Nanomedicine, Volume I: Basic Capabilities
© 1999 Robert A. Freitas Jr. All Rights Reserved.
Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999
126.96.36.199 Nanopendulum Tachometry
The smallest angular velocity detectable using a simple nanopendulum is given approximately by wmin ~ Dx / r tmeas = 2 milliradian/sec (~0.1 deg/sec) for Dx = 1 nm, r = 500 nm, and tmeas = 1 sec. The largest detectable angular velocity wmax ~ 2 p / tmeas if tmeas is the minimum time required to count one complete rotation of the sensor; the smallest tmeas ~ 2 p / wmax ~ 100 nanosec for wmax ~ 49 megaradians/secor ~5% of the bursting speed (Eqn. 4.17) for an r = 1 micron spherical diamondoid nanodevice.
What rotational displacements will normally be encountered by an in vivo medical nanorobot? From Eqn. 3.2, a 1-micron device experiences instantaneous Brownian rotations of wB = (kT / 4 p h R3 t)1/2 radians/sec, where t = (MWkg / 32 p kT NA cH2O R4)1/2, the mean time between molecular collisions with the device (mostly by water molecules), from Eqn. 4.3, where cH2O ~ 3.3 x 1028 molecules/m3. For R = 0.5 microns and T = 310 K, t ~ 6 x 10-20 sec/collision and wB ~ 7 x 109 rad/sec, with each displacement of ~ t wB = 4 x 10-10 radian, none of which are detectable. However, the net of all Brownian rotational displacements is one rotation every 16 sec, ~0.4 radian/sec (Section 3.2.1).
The largest measurable angular displacements in vivo are caused by tumbling due to differential shear forces in the bloodstream. Normal vessel wall shear rates in physiological bloodflow range from 100-1400 radians/sec in the larger arteries to 500-4000 radians/sec in the smaller arteries and capillaries.386 The measured rolling velocity of white cells during vascular "tank tread" locomotion (Section 188.8.131.52) on venule walls is ~40 micron/sec, or ~8 radians/sec, which may be encountered by nanodevices employing cytocarriage (Section 9.4.7). The human vestibular mechanism has a frequency response range of 0.048-260 radians/sec.449
Thus, the maximum range of physiologically relevant rotation rates is 10-2 to 104 radians/sec. Since the natural harmonic frequency of a simple nanopendulum is wres = (g/r)1/2 = 4400 radians/sec for r = 500 nm, the possibility of forced oscillations and resonance need be faced only when measuring the highest likely physiological rotation rates, in which case the addition of a damping force Fdamp ~ m vturn wres during resonance events should eliminate the problem. Fdamp for critical damping peaks at ~mg ~ 0.01 pN for the m = 10-15 kg, r = 500 nm case (vturn ~ 2 mm/sec).
Last updated on 17 February 2003