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
6.5.3 Nanorobot Power Scaling
As an initial, crude order-of-magnitude estimate of nanorobot power consumption, allometric scaling of metabolism in biology for whole organisms follows a 3/4 power law.698,3242 Normalizing to P = 100 watts for an m = 70 kg human body mass and assuming ~water density for nanorobots, then P = (4.13) m3/4 = 23 pW for a 1 micron3 nanorobot, a power density of Dn ~ 2 x 107 watts/m3.
Surface power intensity considerations drive maximum nanorobot onboard power density. For instance, the maximum safe intensity for ultrasound is 100-1000 watts/m2 (pain threshold for human hearing ~100 watts/m2) (Section 6.4.1) and the conservative maximum safe electromagnetic intensity is also ~100 watts/m2 (Section 6.4.2). Additionally, a surface at 373 K (boiling water) relative to a 310 K environment radiates 100-1000 watts/m2 at emissivities er = 0.1-1 (Eqn. 6.19). Conservatively taking 100 watts/m2 as the maximum safe energy flux across the surface of a spherical nanorobot of radius rn, then the maximum nanorobot power density is
for a device ~1 micron in diameter -- the largest safe whole-device power density that should be developed in vivo. A 109 watt/m3 maximum implies a ~1000 pW limit for 1 micron3 nanorobots. Interestingly, the energy dissipation rate required to disrupt the plasma membrane of ~95% of all animal cells transported in forced turbulent capillary flows is on the order of ~108-109 watts/m3.1185
For chemically powered foraging nanorobots, another fundamental constraint on power density is imposed by diffusion limits on fuel molecules. From Eqn. 3.4, the maximum diffusion current of glucose molecules of energy content Efuel = Eglu (4765 zJ; Eqn. 6.16), burned at efficiency e% ~ 0.50 (50%) in oxygen, to the surface of a spherical nanorobot of radius rn will support a maximum onboard power density of:
For D = 7.1 x 10-10 m2/sec for glucose in water at 310 K (Table 3.3), C = (0.67) 3.5 x 1024 molecules/m3 in (newborn and) adult human blood plasma (Appendix B), and rn ~ 0.5 micron, Ddiff = 17 x 1010 watts/m3. However, because oxygen dissolves only slightly in blood plasma and interstitial fluid, the oxyglucose engine is more severely diffusion-limited by its oxygen requirements than by its glucose requirements.* Applying Eqn. 6.58 and using (for oxygen) D = 2.0 x 10-9 m2/sec (Table 3.3), C = 7.3 x 1022 molecules/m3 in arterial blood plasma (Appendix B), Efuel ~ Eglu/6, and rn ~ 0.5 micron, Ddiff = 7 x 108 watts/m3. Once again, ~109 watts/m3 appears to be a correct upper limit for whole nanorobots.**
* Since oxyglucose foraging nanorobots are not seriously glucose-limited, there is little to be gained by enabling energy organs or cooperative nanorobot populations to secrete insulin/glucagon hormones (mimicking the pancreas), cortisol hormones, etc. to artificially manipulate serum glucose levels. Available oxygen also may be artificially manipulated using nanorobotic compressed gas dispensers1400 or by other means (Chapter 22).
** Note that as nanodevices get smaller, their surface/volume ratio expands. Thus the Square/Cube law predicts that smaller nanodevices can admit more energy (e.g., chemical, acoustic, electromagnetic) through their surfaces per unit enclosed volume of working nanomachinery, hence can have higher power densities, as illustrated by Eqn. 6.58.
The disposition of combustion byproducts, particularly CO2, may provide another weak constraint on chemical systems. For example, a 10pW glucose-burning nanorobot generates ~107 molecules/sec of CO2. Pressurized to 1000 atm, this production rate fills ~0.001 micron3/sec of onboard storage space, assuming no venting. If CO2 is vented from a population of 1012 10-pW nanorobots uniformly distributed throughout a ~0.1 m3 human body volume, then local CO2 concentration rises by ~2 x 10-7 M/sec, reaching 0.0003 M after 1 hour of continuous operation assuming no physiological removal -- still well below the normal ~0.001 M blood plasma CO2 concentration.
By implication, these limits also drive the maximum number density of nanodevices deployable in human tissue. For example, at 109 watts/m3 the hottest 1-micron nanorobot develops 1000 pW; assuming a ~0.1 watt power budget when restricted to the thyroid gland (Section 6.5.2), 100 million nanodevices may be deployed in the gland giving a maximum number density of ~1013 nanorobots/m3. Nanorobot power consumption may range from ~0.1 pW for simple respirocytes1400 (Chapter 22) up to ~10,000 pW or more for the largest and most sophisticated repair and defensive in vivo devices (Chapter 21), but the typical simple micron-scale nanorobot may develop ~10 pW (roughly in line with biologically-derived allometric scaling laws giving Dn ~ 107 watts/m3) and thus could safely achieve a number density of ~1015 nanorobots/m3 (~10 micron mean interdevice separation). Recall that the maximum diffusion-limited total power draw for a population of oxygen-unrestricted glucose-energized foraging nanorobots in cyto is ~70,000-300,000 pW (Section 184.108.40.206).
Minimum powerplant size varies with requirements. Assuming >~1012 watts/m3 energy conversion for chemical (Section 6.3.4) or electrical (Section 6.4.1) power transducers, then a 100 pW power supply subsystem inside a working nanorobot may be as small as (~50 nm)3 in size.
Macroscale masses of working nanodevices may grow extremely hot, placing major scaling limits on artificial nano-organs and other large-scale nanomachinery aggregates (Chapter 14). As a somewhat fanciful example, consider a macroscopic ball of radius R consisting of N tightly-packed nanodevices each of mass density r and whole-nanorobot power density Dp ~ 109 watts/m3, of which nanodevices some fraction fn are active, all suspended in mid-air. The ball grows hotter as R (~N1/3) rises, until at some "critical combustible mass" Mcrit = (4/3) p r Rcrit3 the surface temperature exceeds the maximum combustion point for diamond in air (Tburn = 1070 K)691 and the solid ball of nanorobots bursts into flame. (Sapphire devices cannot burn, but have a Tmelt ~ 2310 K;1602 as a practical matter, nanomachinery may fail at temperatures significantly below Tburn.)
From simple geometry and neglecting ~2% air conduction losses, the maximum noncombustible aggregate radius Rcrit is:
For er = 0.97 (e.g., carbon black) to maximize heat emission at the lowest possible temperature and Tenviron = 300 K, then Rcrit = 0.22 mm for fn = 100%. Assuming a full cold start, critical time to incineration is tcrit = CV (Tburn - Tenviron) / fn Dp = 1.4 sec for fn = 100% if nanorobot heat capacity CV = 1.8 x 106 joules/m3-K (~diamondoid). Decreasing Dp to a more reasonable 107 watts/m3 or simply switching off 99% of the nanorobots (fn = 1%) increases Rcrit to ~22 cm.
Last updated on 18 February 2003