**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

**5.2.1 Free-Floating
Solitary Nanodevices**

Free-floating simple nanorobots intended solely as materials delivery or storage devices, or used as omnidirectional communications, navigational or control transponders operating independently in vivo, have no preferred orientation. Thus these nanorobots may be spherically symmetric with rigid surfaces. Spherical particles produce the smallest possible increment in blood viscosity as they tumble, in part because spheres offer the smallest possible interaction surface per unit volume of any geometrical shape. Such simple nanomachines are the kind most likely to be deployed in the greatest numbers, and their reduced surface area minimizes potential biocompatibility problems.

These free-floating nanodevices must have ready access to all tissues via blood vessels. Since they are nonmotile machines, to avoid getting stuck they must not physically extend wider across their longest axis than the width of human capillaries, which average 8 microns in diameter but may be as narrow as 4 microns (Section 8.2.1.2).

Consider a nanodevice of fixed surface area A_{n},
total volume V_{n}, and longest transdevice diameter L_{n}.
Then:

A. For a spherical device of radius r, then
L_{n} = 2 r, A_{n} = 4 p r^{2},
and V_{n} = (4/3) p r^{3}.

B. For a prolate spheroidal (football-shaped)
device of length 2a, width 2b, and eccentricity e = (a^{2} - b^{2})^{1/2}
/ a, then A_{n} = 2 p b^{2} + 2 p
a b (arcsin(e) / e) and V_{n} = (4/3) p a
b^{2}. For an oblate spheroidal device, A_{n} = 2 p
a^{2} + p b^{2} (ln({1+e}/{1-e})
/ e) and V_{n} = (4/3) p a^{2} b.
In both cases, L_{n} = 2a and the maximum enclosed volume per unit area
occurs at a = b (e.g., a sphere).

C. For a right circular disk or cylindrical
device of radius r and height h, then L_{n} = (h^{2} + 4r^{2})^{1/2},
A_{n} = 2 p r (h + r), and V_{n}
= p r^{2} h. Maximum enclosed volume per
unit area occurs at h = 2^{1/2} r.

D. For a right circular conical device of
radius r and height h, then L_{n} = 2r for h __<__ 3^{1/2}
r (but L_{n} = (h^{2} + r^{2})^{1/2} if h >
3^{1/2} r), A_{n} = p ( r^{2}
+ r (h^{2} + r^{2})^{1/2}), and V_{n} = p
r^{2} h / 3. Maximum enclosed volume per unit area occurs at h = 3^{1/2}
r.

E. For a right triangular prismatic device
(Fig.
5.4) with three equal sides of length s and height h, then L_{n}
= (h^{2} + s^{2})^{1/2}, A_{n} = 3 h s + 3^{1/2}
s^{2}/ 2, and V_{n} = 3^{1/2} s^{2} h / 4. Maximum
enclosed volume per unit area occurs at h = s / 2^{1/2}.

F. For a cubical device with three equal sides
of length s, then L_{n} = (3^{1/2}) s, A_{n} = 6 s^{2},
and V_{n} = s^{3}.

G. For a right square prismatic device with
two equal sides of length s and height h, then L_{n} = (h^{2}
+ 2 s^{2})^{1/2}, A_{n} = 2 s^{2} + 4 h s, and
V_{n} = h s^{2}. Maximum enclosed volume per unit area occurs
at h = s.

H. For a right hexagonal prismatic device
(Fig.
5.4) with six equal sides of length s and height h, then L_{n} =
(h^{2} + 4 s^{2})^{1/2}, A_{n} = 6 s (h + 3^{1/2}
s / 2), and V_{n} = 27^{1/2} s^{2} h / 2. Maximum enclosed
volume per unit area occurs at h = 2^{1/2} s.

I. For a truncated octahedral device (Fig.
5.5) of edge s, then L_{n} = (10^{1/2}) s, A_{n}
= (6 + 432^{1/2}) s^{2}, and V_{n} = (128^{1/2})
s^{3}.

J. For a rhombic dodecahedron^{1101}
(Fig.
5.6) of edge s, then L_{n} = (48^{1/2} / 3) s, A_{n}
~ (11.3137) s^{2}, and V_{n} ~ (3.0792) s^{3}.

K. For a nonregular octahedron^{ }(Fig.
5.8) of equatorial edge s and vertex edge (3^{1/2} / 2) s, then
L_{n} = (2^{1/2}) s, A_{n} = (8^{1/2}) s^{2},
and V_{n} = (1/3) s^{3}.

L. For a regular octahedron^{1101}
(Fig.
5.9) of edge s, then L_{n} = (2^{1/2}) s, A_{n }=
(12^{1/2})s^{2}, and V_{n} = (2^{1/2} / 3) s^{3}.

Table 5.1, computed using the above relations, confirms that spheres offer the greatest storage volume per unit surface area and thus are the most efficient shape for this application.

Last updated on 17 February 2003