File:0315bearingSums.gif - Revision history

Apm: /* Associated text from Eric's Website */

2021-09-15T13:50:50Z

Associated text from Eric's Website

← Older revision		Revision as of 15:50, 15 September 2021
Line 6:		Line 6:
	= Associated text from Eric's Website =		= Associated text from Eric's Website =

	== ~~symmetric~~ molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function ==		== Symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function ==

	The potential energy of a single atom moving smoothly past a closely-spaced ring of other atoms is approximately sinusoidal, as shown in the upper diagram. The potential energy function for a ring of atoms moving smoothly past another ring is a sum of near-sinusoids, with the net potential having a spatial frequency that depends on the least common multiple of the symmetry-numbers of the two rings. Intuitively, the energy is at a maximum when (any) two atoms are at their point of closest approach, and from such a position, a small rotation places a different pair of atoms in the same situation (when the LCM is large, as in the example here).		The potential energy of a single atom moving smoothly past a closely-spaced ring of other atoms is approximately sinusoidal, as shown in the upper diagram. The potential energy function for a ring of atoms moving smoothly past another ring is a sum of near-sinusoids, with the net potential having a spatial frequency that depends on the least common multiple of the symmetry-numbers of the two rings. Intuitively, the energy is at a maximum when (any) two atoms are at their point of closest approach, and from such a position, a small rotation places a different pair of atoms in the same situation (when the LCM is large, as in the example here).

	This symmetry property of the potential energy function ensures that only high spatial frequency components of the interatomic potential contribute to rotational energy barriers. Explicit calculations and molecular mechanical models of detailed designs routinely yield rotational barriers of negligible magnitude (< 0.001zJ). These low barriers depend little on parameters such as bond lengths and bond angles, and hold true for any physically reasonable model of the interatomic potential. They do, however, require careful choice of symmetry and avoidance of designs in which elastic instabilities violate the smooth-interaction condition. Barriers for similar bearings of reduced symmetry (slightly deformed sleeves, displaced shafts) are larger, but can still be very low by the relevant mechanical standards.		This symmetry property of the potential energy function ensures that only high spatial frequency components of the interatomic potential contribute to rotational energy barriers. Explicit calculations and molecular mechanical models of detailed designs routinely yield rotational barriers of negligible magnitude (< 0.001zJ). These low barriers depend little on parameters such as bond lengths and bond angles, and hold true for any physically reasonable model of the interatomic potential. They do, however, require careful choice of symmetry and avoidance of designs in which elastic instabilities violate the smooth-interaction condition. Barriers for similar bearings of reduced symmetry (slightly deformed sleeves, displaced shafts) are larger, but can still be very low by the relevant mechanical standards.

Apm: /* Associated text from Eric's Website */

2021-09-15T13:50:22Z

Associated text from Eric's Website

← Older revision		Revision as of 15:50, 15 September 2021
Line 6:		Line 6:
	= Associated text from Eric's Website =		= Associated text from Eric's Website =

	== ~~Symmetric~~ molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function ==		== symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function ==

	The potential energy of a single atom moving smoothly past a closely-spaced ring of other atoms is approximately sinusoidal, as shown in the upper diagram. The potential energy function for a ring of atoms moving smoothly past another ring is a sum of near-sinusoids, with the net potential having a spatial frequency that depends on the least common multiple of the symmetry-numbers of the two rings. Intuitively, the energy is at a maximum when (any) two atoms are at their point of closest approach, and from such a position, a small rotation places a different pair of atoms in the same situation (when the LCM is large, as in the example here).		The potential energy of a single atom moving smoothly past a closely-spaced ring of other atoms is approximately sinusoidal, as shown in the upper diagram. The potential energy function for a ring of atoms moving smoothly past another ring is a sum of near-sinusoids, with the net potential having a spatial frequency that depends on the least common multiple of the symmetry-numbers of the two rings. Intuitively, the energy is at a maximum when (any) two atoms are at their point of closest approach, and from such a position, a small rotation places a different pair of atoms in the same situation (when the LCM is large, as in the example here).

	This symmetry property of the potential energy function ensures that only high spatial frequency components of the interatomic potential contribute to rotational energy barriers. Explicit calculations and molecular mechanical models of detailed designs routinely yield rotational barriers of negligible magnitude (< 0.001zJ). These low barriers depend little on parameters such as bond lengths and bond angles, and hold true for any physically reasonable model of the interatomic potential. They do, however, require careful choice of symmetry and avoidance of designs in which elastic instabilities violate the smooth-interaction condition. Barriers for similar bearings of reduced symmetry (slightly deformed sleeves, displaced shafts) are larger, but can still be very low by the relevant mechanical standards.		This symmetry property of the potential energy function ensures that only high spatial frequency components of the interatomic potential contribute to rotational energy barriers. Explicit calculations and molecular mechanical models of detailed designs routinely yield rotational barriers of negligible magnitude (< 0.001zJ). These low barriers depend little on parameters such as bond lengths and bond angles, and hold true for any physically reasonable model of the interatomic potential. They do, however, require careful choice of symmetry and avoidance of designs in which elastic instabilities violate the smooth-interaction condition. Barriers for similar bearings of reduced symmetry (slightly deformed sleeves, displaced shafts) are larger, but can still be very low by the relevant mechanical standards.

Apm: /* associated text from Eric's Website */

2021-09-15T13:32:03Z

associated text from Eric's Website

← Older revision		Revision as of 15:32, 15 September 2021
Line 4:		Line 4:
	"bearing symmetry"		"bearing symmetry"

	= ~~associated~~ text from Eric's Website =		= Associated text from Eric's Website =

	== Symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function ==		== Symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function ==

Apm at 13:30, 15 September 2021

2021-09-15T13:30:28Z

← Older revision		Revision as of 15:30, 15 September 2021
Line 2:		Line 2:
	Author: Eric K. Drexler		Author: Eric K. Drexler

	* bearing symmetry		"bearing symmetry"
	* "Symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function"
			= associated text from Eric's Website =

			== Symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function ==

			The potential energy of a single atom moving smoothly past a closely-spaced ring of other atoms is approximately sinusoidal, as shown in the upper diagram. The potential energy function for a ring of atoms moving smoothly past another ring is a sum of near-sinusoids, with the net potential having a spatial frequency that depends on the least common multiple of the symmetry-numbers of the two rings. Intuitively, the energy is at a maximum when (any) two atoms are at their point of closest approach, and from such a position, a small rotation places a different pair of atoms in the same situation (when the LCM is large, as in the example here).

			This symmetry property of the potential energy function ensures that only high spatial frequency components of the interatomic potential contribute to rotational energy barriers. Explicit calculations and molecular mechanical models of detailed designs routinely yield rotational barriers of negligible magnitude (< 0.001zJ). These low barriers depend little on parameters such as bond lengths and bond angles, and hold true for any physically reasonable model of the interatomic potential. They do, however, require careful choice of symmetry and avoidance of designs in which elastic instabilities violate the smooth-interaction condition. Barriers for similar bearings of reduced symmetry (slightly deformed sleeves, displaced shafts) are larger, but can still be very low by the relevant mechanical standards.

Apm at 13:21, 15 September 2021

2021-09-15T13:21:14Z

← Older revision		Revision as of 15:21, 15 September 2021
Line 1:		Line 1:
	Source: https://web.archive.org/web/20160314084528/http://e-drexler.com/p/04/02/0315bearingSums.html <br>		Source: https://web.archive.org/web/20160314084528/http://e-drexler.com/p/04/02/0315bearingSums.html <br>
	Author: Eric K. Drexler		Author: Eric K. Drexler

			* bearing symmetry
			* "Symmetric molecular bearings can exhibit low energy barriers that are insensitive to details of the potential energy function"

Apm: Source: https://web.archive.org/web/20160314084528/http://e-drexler.com/p/04/02/0315bearingSums.html
Author: Eric K. Drexler

2021-09-15T12:56:07Z

Source: https://web.archive.org/web/20160314084528/http://e-drexler.com/p/04/02/0315bearingSums.html <br> Author: Eric K. Drexler

New page

Source: https://web.archive.org/web/20160314084528/http://e-drexler.com/p/04/02/0315bearingSums.html <br>
Author: Eric K. Drexler

File:0315bearingSums.gif - Revision history

Apm: /* Associated text from Eric's Website */

Apm: /* Associated text from Eric's Website */

Apm: /* associated text from Eric's Website */

Apm at 13:30, 15 September 2021

Apm at 13:21, 15 September 2021

Apm: Source: https://web.archive.org/web/20160314084528/http://e-drexler.com/p/04/02/0315bearingSums.html Author: Eric K. Drexler

Apm: Source: https://web.archive.org/web/20160314084528/http://e-drexler.com/p/04/02/0315bearingSums.html
Author: Eric K. Drexler