In geometry, property of being directionally dependent
WMAP
image of the (extremely tiny) anisotropies in the
cosmic microwave background radiation
Anisotropy
(
) 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
or
mechanical properties
when measured along different axes, e.g.
absorbance
,
refractive index
,
conductivity
, and
tensile strength
.
An example of anisotropy is light coming through a
polarizer
. Another is
wood
, which is easier to split along its
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
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Computer graphics
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In the field of
computer graphics
, an anisotropic surface changes in appearance as it rotates about its
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 steeply angled with respect to the point of view. Older techniques, such as
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
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A chemical anisotropic
filter
, as used to filter particles, is a filter with increasingly smaller interstitial spaces in the direction of filtration so that the
proximal
regions filter out larger particles and
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
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
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
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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
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A
plasma globe
displaying the nature of
plasmas
, in this case, the phenomenon of "filamentation"
Physicists
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
early Universe matter
, the source of the radiation.
[1]
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 a
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.
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 crystals
are examples of anisotropic liquids.
Some materials
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.
[2]
Many
crystals
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
optical axes
.
Geophysics and geology
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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
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
,
mantle
, and
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.
[3]
The
hydraulic conductivity
of
aquifers
is often anisotropic for the same reason. When calculating groundwater flow to
drains
[4]
or to
wells
,
[5]
the difference between horizontal and vertical permeability must be taken into account; otherwise the results may be subject to error.
Most common rock-forming
minerals
are anisotropic, including
quartz
and
feldspar
. Anisotropy in minerals is most reliably seen in their
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.
[6]
Medical acoustics
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Anisotropy is also a well-known property in
medical ultrasound
imaging describing a different resulting
echogenicity
of soft tissues, such as
tendons
, 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
]
Materials science and engineering
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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
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.
[7]
[8]
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
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 rate
, are often dependent on the direction of measurement.
[9]
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,
![{\displaystyle G=E/[2(1+\nu )],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b008fdf44559a0ecfc6e7503e8f6d4ed97ed2cdb)
where
![{\displaystyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
is the
shear modulus
,
![{\displaystyle E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
is the
Young's modulus
, and
![{\displaystyle \nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468)
is the material's
Poisson's ratio
. Therefore, for cubic materials, we can think of anisotropy,
![{\displaystyle a_{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e315f1fc37a1d5b4da31f47d4e052c4a810d0f74)
, as the ratio between the empirically determined shear modulus for the cubic material and its (isotropic) equivalent:
![{\displaystyle a_{r}={\frac {G}{E/[2(1+\nu )]}}={\frac {2(1+\nu )G}{E}}\equiv {\frac {2C_{44}}{C_{11}-C_{12}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f41d6821eb2a7288d1421770c47c1ff4ae80191)
The latter expression is known as the
Zener ratio
,
, where
refers to
elastic constants
in
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
T
[10]
that takes into consideration all the 27 components of the fully anisotropic stiffness tensor. It is composed of two major parts
and
, the former referring to components existing in cubic tensor and the latter in anisotropic tensor so that
This first component includes the modified Zener ratio and additionally accounts for directional differences in the material, which exist in
orthotropic
material, for instance. The second component of this index
covers the influence of stiffness coefficients that are nonzero only for non-cubic materials and remains zero otherwise.
Fiber-reinforced or layered
composite materials
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.
[11]
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
macromolecules
in polymeric materials,
polymers
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
hydrogels
, in order to mimic hierarchically ordered biological soft matter.
[12]
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.
[13]
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
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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
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Diffusion tensor imaging
is an
MRI
technique that involves measuring the fractional anisotropy of the random motion (
Brownian motion
) of water molecules in the brain. Water molecules located in
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
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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
BRDF
be
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.
![{\displaystyle P(\Omega _{i})=\int _{\Omega _{v}}\gamma (\Omega _{i},\Omega _{v}){\hat {n}}\cdot d{\hat {\Omega }}_{v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90bff0bbd8e083828d73e77c62a13d99a8121ef2)
![{\displaystyle A(\Omega _{i},\Omega _{v})={\frac {\gamma (\Omega _{i},\Omega _{v})}{P(\Omega _{i})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffefa34a1e1861761bdd2bb49159100ac276f00)
It is of interest because, with knowledge of the anisotropy function as defined, a measurement of the
BRDF
from a single viewing direction (say,
) yields a measure of the total scene reflectance (planar
albedo
) for that specific incident geometry (say,
).
See also
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References
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- ^
Smoot G. F.; Gorenstein M. V. &
Muller R. A.
(5 October 1977).
"Detection of Anisotropy in the Cosmic Blackbody Radiation"
(PDF)
.
Lawrence Berkeley Laboratory
and
Space Sciences Laboratory
,
University of California, Berkeley
.
Archived
(PDF)
from the original on 9 October 2022
. Retrieved
15 September
2013
.
- ^
Tian, Xiaojuan; Itkis, Mikhail E; Bekyarova, Elena B; Haddon, Robert C (8 April 2013).
"Anisotropic Thermal and Electrical Properties of Thin Thermal Interface Layers of Graphite Nanoplatelet-Based Composites"
.
Scientific Reports
.
3
: 1710.
Bibcode
:
2013NatSR...3E1710T
.
doi
:
10.1038/srep01710
.
PMC
3632880
.
- ^
Saurabh, Suman; Harpalani, Satya (2 January 2019). "Anisotropy of coal at various scales and its variation with sorption".
International Journal of Coal Geology
.
201
: 14?25.
doi
:
10.1016/j.coal.2018.11.008
.
S2CID
133624963
.
- ^
Oosterbaan, R. J. (1997).
"The Energy Balance of Groundwater Flow Applied to Subsurface Drainage in Anisotropic Soils by Pipes or Ditches With Entrance Resistance"
(PDF)
.
Archived
(PDF)
from the original on 19 February 2009.
The corresponding free EnDrain program can be downloaded from:
[1]
.
- ^
Oosterbaan, R. J. (2002).
"Subsurface Land Drainage By Tube Wells"
(PDF)
.
9 pp. The corresponding free WellDrain program can be downloaded from:
[2]
- ^
MAT, Mahmut (19 April 2018).
"Granite | Properties, Formation, Composition, Uses ≫ Geology Science"
.
Geology Science
. Retrieved
16 February
2024
.
- ^
Newnham, Robert E.
Properties of Materials: Anisotropy, Symmetry, Structure
(1st ed.). Oxford University Press.
ISBN
978-0198520764
.
- ^
Nye, J.F.
Physical Properties of Crystals
(1st ed.). Clarendon Press.
- ^
Courtney, Thomas H. (2005).
Mechanical Behavior of Materials
(2nd ed.). Waveland Pr Inc.
ISBN
978-1577664253
.
- ^
Sokołowski, Damian; Kami?ski, Marcin (1 September 2018).
"Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects"
.
Acta Mechanica
.
229
(9): 3727?3765.
doi
:
10.1007/s00707-018-2174-7
.
ISSN
1619-6937
.
S2CID
126198766
.
- ^
"Fabric Weave Styles"
.
Composite Envisions
. Retrieved
23 May
2019
.
- ^
Sano, Koki; Ishida, Yasuhiro; Aida, Tazuko (16 October 2017). "Synthesis of Anisotropic Hydrogels and Their Applications".
Angewandte Chemie International Edition
.
57
(10): 2532?2543.
doi
:
10.1002/anie.201708196
.
PMID
29034553
.
- ^
Wang, Xin; Jiang, Man; Gou, Jihua; Hui, David (1 February 2017). "3D printing of polymer matrix composites: A review and prospective".
Composites Part B: Engineering
.
110
: 442?458.
doi
:
10.1016/j.compositesb.2016.11.034
.
External links
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