© 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 r₁, r₂,..., r_N, Murray¹²²⁰ found that shear stresses are equalized and flow impedance is minimized when:

{Eqn. 9.41}

In the special case where r = r₁ = r₂ = ... = r_N, which is quite common in biological systems, Murray's law reduces to R³ = N r³. 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 R² = N r² 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_valve³. If each independent fluidic pathway is gated, on average, by a single valve, then N_tube = N_valve = N = f_valve V_circuit / L_valve³ and l_tube = (f_tube / p f_valve) (L_valve³ / r_tube²). Ignoring reservoirs, inlet and outlet manifolds and support mechanisms, and taking V_circuit = 1 micron³, 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 micron³/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 each²⁹⁰² and micron-scale electrorheological diodes⁴⁹⁸ 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 Silver²⁸⁹ 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