{Eqn. 9.18}
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
9.2.4 Capillarity and Nanoscale Fluid Flow
In vivo nanodevices will often find it necessary to inject or to extract small aliquots of fluid from organelles, cells, tissues, or the environment. Internal gas or liquid transfers will also be commonplace in fluidic tethers, hydraulic communication and power conduits, nanohydraulic pistons, manipulators and metamorphic bumper drivers, and in bulk transfers into sensor cavities or chemical reaction chambers inside nanofactories (Chapter 19). Nanopipes may also be used in biomimetic nanosystems such as artificial organs and artificial cell components such as tubulin substitutes (Chapter 21). Hence it is essential to examine the nature of fluid flows and "wicking" in nanocapillary vessels, and the likely behavior of fluids in nanomachines.
The theory of macroscale capillarity is well-studied.1163 The concept of "wetting" is basic. Consider a fluid in a vessel. At the solid-liquid-gas interface (the vessel wall) there is a characteristic contact angle q, the angle at which the meniscus contacts the container wall. This angle is indicative of the balance between the forces of (1) adhesion between the liquid and the solid wall and (2) cohesion within the surface of the liquid. A contact angle <90° means that adhesion is strong and the liquid is "wetting" the tube; an angle >90° implies that adhesion is relatively weak and the liquid is "nonwetting." In air against glass, the liquids alcohol, glycerol, and pure water have q ~ 0°, while turpentine has q ~ 17° and impure water has q ~25°;1164 all are wetting. On the other hand, mercury on glass has q = 140° and water on paraffin has q = 109°;1164 both are nonwetting.
In the classical example of capillarity, one end of a narrow-bore open-ended tube is dipped vertically into a liquid. The liquid wets the tube, and the liquid rises until the force due to surface tension (Fcap) pulling the liquid upward equals the force of gravity (Fgrav) pulling the column of liquid downward at some height hcap, given by:1163
with wetting coefficient cos(q) = (gsg - gsl) / glg (the Young equation1163) where g is surface tension at the solid-gas (sg), solid-liquid (sl), or liquid-gas (lg) interfaces; rl and rg are liquid and gas density; g = 9.81 m/sec (acceleration of gravity) and rcap is the inside diameter of the tube. Thus pure water with air at STP in a glass capillary with rcap = 1 micron can rise hcap = 15 meters against gravity, representing a mean pressure pcap = hcap g (rl - rg) = 1.5 atm. A nonwetting liquid does not rise up the tube, but falls instead.
In the more general case of a linear capillary tube whose flow vector makes an angle jgrav with the ambient gravity field (e.g., jgrav = p is a tube pointing straight up), then the force on the fluid column is:
Taking the largest reasonable tube that might fit inside an in vivo nanorobot, hcap = 1 micron, rcap = 0.1 micron, rl - rg ~ 1000 kg/m3, glg ~ 72 x 10-3 N/m for pure water, and taking cos(q) ~ cos(jgrav) ~ 1, then Fcap/Fgrav ~ 108, giving the familiar result that gravity can usually be ignored in nanoscale systems. For this example, Fcolumn ~ Fcap = 44 nN and column fluid pressure pcolumn ~ 2 glg cos(q) / r = 14 atm. Taking instead the smallest reasonable nanocapillary tube with hcap = 20 nm and r = 2 nm, then Fcap/Fgrav ~ 1011, Fcolumn ~ 0.9 nN and pcolumn ~ 700 atm, making a quite substantial wicking force. Of course, the frictional effect of the interface region requires that a finite pressure P0 ~ 2 a / rcap must be applied before fluid motion will begin,1188 where a = 0.01-0.1 N/m (experimental); taking a ~ 0.01 N/m, then P0 ~ 2 atm for r = 0.1 micron, or ~100 atm for r = 2 nm.
The capillary force may also be compared to the simple hydrostatic suction force:
A nanopipette with inside radius rcap = 0.1 micron which is pressed against a plasma membrane surface and is then evacuated behind the attachment point sufficiently to create a surfacial pressure differential of Dp = 0.3 atm produces a purely hydrostatic suction force of Fsuck = 1 nN.
Such classical continuum models assume, among other things, that the molecular graininess of the fluid can be ignored. This assumption fails when tube dimensions (e.g., rcap) are comparable to the characteristic molecular length scale (l) of the fluid.10 In a gas, lgas is the mean free path between collisions. For Tgas = 310 K air with ngas = 2.4 x 1025 molecules/m3 at pgas = ngas kTgas = 1 atm pressure, the free path is given by:
under ideal gas conditions, if we take the effective molecular diameter as dgas ~ 0.2 nm. At high pressure, van der Waals equation must be used (Section 10.3.2), but for pgas = 1000 atm with ngas = 1.3 x 1028 molecules/m3, then lgas ~ 0.4 nm. In a liquid, lliq ~ dliq, the molecular diameter; for water molecules, lliq ~ 0.3 nm.
A ratio of rcap/l ~ 1 marks an important transition because it implies that molecules are colliding with the walls about as often as they are colliding with each other. If rcap << l, then intermolecular interaction is rare, wall collisions are relatively frequent, and molecular motion is largely ballistic between wall impacts. But if rcap >> l, then molecule/wall interactions are relatively rare and intermolecular collisions are relatively frequent. In the latter situation the continuum flow relations should become valid, for example when rcap >> 200 nm for gases at ~1 atm or when rcap >> 0.4 nm for gases at ~1000 atm, or when rcap >> 0.3 nm for liquids.
The range of non-continuum skin layer effects appears quite narrow in many cases. For instance, two smooth mica surfaces that are slowly brought together while immersed in various nonaqueous fluids display oscillating attractive and repulsive forces as a function of separation when the gap is less than 6-10 molecular diameters; periodic jump distances are about equal to the molecular diameter of the fluid.1165 Oscillations become smaller between rough surfaces and in liquids that mix molecules of different sizes.1149 Similar experiments in water with electrolyte produce similar behavior, but with ionic double layer repulsions superimposed -- oscillatory solvent forces become significant only at zsep < 2 nm.1166 Elastohydrodynamic simulations suggest that flat gold surfaces separated by 2.3 nm with 0.9-nm asperities (the "near-overlap" case) should slide smoothly on a thin film of liquid lubricant molecules, albeit with nanometer-scale cavitated zones extending ~3 nm downstream persisting for ~0.1 nanosec at a sliding velocity of 10 m/sec.1172 Solvation forces may also appear at separations below a few molecular diameters.1149
These results suggest that water-carrying nanocapillaries >2-4 nm in internal diameter should function largely in line with continuum models. Early theoretical calculations predicted that open-ended carbon nanotubes as small as 0.8 nm in diameter might act as "nanostraws" and could wick in molecules from vapor or fluid phases.1167 Subsequent experiments1168-1170 confirmed that fluids with glg <~ 190 x 10-3 N/m, including liquid sulfur, liquid selenium and nitric acid, are readily drawn into the inner cavity of carbon nanotubes with inside diameters of 48 nm through capillarity.1169 This limit is sufficiently high to allow nanotube wetting by water (glg ~ 72 x 10-3 N/m), most organic solvents (glg < 72 x 10-3 N/m), and liquid acids (e.g., glg ~ 43 x 10-3 N/m for HNO3) which can then be used as low surface tension carriers to introduce dissolved solute into the nanotubes.1170 Bulk nanotubes are readily wetted by water.1168 Good wetting is favored when the polarizability of the tube material is higher than that of the liquid.1171 Pure metals like molten lead or mercury with glg >~ 190 x 10-3 N/m do not wet carbon nanotubes and are not drawn in by capillarity. Such high-glg liquids may be forced into the nanotubes by applying a hydrostatic pressure pforce given by the Laplace equation:1168
Thus, liquid mercury with glg = 490 x 10-3 N/m can be pushed into an rcap = 0.5 micron tube by applying pforce ~ 20 atm to the fluid; pforce ~ 2000 atm for rcap = 5 nm.
By 1998, detailed molecular dynamics simulation of fluid flow inside 1.3-1.6 nm diameter carbon nanotubes had been a subject of active research.1173,1174 One major difference between macroscale and nanoscale fluidic systems is that in nanomachines the tube walls may be considerably less rigid* and may flex and resonate. Nanomachine designers must take into account the effects of these vibrations as they are transmitted through nanoscale structural elements attached to the tubes. Furthermore, tube wall motions are sometimes strongly size-dependent. The simulations confirmed that fluids flow faster through rigid tubes than through floppy tubes, as predicted by classical hydrodynamics theory (Section 9.2.5). At high flow velocities we may expect nanotubes to exhibit buckling modes, variable patency, progressive wave propagation, sluicing, flow-limiting flutter, and stall, by analogy to the fluid mechanics of compliant tube flow in human veins.361 Carbon nanotubes may be rigidified by imposing a stretching tension, or possibly may be replaced by stiffer diamondoid tubes.1173 If the fluid has more than one kind of atom, size-mass effects and differing interaction ranges or strengths may cause one or more species to segregate at the walls or to flow at different rates.1174
* Approximately 100 nN of force are required to produce the first buckling2659 of a carbon nanotube pressed perpendicularly onto a diamond surface. Nanotube compressive strength is ~1.5 x 1011 N/m2 ~ 1.5 million atm, easily sufficient to penetrate any biological surface.
Material flow through nanotubes has been well-exploited by biology, including the bacterial sex pili3549 (Section 5.4.2), bacterial anal pores and cytoprocts,3550 cellular excretory canals (e.g., in the glandular cells of the pancreas), 10-100 nm intranuclear nucleolar canals (Section 8.5.4.5), the 20-40 nm wide hollow tubular fluid transport network that forms the smooth intracellular endoplasmic reticulum (Section 8.5.3.5), the 75 nm diameter nutrient-transporting tubovesicular membrane network of the human malaria parasite,1183 the >100-nm wide fluid-carrying human bone canaliculi (Section 8.2.4), and prelymphatic tissue channels (Section 8.2.1.3).
The syringelike T4 bacteriophage tail assembly is perhaps the best-studied example of biological nanotube flow.1179,1180 The 100-nm long, 20-nm wide cylindrical T4 tail assembly, made of 15 different proteins joined in 24 annular segments with an 8-nm inside bore, has a set of small fibers near the tip that attach to the plasma membrane of the host cell. After a lysozyme-like enzyme opens a breach in the host cell wall, the tail sheath thickens and contracts,1181 inserting a ~2.8 megadalton hollow core protein nanotube (80 nm long, 7 nm wide, 2.5 nm inside bore) through the host cell integument. The protein nanotube is then uncorked in response to chemical signals, and a large-molecule (mostly putrescine and spermadine) pressure of ~30 atm ejects a single 70-micron long, 2-nm wide DNA thread (~50% of head volume) through the 2.5-nm nanotube aperture and out into the cell typically in ~3 seconds, a mean flow velocity of ~23 microns/sec.1178 (DNA packing into the capsids of some phages is purely mechanical.1723) The double-stranded DNA thread rotates ~4000 times around its axis as it emerges.1182 Under optimum conditions, initial injection velocity may start as high as 360 micron/sec with a minimum injection time of 0.23 sec.1178
The tensile strength of liquids may also be important in nanorobot internal fluid flows, even in the complete absence of capillarity. For instance, completely liquid-filled wetted tubes with no gas pockets make a continuous liquid column that can exhibit a large internal tensile strength Kliquid ~ 4 glg / xmolec,1163 where xmolec ~ 10 nm is the approximate maximum range of intermolecular forces. Taking glg ~ 72 x 10-3 N/m for water, Kliquid ~ 300 atm. There is experimental evidence that water saturated with air but denucleated by high pressures does exhibit a tensile strength on the order of ~300 atm.1175 This tensile strength, and not capillarity, is credited for the ability of tall trees to pull sap up to 115 meters high,1176 far exceeding the maximum 10.33 meters that water can be pulled vertically at the Earth's surface using a vacuum pump. M. Zimmermann2031 reviews the sap transport system; the maximum pull recorded experimentally in trees is 120 atm.2032
Last updated on 20 February 2003
{Eqn. 9.19}
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{Eqn. 9.24}