http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Daniel&feedformat=atomSoft-Matter - User contributions [en]2022-12-03T16:51:23ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22732FRET2011-12-04T23:47:19Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
Figure 2. Schematic of resonance energy transfer<br />
<br />
The principle of resonance energy transfer was first elucidated in the late 1940s by Theodor Förster. The idea was that if the fluorescent emission spectrum of the donor fluorophore overlaps with the absorption spectrum of the acceptor chromophore, and the two are near enough to each other, the donor fluorophore can transfer its excitation energy through long-range dipole-dipole interactions. This is a quantum mechanical process and does not require a collision and produces no heat. This is schematically shown in figure 2. Typically, the two chromophores have to be between 1-10 nm apart. At distance below 1 nm, other modes of electron and energy transfer becomes possible and beyond 10 nm, the probability of resonance energy transfer occuring becomes minimal. <br />
<br />
In fact, the quantum yield of FRET process is given by <br />
<br />
: <math>E=\frac{1}{1+(r/R_0)^6}\!</math><br />
<br />
where <math>R_0</math> is the förster radius and the distance for which the efficiency falls to half. Typically, for fluorophores, <math>R_0</math> is of the order of 6 nm. <br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
Figure 2. FRET can be used to study conformation of protein. <br />
<br />
FRET has many biological applications. It can be used to study the conformation of protein and other biological structures, where the desired resolution is of the order of 10 nm. FRET has also been used extensively in the study of lipid rafts, whose very existance is still an open question in biology. <br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22731FRET2011-12-04T23:44:18Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
Figure 2. Schematic of resonance energy transfer<br />
<br />
The principle of resonance energy transfer was first elucidated in the late 1940s by Theodor Förster. The idea was that if the fluorescent emission spectrum of the donor fluorophore overlaps with the absorption spectrum of the acceptor chromophore, and the two are near enough to each other, the donor fluorophore can transfer its excitation energy through long-range dipole-dipole interactions. This is a quantum mechanical process and does not require a collision and produces no heat. This is schematically shown in figure 2. Typically, the two chromophores have to be between 1-10 nm apart. At distance below 1 nm, other modes of electron and energy transfer becomes possible and beyond 10 nm, the probability of resonance energy transfer occuring becomes minimal. <br />
<br />
In fact, the quantum yield of FRET process is given by <br />
<br />
: <math>E=\frac{1}{1+(r/R_0)^6}\!</math><br />
<br />
where <math>R_0</math> is the förster radius and the distance for which the efficiency falls to half. Typically, for fluorophores, <math>R_0</math> is of the order of 6 nm. <br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
Figure 2. FRET can be used to study conformation of protein. <br />
<br />
FRET<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22730FRET2011-12-04T23:43:46Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
Figure 2. Schematic of resonance energy transfer<br />
<br />
The principle of resonance energy transfer was first elucidated in the late 1940s by Theodor Förster. The idea was that if the fluorescent emission spectrum of the donor fluorophore overlaps with the absorption spectrum of the acceptor chromophore, and the two are near enough to each other, the donor fluorophore can transfer its excitation energy through long-range dipole-dipole interactions. This is a quantum mechanical process and does not require a collision and produces no heat. This is schematically shown in figure 2. Typically, the two chromophores have to be between 1-10 nm apart. At distance below 1 nm, other modes of electron and energy transfer becomes possible and beyond 10 nm, the probability of resonance energy transfer occuring becomes minimal. <br />
<br />
In fact, the quantum yield of FRET process is given by <br />
<br />
: <math>E=\frac{1}{1+(r/R_0)^6}\!</math><br />
<br />
where <math>R_0</math> is the förster radius and the distance for which the efficiency falls to half. Typically, for fluorophores, <math>R_0</math> is of the order of 6 nm. <br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22729FRET2011-12-04T23:42:09Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
Figure 2. Schematic of resonance energy transfer<br />
<br />
The principle of resonance energy transfer was first elucidated in the late 1940s by Theodor Förster. The idea was that if the fluorescent emission spectrum of the donor fluorophore overlaps with the absorption spectrum of the acceptor chromophore, and the two are near enough to each other, the donor fluorophore can transfer its excitation energy through long-range dipole-dipole interactions. This is a quantum mechanical process and does not require a collision and produces no heat. This is schematically shown in figure 2. Typically, the two chromophores have to be between 1-10 nm apart. At distance below 1 nm, other modes of electron and energy transfer becomes possible and beyond 10 nm, the probability of resonance energy transfer occuring becomes minimal. <br />
<br />
In fact, the quantum yield of FRET process is given by <br />
<br />
: <math>E=\frac{1}{1+(r/R_0)^6}\!</math><br />
<br />
where <math>R_0</math><br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22728FRET2011-12-04T23:40:50Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
The principle of resonance energy transfer was first elucidated in the late 1940s by Theodor Förster. The idea was that if the fluorescent emission spectrum of the donor fluorophore overlaps with the absorption spectrum of the acceptor chromophore, and the two are near enough to each other, the donor fluorophore can transfer its excitation energy through long-range dipole-dipole interactions. This is a quantum mechanical process and does not require a collision and produces no heat. This is schematically shown in figure 2. Typically, the two chromophores have to be between 1-10 nm apart. At distance below 1 nm, other modes of electron and energy transfer becomes possible and beyond 10 nm, the probability of resonance energy transfer occuring becomes minimal. <br />
<br />
In fact, the quantum yield of FRET process is given by <br />
<br />
: <math>E=\frac{1}{1+(r/R_0)^6}\!</math><br />
<br />
where<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22727FRET2011-12-04T23:40:27Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
The principle of resonance energy transfer was first elucidated in the late 1940s by Theodor Förster. The idea was that if the fluorescent emission spectrum of the donor fluorophore overlaps with the absorption spectrum of the acceptor chromophore, and the two are near enough to each other, the donor fluorophore can transfer its excitation energy through long-range dipole-dipole interactions. This is a quantum mechanical process and does not require a collision and produces no heat. This is schematically shown in figure 2. Typically, the two chromophores have to be between 1-10 nm apart. At distance below 1 nm, other modes of electron and energy transfer becomes possible and beyond 10 nm, the probability of resonance energy transfer occuring becomes minimal. <br />
<br />
In fact, the quantum yield of FRET process is given by <br />
<br />
: <math>E=\frac{1}{1+(r/R_0)^6}\!</math><br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22726FRET2011-12-04T23:40:10Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
The principle of resonance energy transfer was first elucidated in the late 1940s by Theodor Förster. The idea was that if the fluorescent emission spectrum of the donor fluorophore overlaps with the absorption spectrum of the acceptor chromophore, and the two are near enough to each other, the donor fluorophore can transfer its excitation energy through long-range dipole-dipole interactions. This is a quantum mechanical process and does not require a collision and produces no heat. This is schematically shown in figure 2. Typically, the two chromophores have to be between 1-10 nm apart. At distance below 1 nm, other modes of electron and energy transfer becomes possible and beyond 10 nm, the probability of resonance energy transfer occuring becomes minimal. <br />
<br />
In fact, the quantum yield of FRET process is given by <br />
<br />
: <math>E=\frac{1}{1+(r/R_0)^6}\!</math><br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22725FRET2011-12-04T23:30:06Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22724FRET2011-12-04T23:28:24Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22723FRET2011-12-04T23:28:10Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
<br />
[[image:FRET3.png]]<br />
Figure 1. Schematic of FRET between the donor chromophore, cyan fluorescent protein (CFP), and the acceptor chromophore, yellow fluorescent protein (YFP). FRET will only occur when the two chromophores are sufficiently close to each other.<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22722FRET2011-12-04T23:25:45Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Förster resonance energy transfer (FRET) is a mechanism that describes the transfer of energy between chromophores. A donor chromophore, which is in its electronically excited state, can transfer the energy non-radiatively to a acceptor chromophore that is nearby (typically less than 10 nm away) through dipole-dipole coupling. This process is known as FRET and it is exquisitely dependent on the distance between the two chromophores. FRET is often referred to as fluorescence resonance energy transfer, though this is a misnomer since the energy transfer is non-radiative and does not involve fluorescence. FRET allows imaging of objects that is separated by a distance of the order of 10 nm, which is way below the abbe diffraction limit of a microscope of 200 nm. It is one of the modern imaging techniques that allows us to surpass the diffraction limit that is characteristic of wide-field microscopy. <br />
<br />
[[image:FRET3.png]]<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22721FRET2011-12-04T23:14:50Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
[[image:FRET3.png]]<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Acknowledgements==<br />
<br />
Figures 2 and 3 are taken from http://www.olympusfluoview.com/applications/fretintro.html<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22720FRET2011-12-04T23:14:02Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
[[image:FRET3.png]]<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=File:FRET3.png&diff=22719File:FRET3.png2011-12-04T23:13:45Z<p>Daniel: </p>
<hr />
<div></div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22718FRET2011-12-04T23:12:54Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
[[image:FRET3.jpg]]<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
==Applications==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22717FRET2011-12-04T23:12:03Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
[[image:FRET1.jpg]]<br />
<br />
==Theoretical Basis==<br />
<br />
[[image:FRET2.jpg]]<br />
<br />
==Applications==<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=File:FRET2.jpg&diff=22716File:FRET2.jpg2011-12-04T23:11:30Z<p>Daniel: </p>
<hr />
<div></div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=File:FRET1.jpg&diff=22715File:FRET1.jpg2011-12-04T23:11:06Z<p>Daniel: </p>
<hr />
<div></div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22714FRET2011-12-04T23:10:56Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
<br />
==Theoretical Basis==<br />
<br />
==Applications==<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22705Biomimetics2011-12-04T20:28:41Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. There are many early examples of biomimetics, such as the invention of velcro, which was inspired by tiny hooks found on the surface of burs and the cat's eye reflectors which were the results of studying the mechanism of cat's eyes. <br />
<br />
[[image:biomimetics1.jpg]]<br />
Figure 1. Tiny hooks found on the surface of burs.<br />
<br />
Examples of biomimetic systems can be found in the wikipedia article on bionics. <br />
http://en.wikipedia.org/wiki/Bionics<br />
<br />
==Biomimetics Chemistry==<br />
<br />
From the point of view of chemistry, biological systems are able to synthesize complex chemical compounds efficiently at relatively low temperature (e.g. human body's temperature ~37 degrees celsius), whereas we often requires the use of high temperature, high energy and huge reactors. Biological systems often achieve these through enzymatic reactions and it will interesting to study the way biological systems snynthesize chemical compounds to better optimize the way we do chemistry. <br />
<br />
A good article discussing this by Ronald Breslow in the Journal of Biological Chemistry. <br />
http://www.jbc.org/content/284/3/1337.full<br />
<br />
==Difference between biological systems and artificial systems==<br />
<br />
One main difference between biological and artificial system is that the former is responsive to the environment, exhibit homeostasis and self-repair properties, while the latter is often static, lacks self-regulatory abilities and is relatively unresponsive. The interest in studying biological systems is in part hoping to incorporate their design principles in smart material in the future which can responds to different environments appropriately. Examples include glass windows that can regulate the amount of sunlight entering the room to optimize energy efficiency.<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22702Biomimetics2011-12-04T20:19:55Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. There are many early examples of biomimetics, such as the invention of velcro, which was inspired by tiny hooks found on the surface of burs and the cat's eye reflectors which were the results of studying the mechanism of cat's eyes. <br />
<br />
[[image:biomimetics1.jpg]]<br />
Figure 1. Tiny hooks found on the surface of burs.<br />
<br />
==Biomimetics Chemistry==<br />
<br />
From the point of view of chemistry, biological systems are able to synthesize complex chemical compounds efficiently at relatively low temperature (e.g. human body's temperature ~37 degrees celsius), whereas we often requires the use of high temperature, high energy and huge reactors. Biological systems often achieve these through enzymatic reactions and it will interesting to study the way biological systems snynthesize chemical compounds to better optimize the way we do chemistry. <br />
<br />
A good article discussing this by Ronald Breslow in the Journal of Biological Chemistry. <br />
http://www.jbc.org/content/284/3/1337.full<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22699Biomimetics2011-12-04T20:07:58Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. There are many early examples of biomimetics, such as the invention of velcro, which was inspired by tiny hooks found on the surface of burs and the cat's eye reflectors which were the results of studying the mechanism of cat's eyes. <br />
<br />
[[image:biomimetics1.jpg]]<br />
Figure 1. Tiny hooks found on the surface of burs.<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22698Biomimetics2011-12-04T20:06:29Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. There are many early examples of biomimetics, such as the invention of velcro, which was inspired by tiny hooks found on the surface of burs. <br />
<br />
[[image:biomimetics1.jpg]]<br />
Figure 1. Tiny hooks found on the surface of burs.<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=File:Biomimetics2.jpg&diff=22697File:Biomimetics2.jpg2011-12-04T20:05:41Z<p>Daniel: </p>
<hr />
<div></div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=File:Biomimetics1.jpg&diff=22696File:Biomimetics1.jpg2011-12-04T20:05:34Z<p>Daniel: </p>
<hr />
<div></div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22695Biomimetics2011-12-04T20:05:24Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. There are many early examples of biomimetics, such as the invention of velcro, which was inspired by tiny hooks found on the surface of burs. <br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22694Biomimetics2011-12-04T20:02:02Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. <br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22692Biomimetics2011-12-04T20:00:26Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. <br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22691Biomimetics2011-12-04T19:59:20Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. <br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22690Biomimetics2011-12-04T19:56:23Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of natural structures<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=FRET&diff=22689FRET2011-12-04T19:52:02Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Crosslinking of cell-derived 3D scaffolds up-regulates the stretching and unfolding of new extracellular matrix assembled by reseeded cells]]<br />
<br />
[[FRET and FCS Imaging Techniques]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=22688Biomimetics2011-12-04T19:51:21Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Daniel_Daniel&diff=22687Daniel Daniel2011-12-04T19:47:29Z<p>Daniel: </p>
<hr />
<div>[[Pitcher plant inspired non-stick surface]]<br />
<br />
[[Control of Shape and Size of Nanopillar Assembly by Adhesion-Mediated Elastocapillary Interaction]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Microfluidic fabrication of smart micro gels from macromolecular precursors]]<br />
<br />
[[Semi-permeable vesicles composed of natural clay]]<br />
<br />
[[How non-iridescent colors are generated by quasi-ordered structures of bird feathers]]<br />
<br />
[[The Determination of the Location of Contact Electrification-Induced Discharge Events]]<br />
<br />
[[Contact angle associated with thin liquid films in emulsions]]<br />
<br />
[[Single-particle Brownian dynamics for characterizing the rheology of fluid Langmuir monolayers]]<br />
<br />
[[biomimetics]]<br />
<br />
[[Fractal Dimension]]<br />
<br />
[[FRET]]<br />
<br />
[[Soap films]]<br />
<br />
[[Quantum dot]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Quantum_dot&diff=22686Quantum dot2011-12-04T19:46:33Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Quantum dot is a semiconductor crystal, which has electronic properties intermediate between those of bulk semiconductor and those of discrete molecules. The electronic properties of quantum dot are closely related to the size and shape of the individual crystal. Generally, the smaller the crystal, the larger the band gap and the larger is the energy transition between the highest valence band and the lowest conducting band. A main advantage of quantum dot is that because the size of the crystal can be exquisitely controlled, the electronic properties of quantum dots can similarly be tuned precisely. <br />
<br />
==Physics of quantum dot==<br />
<br />
In bulk semiconductor, an electron-hole pair is typically bound with a characteristic length, called the exciton bohr radius. The electron-hole pair forms an unconfined hydrogen-like energy states. When the size of the crystal approaches that of the bohr radius, the electron and hole pairs become confined and the properties of the semiconductors change dramatically. A typical size of a quantum dot is of the order of several nms and there is confinement in all three spatial dimensions. <br />
<br />
==Applications==<br />
<br />
[[image:QD1.jpg]]<br />
<br />
Quantum dots can be used for various optical devices such as dye because of its superior optical properties. Quantum dots are much brighter than conventional fluorescent dyes (up to 20 times brighter) and are resistant to photobleaching. Though, one persistant issue that prevents its use in vivo is its toxicity. There have also been developments to use quantum dots as photovoltaic and light emitting devices. Quantum dots have also been suggested to be used as qubits in quantum computing. <br />
<br />
== Keyword in references: ==<br />
<br />
[[Nanocrystal Inks without Ligands: Stable Colloids of Bare Germanium Nanocrystals]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=File:QD1.jpg&diff=22684File:QD1.jpg2011-12-04T19:42:35Z<p>Daniel: </p>
<hr />
<div></div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Quantum_dot&diff=22683Quantum dot2011-12-04T19:42:27Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Quantum dot is a semiconductor crystal, which has electronic properties intermediate between those of bulk semiconductor and those of discrete molecules. The electronic properties of quantum dot are closely related to the size and shape of the individual crystal. Generally, the smaller the crystal, the larger the band gap and the larger is the energy transition between the highest valence band and the lowest conducting band. A main advantage of quantum dot is that because the size of the crystal can be exquisitely controlled, the electronic properties of quantum dots can similarly be tuned precisely. <br />
<br />
==Physics of quantum dot==<br />
<br />
In bulk semiconductor, an electron-hole pair is typically bound with a characteristic length, called the exciton bohr radius. The electron-hole pair forms an unconfined hydrogen-like energy states. When the size of the crystal approaches that of the bohr radius, the electron and hole pairs become confined and the properties of the semiconductors change dramatically. A typical size of a quantum dot is of the order of several nms and there is confinement in all three spatial dimensions. <br />
<br />
==Applications==<br />
<br />
[[image:QD1.jpg]]<br />
<br />
Quantum dots can be used for various optical devices such as dye because of its superior optical properties. Quantum dots are <br />
<br />
== Keyword in references: ==<br />
<br />
[[Nanocrystal Inks without Ligands: Stable Colloids of Bare Germanium Nanocrystals]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Quantum_dot&diff=22681Quantum dot2011-12-04T19:33:47Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
Quantum dot is a semiconductor crystal, which has electronic properties intermediate between those of bulk semiconductor and those of discrete molecules. The electronic properties of quantum dot are closely related to the size and shape of the individual crystal. Generally, the smaller the crystal, the larger the band gap and the larger is the energy transition between the highest valence band and the lowest conducting band. A main advantage of quantum dot is that because the size of the crystal can be exquisitely controlled, the electronic properties of quantum dots can similarly be tuned precisely. <br />
<br />
==Physics of <br />
== Keyword in references: ==<br />
<br />
[[Nanocrystal Inks without Ligands: Stable Colloids of Bare Germanium Nanocrystals]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22679Fractal Dimension2011-12-04T19:26:52Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size l required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by <math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by <math>R^2 = N l^2</math>, giving a fractal dimension of 2 using the definition in eq. (1) whatever dimension of space it is occupying. Flory has shown that a polymer can have D < 2 in particular when self-avoiding walk is accounted for, in which case D = 1.66. A collapsed polymer has D=3 and fills space completely. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Quantum_dot&diff=22678Quantum dot2011-12-04T19:26:31Z<p>Daniel: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
==Introduction==<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Nanocrystal Inks without Ligands: Stable Colloids of Bare Germanium Nanocrystals]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22677Fractal Dimension2011-12-04T18:06:48Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size l required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by <math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by <math>R^2 = N l^2</math>, giving a fractal dimension of 2 using the definition in eq. (1) whatever dimension of space it is occupying. Flory has shown that a polymer can have D < 2 in particular when self-avoiding walk is accounted for, in which case D = 1.66. A collapsed polymer has D=3 and fills space completely. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22676Fractal Dimension2011-12-04T18:05:50Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by <math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by <math>R^2 = N l^2</math>, giving a fractal dimension of 2 using the definition in eq. (1) whatever dimension of space it is occupying. Flory has shown that a polymer can have D < 2 in particular when self-avoiding walk is accounted for, in which case D = 1.66. A collapsed polymer has D=3 and fills space completely. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22675Fractal Dimension2011-12-04T18:00:16Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by <math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by <math>R^2 = N l^2</math> <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22674Fractal Dimension2011-12-04T17:59:56Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by :<math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by <math>R^2 = N l^2</math> <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22673Fractal Dimension2011-12-04T17:59:10Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by :<math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by :<math>R^2 = N l^2<\math> <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22672Fractal Dimension2011-12-04T17:58:58Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by :<math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by <math>R^2 = N l^2<\math> <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22671Fractal Dimension2011-12-04T17:56:42Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Eq(1)<br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by :<math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22670Fractal Dimension2011-12-04T17:56:23Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by :<math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22669Fractal Dimension2011-12-04T17:56:01Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
Equation (1)<br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by :<math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1). <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22668Fractal Dimension2011-12-04T17:54:48Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by <br />
<br />
<br />
:<math>L =M l^{(1-D)} </math><br />
<br />
<br />
where M is some positive constant and D is the fractal dimension. <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Danielhttp://soft-matter.seas.harvard.edu/index.php?title=Fractal_Dimension&diff=22667Fractal Dimension2011-12-04T17:54:28Z<p>Daniel: </p>
<hr />
<div>==Introduction==<br />
<br />
A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions. <br />
<br />
One possible working definition of the fractal dimension D is <br />
<br />
:<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math><br />
<br />
where N(l) is the number of self-similar structures of linear size(l) required to cover the original object. <br />
<br />
== Examples of fractals==<br />
<br />
[[image:Fractaldimensions1.png]]<br />
Figure 1. Koch snowflake<br />
<br />
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object. <br />
<br />
[[image:Fractaldimension2.jpg]]<br />
<br />
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.<br />
<br />
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by <br />
<br />
:<math>L =M l^{(1-D)} </math><br />
<br />
where M is some positive constant and D is the fractal dimension. <br />
<br />
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension<br />
<br />
== Fractals and Polymers==<br />
<br />
A polymer is an example of a fractal structure. <br />
<br />
== Keyword in references: ==<br />
<br />
[[G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]</div>Daniel