects the [[elution time]] of the analytes.

			It is projected that micro fluidic devices utilizing electroosmotic flow will have great application with medical research. Once controlling this flow is better understood and implemented being able to separate fluids on the atomic level will be a vital component for drug dischargers.<ref name=Ducree>{{cite book \| author=Ducree, Jen \| url=http://myfluidix.com/}}</ref> Mixing fluids at the micro scale is currently troublesome. It is believed that electrically controlling fluids will be the method in which small fluids are mixed.<ref name=Ducree/>

			=== Physics ===
			In [[fuel cell]]s, electro-osmosis causes [[proton]]s moving through a [[Proton Exchange Membrane\|proton exchange membrane]] (PEM) to drag water molecules from one side ([[anode]]) to the other ([[cathode]]).

			=== Vascular plant biology ===
			In vascular plant biology, electro-osmosis is also used as an alternative or supplemental explanation for the movement of polar liquids via the [[phloem]] that differs from the [[cohesion-tension theory]] supplied in the mass flow hypothesis and others, such as [[cytoplasmic streaming]].<ref name="Clegg, C.J. 2006 p340-343">Clegg, C.J., Mackean, D. G. (2006) "''Advanced Biology – principles & applications''" Hodder Stoughton Publhishers, pp. 340–343</ref> [[Phloem#Companion cells\|Companion cells]] are involved in the "cyclic" withdrawal of ions (<math>K^+</math>) from sieve tubes, and their secretion parallel to their position of withdrawal between sieve plates, resulting in polarisation of sieve plate elements alongside [[potential difference]] in pressure, and results in polar water molecules and other solutes present moved upward through the phloem.<ref name="Clegg, C.J. 2006 p340-343"/>

			In 2003, [[St Petersburg University]] graduates applied direct [[electric current]] to 10 mm segments of mesocotyls of maize seedlings alongside one-year linden shoots; electrolyte solutions present in the tissues moved toward the cathode that was in place, suggesting that electro-osmosis might play a role in solution transport through conductive plant tissues.<ref>{{cite journal\|last1=Polevoi\|first1=V. V.\|journal=Biology Bulletin\|volume=30\|pages=133\|year=2003\|doi=10.1023/A:1023285121361\|issue=2}}</ref>

			==Disadvantages==

			Maintaining an electric field in an electrolyte requires [[Faradaic current\|Faradaic]] reactions to occur at the anode and cathode. This is typically [[electrolysis of water]], which generates [[hydrogen peroxide]], [[hydrogen ion]]s (acid) and [[hydroxide]] (base) as well as [[oxygen]] and [[hydrogen]] gas bubbles. The hydrogen peroxide and/or pH changes generated can adversely affect biological cells and biomolecules such as proteins, while gas bubbles tend to "clog" [[Microfluidics\|microfluidic]] systems. These problems can be alleviated by using alternative electrode materials such as [[Conjugated system\|conjugated polymers]] which can undergo the Faradaic reactions themselves, dramatically reducing electrolysis.<ref>{{cite journal\|last1=Erlandsson\|first1=P.G.\|last2=Robinson\|first2=N.D.\|journal=Electrophoresis\|volume=32\|pages=784–790\|doi=10.1002/elps.201000617\|title=Electrolysis-reducing electrodes for electrokinetic devices\|year=2011\|issue=6–7\|pmid=21425174}}</ref>

			==See also==
			{{Commons category\|Electro-osmosis}}
			* [[Capillary electrophoresis]]
			* [[Double layer]]
			* [[Streaming current]]
			* [[Streaming potential]]
			* [[Zeta potential]]
			* [[Electroosmotic pump]]
			* [[electrical double layer]]
			* [[microfluidics]]
			* [[electrochemistry]]

			==References==
			{{Reflist\|2}}

			==Further reading==
			* {{cite book \| author=Bell, F.G. \| title=Engineering Properties of Soils and Rocks, 4th ed. \| year= 2000}}
			* {{cite book \| author=Chang, H.C. and Yao, L. \| title=Electrokinetically Driven Microfluidics and Nanofluidics \| year= 2009}}
			* {{cite book \| author=Levich, V. \| title=Physicochemical Hydrodynamics \| year= 1962 \| isbn=0-903012-40-5}}
			* {{cite book \| author=Probstein, R.F. \| title=Physicochemical Hydrodynamics: an introduction, 2nd ed. \| year= 2003}}

			{{DEFAULTSORT:Electro-Osmosis}}
			[[Category:Fluid dynamics]]
			[[Category:Electrochemistry]]

89.12.0.165 at 15:58, 14 June 2012

2012-06-14T15:58:57Z

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