Editing Anisotropy

{{short description|In geometry, property of being directionally dependent}}
{{Use dmy dates|date=December 2020}}
[[File:WMAP 2010.png|thumb|upright=1.36|[[Wilkinson Microwave Anisotropy Probe|WMAP]] image of the tiny anisotropies in the [[cosmic microwave background radiation]]]]
'''Anisotropy''' ({{IPAc-en|ˌ|ae|n|aɪ|ˈ|s|ɒ|t|r|ə|p|i|,_|ˌ|æ|n|ɪ|-}}) is the structural property of non-uniformity in different directions, as opposed to [[isotropy]]. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit very different [[physical property|physical]] or [[list of materials properties#Mechanical properties|mechanical properties]] when measured along different axes, e.g. [[absorbance]], [[refractive index]], [[electrical resistivity and conductivity|conductivity]], and [[tensile strength]].

An example of anisotropy is light coming through a [[polarizer]]. Another is [[wood]], which is easier to split along its [[wood grain|grain]] than across it because of the directional non-uniformity of the grain (the grain is the same in one direction, not all directions).

==Fields of interest==

===Computer graphics===
In the field of [[computer graphics]], an anisotropic surface changes in appearance as it rotates about its [[normal (geometry)|geometric normal]], as is the case with [[velvet]].

[[Anisotropic filtering]] (AF) is a method of enhancing the image quality of textures on surfaces that are far away and viewed at a shallow angle. Older techniques, such as [[bilinear filtering|bilinear]] and [[trilinear filtering]], do not take into account the angle a surface is viewed from, which can result in [[aliasing]] or blurring of textures. By reducing detail in one direction more than another, these effects can be reduced easily.

===Chemistry===
A chemical anisotropic [[filtration|filter]], as used to filter particles, is a filter with increasingly smaller interstitial spaces in the direction of filtration so that the [[anatomical terms of location#Proximal and distal|proximal]] regions filter out larger particles and [[anatomical terms of location#Proximal and distal|distal]] regions increasingly remove smaller particles, resulting in greater flow-through and more efficient filtration.

In [[fluorescence spectroscopy]], the [[fluorescence anisotropy]], calculated from the [[polarization (physics)|polarization]] properties of fluorescence from samples excited with plane-polarized light, is used, e.g., to determine the shape of a macromolecule. Anisotropy measurements reveal the average angular displacement of the fluorophore that occurs between absorption and subsequent emission of a photon.

In [[nuclear magnetic resonance spectroscopy|NMR spectroscopy]], the orientation of nuclei with respect to the applied [[magnetic field]] determines their [[chemical shift]]. In this context, anisotropic systems refer to the electron distribution of molecules with abnormally high electron density, like the pi system of [[benzene]]. This abnormal electron density affects the applied magnetic field and causes the observed chemical shift to change.

===Real-world imagery===
Images of a gravity-bound or man-made environment are particularly anisotropic in the orientation domain, with more image structure located at orientations parallel with or orthogonal to the direction of gravity (vertical and horizontal).

===Physics===<!-- This section is linked from [[Birefringence]] -->
[[File:Plasma-lamp 2.jpg|thumb|300px|right|A [[plasma globe]] displaying the nature of [[plasma (physics)|plasmas]], in this case, the phenomenon of "filamentation"]]

[[Physicist]]s from [[University of California, Berkeley]] reported about their detection of the cosmic anisotropy in [[cosmic microwave background radiation]] in 1977. Their experiment demonstrated the [[Doppler shift]] caused by the movement of the earth with respect to the [[Big Bang|early Universe]] matter, the source of the radiation.<ref>{{cite web |title=Detection of Anisotropy in the Cosmic Blackbody Radiation |publisher=[[Lawrence Berkeley Laboratory]] and [[Space Sciences Laboratory]], [[University of California, Berkeley]] |author1=Smoot G. F. |author2=Gorenstein M. V. |author3-link=Richard A. Muller |author3=Muller R. A. |name-list-style=amp |date=5 October 1977 |url=https://muller.lbl.gov/COBE-early_history/anisotropy-PRL.pdf |access-date=15 September 2013 |url-status=live |archive-url=https://ghostarchive.org/archive/20221009/https://muller.lbl.gov/COBE-early_history/anisotropy-PRL.pdf |archive-date=2022-10-09}}</ref> Cosmic anisotropy has also been seen in the alignment of galaxies' rotation axes and polarization angles of quasars.{{fact|date=March 2025}}

Physicists use the term anisotropy to describe direction-dependent properties of materials. [[Magnetic anisotropy]], for example, may occur in a [[plasma (physics)|plasma]], so that its magnetic field is oriented in a preferred direction. Plasmas may also show "filamentation" (such as that seen in [[lightning]] or a [[plasma globe]]) that is directional.{{fact|date=March 2025}}

An ''anisotropic liquid'' has the fluidity of a normal liquid, but has an average structural order relative to each other along the molecular axis, unlike water or [[chloroform]], which contain no structural ordering of the molecules. [[Liquid crystal]]s are examples of anisotropic liquids.{{fact|date=March 2025}}

Some materials [[heat conduction|conduct heat]] in a way that is isotropic, that is independent of spatial orientation around the heat source.  Heat conduction is more commonly anisotropic, which implies that detailed geometric modeling of typically diverse materials being thermally managed is required. The materials used to transfer and reject heat from the heat source in [[electronics]] are often anisotropic.<ref name="Nature8April2013">{{cite journal |last1=Tian |first1=Xiaojuan |last2=Itkis |first2=Mikhail E |last3=Bekyarova |first3=Elena B |last4=Haddon |first4=Robert C |title=Anisotropic Thermal and Electrical Properties of Thin Thermal Interface Layers of Graphite Nanoplatelet-Based Composites |journal=Scientific Reports |volume=3 |pages=1710 |date=8 April 2013 |doi=10.1038/srep01710 |pmc=3632880 |bibcode=2013NatSR...3.1710T}}</ref>

Many [[crystal]]s are anisotropic to [[light]] ("optical anisotropy"), and exhibit properties such as [[birefringence]]. [[Crystal optics]] describes light propagation in these media. An "axis of anisotropy" is defined as the axis along which isotropy is broken (or an axis of symmetry, such as normal to crystalline layers). Some materials can have multiple such [[optic axis of a crystal|optical axes]].{{fact|date=March 2025}}

===Geophysics and geology===
[[Seismic anisotropy]] is the variation of seismic wavespeed with direction. Seismic anisotropy is an indicator of long range order in a material, where features smaller than the seismic [[wavelength]] (e.g., crystals, cracks, pores, layers, or inclusions) have a dominant alignment. This alignment leads to a directional variation of [[elasticity (physics)|elasticity]] wavespeed. Measuring the effects of anisotropy in seismic data can provide important information about processes and mineralogy in the Earth; significant seismic anisotropy has been detected in the Earth's [[crust (geology)|crust]], [[mantle (geology)|mantle]], and [[Earth's inner core|inner core]].

[[Geological]] formations with distinct layers of [[sedimentary]] material can exhibit electrical anisotropy; [[electrical conductivity]] in one direction (e.g. parallel to a layer), is different from that in another (e.g. perpendicular to a layer). This property is used in the gas and [[oil exploration]] industry to identify [[hydrocarbon]]-bearing sands in sequences of [[sand]] and [[shale]]. Sand-bearing hydrocarbon assets have high [[resistivity]] (low conductivity), whereas shales have lower resistivity. [[Formation evaluation]] instruments measure this conductivity or resistivity, and the results are used to help find oil and gas in wells. The mechanical anisotropy measured for some of the sedimentary rocks like coal and shale can change with corresponding changes in their surface properties like sorption when gases are produced from the coal and shale reservoirs.<ref>{{cite journal |last1=Saurabh |first1=Suman |last2=Harpalani |first2=Satya |title=Anisotropy of coal at various scales and its variation with sorption |journal=International Journal of Coal Geology |date=2 January 2019 |volume=201 |pages=14–25 |doi=10.1016/j.coal.2018.11.008 |bibcode=2019IJCG..201...14S |s2cid=133624963}}</ref>

The [[hydraulic conductivity]] of [[aquifer]]s is often anisotropic for the same reason. When calculating groundwater flow to [[drainage|drains]]<ref>{{cite web |author=Oosterbaan, R. J. |year=1997 |title=The Energy Balance of Groundwater Flow Applied to Subsurface Drainage in Anisotropic Soils by Pipes or Ditches With Entrance Resistance |url=https://www.waterlog.info/pdf/enerart.pdf |url-status=live |archive-url=https://web.archive.org/web/20090219221547/http://www.waterlog.info/pdf/enerart.pdf |archive-date=19 February 2009}} The corresponding free EnDrain program can be downloaded from: [https://www.waterlog.info/endrain.htm].</ref> or to [[water well|wells]],<ref>{{cite web |author=Oosterbaan, R. J. |year=2002 |title=Subsurface Land Drainage By Tube Wells |url=https://www.waterlog.info/pdf/wellspac.pdf}} 9 pp. The corresponding free WellDrain program can be downloaded from: [https://www.waterlog.info/weldrain.htm]</ref> the difference between horizontal and vertical permeability must be taken into account; otherwise the results may be subject to error.

Most common rock-forming [[mineral]]s are anisotropic, including [[quartz]] and [[feldspar]]. Anisotropy in minerals is most reliably seen in their [[optical mineralogy|optical properties]]. An example of an isotropic mineral is [[garnet]].

Igneous rock like granite also shows the anisotropy due to the orientation of the minerals during the solidification process.<ref>{{Cite web |last=MAT |first=Mahmut |date=2018-04-19 |title=Granite {{!}} Properties, Formation, Composition, Uses » Geology Science |url=https://geologyscience.com/rocks/granite/ |access-date=2024-02-16 |website=Geology Science |language=en-US}}</ref>

===Medical acoustics===
Anisotropy is also a well-known property in [[medical ultrasound]] imaging describing a different resulting [[echogenicity]] of soft tissues, such as [[tendon]]s, when the angle of the [[transducer]] is changed. Tendon fibers appear hyperechoic (bright) when the transducer is perpendicular to the tendon, but can appear hypoechoic (darker) when the transducer is angled obliquely. This can be a source of interpretation error for inexperienced practitioners.{{citation needed|date=May 2024}}

===Materials science and engineering===
Anisotropy, in [[materials science]], is a material's directional dependence of a [[physical property]]. This is a critical consideration for [[materials selection]] in engineering applications. A material with physical properties that are symmetric about an axis that is normal to a plane of isotropy is called a [[transverse isotropy|transversely isotropic material]]. [[Tensor]] descriptions of material properties can be used to determine the directional dependence of that property. For a [[monocrystalline]] material, anisotropy is associated with the crystal symmetry in the sense that more symmetric crystal types have fewer independent coefficients in the tensor description of a given property.<ref>{{cite book |last1=Newnham |first1=Robert E. |title=Properties of Materials: Anisotropy, Symmetry, Structure |publisher=Oxford University Press |isbn=978-0198520764 |edition=1st}}</ref><ref>{{cite book |last1=Nye |first1=J.F. |title=Physical Properties of Crystals |publisher=Clarendon Press |edition=1st}}</ref> When a material is [[polycrystalline]], the directional dependence on properties is often related to the processing techniques it has undergone. A material with randomly oriented grains will be isotropic, whereas materials with [[texture (crystalline)|texture]] will be often be anisotropic. Textured materials are often the result of processing techniques like [[cold rolling]], [[wire drawing]], and [[heat treatment]].

Mechanical properties of materials such as [[Young's modulus]], [[ductility]], [[yield strength]], and high-temperature [[creep (deformation)|creep rate]], are often dependent on the direction of measurement.<ref>{{cite book |last1=Courtney |first1=Thomas H. |title=Mechanical Behavior of Materials |publisher=Waveland Pr Inc |isbn=978-1577664253 |edition=2nd |year=2005}}</ref> Fourth-rank [[tensor]] properties, like the elastic constants, are anisotropic, even for materials with cubic symmetry. The Young's modulus relates stress and strain when an isotropic material is elastically deformed; to describe elasticity in an anisotropic material, [[stiffness]] (or compliance) tensors are used instead.

In metals, anisotropic elasticity behavior is present in all single crystals with three independent coefficients for cubic crystals, for example. For face-centered cubic materials such as nickel and copper, the stiffness is highest along the <111> direction, normal to the close-packed planes, and smallest parallel to <100>. Tungsten is so nearly isotropic at room temperature that it can be considered to have only two stiffness coefficients; aluminium is another metal that is nearly isotropic.

For an isotropic material, <math display="block">G = E/[2(1 + \nu)], </math> where <math> G </math> is the [[shear modulus]], <math> E </math> is the [[Young's modulus]], and <math> \nu </math> is the material's [[Poisson's ratio]]. Therefore, for cubic materials, we can think of anisotropy, <math> a_r </math>, as the ratio between the empirically determined shear modulus for the cubic material and its (isotropic) equivalent:
<math display="block">a_r = \frac{G}{E/[2(1 + \nu)]} = \frac{2(1+\nu)G}{E} \equiv \frac{2 C_{44}}{C_{11} - C_{12}}.</math>

The latter expression is known as the [[Zener ratio]], <math> a_r </math>, where <math>C_{ij}</math> refers to [[Hooke's Law|elastic constants]] in [[Voigt notation|Voigt (vector-matrix) notation]]. For an isotropic material, the ratio is one.

Limitation of the [[Zener ratio]] to cubic materials is waived in the Tensorial anisotropy index A<sup>T</sup> <ref>{{cite journal |last1=Sokołowski |first1=Damian |last2=Kamiński |first2=Marcin |date=2018-09-01 |title=Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects |journal=Acta Mechanica |language=en |volume=229 |issue=9 |pages=3727–3765 |doi=10.1007/s00707-018-2174-7 |s2cid=126198766 |issn=1619-6937 |doi-access=free}}</ref> that takes into consideration all the 27 components of the fully anisotropic stiffness tensor. It is composed of two major parts <math>A^I</math>and <math>A^A </math>, the former referring to components existing in cubic tensor and the latter in anisotropic tensor so that <math>A^T = A^I+A^A .</math> This first component includes the modified Zener ratio and additionally accounts for directional differences in the material, which exist in [[Orthotropic material|orthotropic]] material, for instance. The second component of this index <math>A^A </math> covers the influence of stiffness coefficients that are nonzero only for non-cubic materials and remains zero otherwise.

Fiber-reinforced or layered [[composite material]]s exhibit anisotropic mechanical properties, due to orientation of the reinforcement material. In many fiber-reinforced composites like carbon fiber or glass fiber based composites, the weave of the material (e.g. unidirectional or plain weave) can determine the extent of the anisotropy of the bulk material.<ref>{{cite web |title=Fabric Weave Styles |url=https://compositeenvisions.com/fabric-weave-styles/ |website=Composite Envisions |access-date=23 May 2019}}</ref> The tunability of orientation of the fibers allows for application-based designs of composite materials, depending on the direction of stresses applied onto the material.

Amorphous materials such as glass and polymers are typically isotropic. Due to the highly randomized orientation of [[macromolecule]]s in polymeric materials, [[polymer]]s are in general described as isotropic. However, [[mechanically gradient polymers]] can be engineered to have directionally dependent properties through processing techniques or introduction of anisotropy-inducing elements. Researchers have built composite materials with aligned fibers and voids to generate anisotropic [[hydrogel]]s, in order to mimic hierarchically ordered biological soft matter.<ref>{{cite journal |last1=Sano |first1=Koki |last2=Ishida |first2=Yasuhiro |last3=Aida |first3=Tazuko |title=Synthesis of Anisotropic Hydrogels and Their Applications |journal=Angewandte Chemie International Edition |date=16 October 2017 |volume=57 |issue=10 |pages=2532–2543 |doi=10.1002/anie.201708196 |pmid=29034553}}</ref> 3D printing, especially Fused Deposition Modeling, can introduce anisotropy into printed parts. This is because FDM is designed to extrude and print layers of thermoplastic materials.<ref>{{cite journal |last1=Wang |first1=Xin |last2=Jiang |first2=Man |last3=Gou |first3=Jihua |last4=Hui |first4=David |title=3D printing of polymer matrix composites: A review and prospective |journal=Composites Part B: Engineering |date=1 February 2017 |volume=110 |pages=442–458 |doi=10.1016/j.compositesb.2016.11.034}}</ref> This creates materials that are strong when tensile stress is applied in parallel to the layers and weak when the material is perpendicular to the layers.

===Microfabrication===

Anisotropic etching techniques (such as [[deep reactive-ion etching]]) are used in [[microfabrication]] processes to create well defined microscopic features with a high [[aspect ratio]]. These features are commonly used in [[MEMS]] (microelectromechanical systems) and [[microfluidic]] devices, where the anisotropy of the features is needed to impart desired optical, electrical, or physical properties to the device. Anisotropic etching can also refer to certain chemical etchants used to etch a certain material preferentially over certain crystallographic planes (e.g., KOH etching of [[silicon]] [100] produces pyramid-like structures)

===Neuroscience===

[[Diffusion tensor imaging]] is an [[magnetic resonance imaging|MRI]] technique that involves measuring the fractional anisotropy of the random motion ([[Brownian motion]]) of water molecules in the brain. Water molecules located in [[white matter|fiber tracts]] are more likely to move anisotropically, since they are restricted in their movement (they move more in the dimension parallel to the fiber tract rather than in the two dimensions orthogonal to it), whereas water molecules dispersed in the rest of the brain have less restricted movement and therefore display more isotropy. This difference in fractional anisotropy is exploited to create a map of the fiber tracts in the brains of the individual.

===Remote sensing and radiative transfer modeling===
[[Radiance]] fields (see [[Bidirectional reflectance distribution function]] (BRDF)) from a reflective surface are often not isotropic in nature. This makes calculations of the total energy being reflected from any scene a difficult quantity to calculate. In [[remote sensing]] applications, anisotropy functions can be derived for specific scenes, immensely simplifying the calculation of the net reflectance or (thereby) the net [[irradiance]] of a scene.
For example, let the [[bidirectional reflectance distribution function|BRDF]] be <math>\gamma(\Omega_i, \Omega_v)</math> where 'i' denotes incident direction and 'v' denotes viewing direction (as if from a satellite or other instrument). And let P be the Planar Albedo, which represents the total reflectance from the scene.
<math display="block">P(\Omega_i) = \int_{\Omega_v} \gamma(\Omega_i, \Omega_v)\hat{n} \cdot d\hat\Omega_v</math>
<math display="block">A(\Omega_i, \Omega_v) = \frac{\gamma(\Omega_i, \Omega_v)}{P(\Omega_i)}</math>

It is of interest because, with knowledge of the anisotropy function as defined, a measurement of the [[bidirectional reflectance distribution function|BRDF]] from a single viewing direction (say, <math>\Omega_v</math>) yields a measure of the total scene reflectance (planar [[albedo]]) for that specific incident geometry (say, <math>\Omega_i</math>).

==See also==
* [[Circular symmetry]]
* {{In title|anisotropy}}
* {{In title|anisotropic}}

==References==
{{Reflist}}

==External links==
{{Wiktionary|anisotropy|anisotropic}}
* [https://web.archive.org/web/20100303080919/http://aluminium.matter.org.uk/content/html/eng/default.asp?catid=99&pageid=1028022659 "Overview of Anisotropy"]
* [https://www.doitpoms.ac.uk/tlplib/anisotropy/index.php DoITPoMS Teaching and Learning Package: "Introduction to Anisotropy"]
* [https://knitty.com/ISSUEsummer05/FEATsum05TBP.html "Gauge, and knitted fabric generally, is an anisotropic phenomenon"]

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[[Category:Orientation (geometry)]]
[[Category:Asymmetry]]