Anisotropy

WMAPimage of the (extremely tiny) anisotropies in thecosmic microwave background radiation

Anisotropy(/ˌænaɪˈsɒtrəpi,ˌænɪ-/) is the structural property of non-uniformity in different directions, as opposed toisotropy.An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit very differentphysicalormechanical propertieswhen measured along different axes, e.g.absorbance,refractive index,conductivity,andtensile strength.

An example of anisotropy is light coming through apolarizer.Another iswood,which is easier to split along itsgrainthan 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 ofcomputer graphics,an anisotropic surface changes in appearance as it rotates about itsgeometric normal,as is the case withvelvet.

Anisotropic filtering(AF) is a method of enhancing the image quality of textures on surfaces that are far away and steeply angled with respect to the point of view. Older techniques, such asbilinearandtrilinear filtering,do not take into account the angle a surface is viewed from, which can result inaliasingor blurring of textures. By reducing detail in one direction more than another, these effects can be reduced easily.

Chemistry

A chemical anisotropicfilter,as used to filter particles, is a filter with increasingly smaller interstitial spaces in the direction of filtration so that theproximalregions filter out larger particles anddistalregions increasingly remove smaller particles, resulting in greater flow-through and more efficient filtration.

Influorescence spectroscopy,thefluorescence anisotropy,calculated from thepolarizationproperties 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.

InNMR spectroscopy,the orientation of nuclei with respect to the applied magnetic field determines theirchemical shift.In this context, anisotropic systems refer to the electron distribution of molecules with abnormally high electron density, like the pi system ofbenzene.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

Aplasma globedisplaying the nature ofplasmas,in this case, the phenomenon of "filamentation"

PhysicistsfromUniversity of California, Berkeleyreported about their detection of the cosmic anisotropy incosmic microwave background radiationin 1977. Their experiment demonstrated theDoppler shiftcaused by the movement of the earth with respect to theearly Universe matter,the source of the radiation.Cosmic anisotropy has also been seen in the alignment of galaxies' rotation axes and polarization angles of quasars.

Physicists use the term anisotropy to describe direction-dependent properties of materials.Magnetic anisotropy,for example, may occur in aplasma,so that its magnetic field is oriented in a preferred direction. Plasmas may also show "filamentation" (such as that seen inlightningor aplasma globe) that is directional.

Ananisotropic liquidhas the fluidity of a normal liquid, but has an average structural order relative to each other along the molecular axis, unlike water orchloroform,which contain no structural ordering of the molecules.Liquid crystalsare examples of anisotropic liquids.

Some materialsconduct heatin 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 inelectronicsare often anisotropic.

Manycrystalsare anisotropic tolight( "optical anisotropy" ), and exhibit properties such asbirefringence.Crystal opticsdescribes 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 suchoptical axes.

Geophysics and geology

Seismic anisotropyis the variation of seismic wavespeed with direction. Seismic anisotropy is an indicator of long range order in a material, where features smaller than the seismicwavelength(e.g., crystals, cracks, pores, layers, or inclusions) have a dominant alignment. This alignment leads to a directional variation ofelasticitywavespeed. 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'scrust,mantle,andinner core.

Geologicalformations with distinct layers ofsedimentarymaterial can exhibit electrical anisotropy;electrical conductivityin 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 andoil explorationindustry to identifyhydrocarbon-bearing sands in sequences ofsandandshale.Sand-bearing hydrocarbon assets have highresistivity(low conductivity), whereas shales have lower resistivity.Formation evaluationinstruments 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.

Thehydraulic conductivityofaquifersis often anisotropic for the same reason. When calculating groundwater flow todrainsor towells,the difference between horizontal and vertical permeability must be taken into account; otherwise the results may be subject to error.

Most common rock-formingmineralsare anisotropic, includingquartzandfeldspar.Anisotropy in minerals is most reliably seen in theiroptical properties.An example of an isotropic mineral isgarnet.

Igneous rock like granite also shows the anisotropy due to the orientation of the minerals during the solidification process.

Medical acoustics

Anisotropy is also a well-known property inmedical ultrasoundimaging describing a different resultingechogenicityof soft tissues, such astendons,when the angle of thetransduceris 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.

Materials science and engineering

Anisotropy, inmaterials science,is a material's directional dependence of aphysical property.This is a critical consideration formaterials selectionin engineering applications. A material with physical properties that are symmetric about an axis that is normal to a plane of isotropy is called atransversely isotropic material.Tensordescriptions of material properties can be used to determine the directional dependence of that property. For amonocrystallinematerial, 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.When a material ispolycrystalline,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 withtexturewill be often be anisotropic. Textured materials are often the result of processing techniques likecold rolling,wire drawing,andheat treatment.

Mechanical properties of materials such asYoung's modulus,ductility,yield strength,and high-temperaturecreep rate,are often dependent on the direction of measurement.Fourth-ranktensorproperties, 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,$$G=E/[2(1+\nu )],$$where$G$is theshear modulus,$E$is theYoung's modulus,and$\nu$is the material'sPoisson's ratio.Therefore, for cubic materials, we can think of anisotropy,$a_{r}$,as the ratio between the empirically determined shear modulus for the cubic material and its (isotropic) equivalent: $$a_{r}={\frac {G}{E/[2(1+\nu )]}}={\frac {2(1+\nu )G}{E}}\equiv {\frac {2C_{44}}{C_{11}-C_{12}}}.$$

The latter expression is known as theZener ratio,$a_{r}$,where$C_{ij}$refers toelastic constantsinVoigt (vector-matrix) notation.For an isotropic material, the ratio is one.

Limitation of theZener ratioto cubic materials is waived in the Tensorial anisotropy index A^Tthat takes into consideration all the 27 components of the fully anisotropic stiffness tensor. It is composed of two major parts$A^{I}$and$A^{A}$,the former referring to components existing in cubic tensor and the latter in anisotropic tensor so that$A^{T}=A^{I}+A^{A}.$This first component includes the modified Zener ratio and additionally accounts for directional differences in the material, which exist inorthotropicmaterial, for instance. The second component of this index$A^{A}$covers the influence of stiffness coefficients that are nonzero only for non-cubic materials and remains zero otherwise.

Fiber-reinforced or layeredcomposite materialsexhibit 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.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 ofmacromoleculesin polymeric materials,polymersare in general described as isotropic. However,mechanically gradient polymerscan 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 anisotropichydrogels,in order to mimic hierarchically ordered biological soft matter.3D printing, especially Fused Deposition Modeling, can introduce anisotropy into printed parts. This is due to the fact that FDM is designed to extrude and print layers of thermoplastic materials.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 asdeep reactive-ion etching) are used inmicrofabricationprocesses to create well defined microscopic features with a highaspect ratio.These features are commonly used inMEMS(microelectromechanical systems) andmicrofluidicdevices, 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 ofsilicon[100] produces pyramid-like structures)

Neuroscience

Diffusion tensor imagingis anMRItechnique that involves measuring the fractional anisotropy of the random motion (Brownian motion) of water molecules in the brain. Water molecules located infiber tractsare 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

Radiancefields (seeBidirectional 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. Inremote sensingapplications, anisotropy functions can be derived for specific scenes, immensely simplifying the calculation of the net reflectance or (thereby) the netirradianceof a scene. For example, let theBRDFbe$\gamma (\Omega _{i},\Omega _{v})$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. $$P(\Omega _{i})=\int _{\Omega _{v}}\gamma (\Omega _{i},\Omega _{v}){\hat {n}}\cdot d{\hat {\Omega }}_{v}$$ $$A(\Omega _{i},\Omega _{v})={\frac {\gamma (\Omega _{i},\Omega _{v})}{P(\Omega _{i})}}$$

It is of interest because, with knowledge of the anisotropy function as defined, a measurement of theBRDFfrom a single viewing direction (say,$\Omega _{v}$) yields a measure of the total scene reflectance (planaralbedo) for that specific incident geometry (say,$\Omega _{i}$).

See also

• Circular symmetry
• All pages with titles containinganisotropy
• All pages with titles containinganisotropic

External links

• "Overview of Anisotropy"
• DoITPoMS Teaching and Learning Package: "Introduction to Anisotropy"
• "Gauge, and knitted fabric generally, is an anisotropic phenomenon"