**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

**9.2.7.6 Nanoplumbing
and Fluidic Circuits**

In fluid distribution systems that minimize total energy dissipation
in laminar nonpulsatile flow, space-filling fractal networks of branching tubes
are most efficient.^{3242} At each
branch point in such a network, where a single large tube of radius R bifurcates
into N branches with each branch tube having radii r_{1}, r_{2},...,
r_{N}, Murray^{1220} found
that shear stresses are equalized and flow impedance is minimized when:

In the special case where r = r_{1} = r_{2}
= ... = r_{N}, which is quite common in biological systems, Murray's
law reduces to R^{3} = N r^{3}. For example, the human bronchial
system typically has N = 2, so the ideal fractal network has R/r ~ N^{1/3}
= 1.26; from Table
8.7, the actual value from trachea (generation 0) through the terminal bronchiole
in generation 16 (after which alveoli begin to appear irregularly) is R/r =
1.24, in good agreement with Murray's law. In turbulent flow regimes, the exponent
on Murray's law becomes 2.33, rather than 3.^{1615}
And in pulsatile flow through elastic tubes, which dominates the aorta and major
arteries in the human circulatory system, the energy minimization principle
requires area-preserving branching, or R^{2} = N r^{2} for the
ideal network.^{698}

Complex nanoscale fluidic logic devices are readily imagined.
While fractal geometries are likely, for simplicity consider a three-dimensional
fluidic circuit of volume V_{circuit}. A volume fraction f_{tube}
consists of N_{tube} independent fluidic pathways each of length l_{tube}
and radius r_{tube}, and a second volume fraction f_{valve}
consists of N_{valve} fluidic gates each of volume L_{valve}^{3}.
If each independent fluidic pathway is gated, on average, by a single valve,
then N_{tube} = N_{valve} = N = f_{valve} V_{circuit}
/ L_{valve}^{3} and l_{tube} = (f_{tube} / p
f_{valve}) (L_{valve}^{3} / r_{tube}^{2}).
Ignoring reservoirs, inlet and outlet manifolds and support mechanisms, and
taking V_{circuit} = 1 micron^{3}, f_{tube} = 0.9, f_{valve}
= 0.1, L_{valve} = 20 nm, r_{tube} = 10 nm, and h
= 0.6915 x 10^{-3} kg/m-sec for water at 310 K, then the fluidic circuit
includes N_{valve} = 12,500 fluidic gates and N_{tube} = 12,500
independent fluidic pathways each of length l_{tube} = 230 nm. If typical
flow velocity v_{flow} = 1 mm/sec,^{1228}
then from Section 9.2.5 the total volume flow rate through
the circuit is N 'V_{HP} ~ 4000 micron^{3}/sec at a pressure
differential Dp = 0.1 atm. Circuit power dissipation
(ignoring valve dissipation) is N P_{flow} ~50 pW and t_{flow}
~0.2 millisec assuming fully parallel operation (e.g., a path length l_{tube}),
allowing a circuit operating frequency of t_{flow}^{-1} ~ 5000
Hz.

Capillary networks are readily gated by applying appropriate
voltages, allowing valveless switching of liquid flow among various fluidic
pathways. Purely electrical valving of fluid flows has been common practice
in neurobiological research for decades. For example, the flow of acetylcholine
from the open end of a micronsized pipette (as it is inserted into tissue) is
prevented by applying a negative bias or braking current of 3 nanoamps at the
tip; flow resumes when the braking current is removed.^{803}
Constant-discharge flow-control valves using opposing polymer brushes with ~25
chains of ~40 monomers each^{2902}
and micron-scale electrorheological diodes^{498}
have been investigated. Macroscale fluidic NOR logic elements that can be used
to construct arbitrary Boolean logic circuits* (for controlling materials flow)
are widely available commercially,^{1227}
although these often employ the Coanda effect, vortices, or turbulence effects,
which generally don't scale well to micron and submicron devices. In 1997, a
microfluidic chip-based system for the integration of high-throughput drug discovery
efforts was demonstrated by SmithKline Beecham and Orchid Biocomputer.^{1222
}This microfluidic chip incorporated microfabricated components for valving
and pumping of organic solvents using electrokinetic transport within a three-dimensional
fluidic network. The pumping and valving mechanisms had no moving parts, "making
large scale integration feasible and inherently reliable." The study of microfluidic
networks^{1228,2697,2698
}using hundreds of micron-wide channels was an active research area in
1998, and integrated chemical systems were being widely discussed.^{121}

* In the 1950s, Marvin Minsky and Rollo Silver^{289}
built a "hydroflip computer" using hydraulic logic elements consisting
of millimeter-wide grooves and holes in multiple layers of plastic sheets with
small rods and balls inserted in some of the grooves. When the assembly was
pressed together and connected to a water supply, it became a hydraulic computer
powered by a 3-inch high column of water, operating at ~30 Hz.

Last updated on 20 February 2003