Nanomedicine, Volume I: Basic Capabilities

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

9.4.1.1 Biofluid Viscosity

Viscosity is a measure of the resistance of a fluid to shearing when the fluid is in motion. Consider a plate of surface area A moving parallel to a fixed plane surface with constant velocity v, being pushed laterally by a constant force F. A fluid of viscosity h fills the volume between the two surfaces, which are separated by a distance d. The fluid layer nearest the moving plate also moves at velocity v and the layer nearest the stationary plane remains stationary. In between the two surfaces, the velocity increases linearly with distance, establishing a constant gradient. This gradient is usually called the shear rate (essentially a size-normalized velocity), or 'g = v/d (m/sec-m, or sec-1). The absolute viscosity may then be defined by:

{Eqn. 9.58}

where h has MKS units of N-sec/m2, Pascal-sec, or, more simply, kg/m-sec. In a Newtonian fluid, the shear stress F/A (N/m2) increases linearly with shear rate 'g, so that viscosity h is constant over a wide range of shear rates. Many common fluids such as air, water, saline and blood serum closely approximate the ideal Newtonian fluid. The viscosity of solutions of molecules is related to the diffusion coefficient; see Eqn. 3.5.

The viscosities of some common materials are given in Table 9.4. The viscosity of an ideal gas is independent of density and independent of pressure between 0.01-10 atm; at higher pressures, intermolecular interactions lead to higher viscosities.390 Theory predicts that gas viscosity ~ T1/2 (e.g., rises with the square-root of temperature T), but a somewhat larger temperature exponent is obtained experimentally for real gases. In contrast, the viscosity of liquids increases with rising pressure (typically by a factor of 2-3 after moving from 1 atm up to 1000 atm for organic liquids) and decreases with rising temperature. In normal liquids the temperature dependence is approximated by Andrade's formula which gives:

{Eqn. 9.59}

where for example the activation energy for viscosity Ev ~ 25 zJ and the constant kv ~ 2.1 x 10-6 kg/m-sec for pure water at 1 atm. Nonelectrolytes dissolved in water generally cause viscosity to rise, while solvation of electrolytes may increase or decrease viscosity (the effect is typically ~10% or less for a 1M solution). Large asymmetric solute molecules increase viscosity more than an equal mass of small spherical molecules. The threshold between solid and liquid is generally taken as h ~ 1014 kg/m-sec.364

Most biofluids are viscoelastic and non-Newtonian, with apparent viscosity ha varying with shear rate and displaying other nonlinear characteristics such as hysteresis, relaxation, and creep.362 Saliva behaves more like an elastic body than like water. Because of its high molecular weight, DNA solution is viscoelastic even at low concentrations. Sex glands produce viscoelastic fluids, including semen and uterine cervical mucus1392 (ha varies ~20% during the menstrual cycle).3575-3577 Human synovial fluid becomes increasingly incompressible at higher pressures, with ha ~ 10 kg/m-sec at 'g ~ 0.1 sec-1 (e.g., knee flexion during very slow walking) declining to ha ~ 0.1 kg/m-sec at 'g ~ 10 sec-1 (very fast running), and to ha ~ 0.001 kg/m-sec at 'g ~ 10,000 sec-1 (experimental).362 Viscoelasticity is also an important property of respiratory tract mucus, which typically shows an ha ~ 1 kg/m-sec at low shear rates near 'g ~ 0.1-1 sec-1, falling to ha ~ 0.01 kg/m-sec at high shear rates near 'g ~ 100-1000 sec-1.362 Even ice is a viscoelastic material, with ha ~ 1010­1013 kg/m-sec (estimated as maximum shear stress divided by shear rate from data in Sinha et al1609) for 'g ~ 10-3-10-7 sec-1 at 262 K; ice viscosity varies with temperature (Table 9.4) as described by Andrade's formula, with Ev ~ 110 zJ as determined experimentally for pure ice,1609 and taking kv ~ 6.3 x 10-4 kg/m-sec at high shear rate.

Cytoplasm is a complex viscoelastic material having a continuous liquid phase (the cytosol) plus various suspended particles, granules, and membranous structures. More precisely, the cytomatrix is a mixed-phase body composed of a fibrillar network penetrated by a solution.1408 As a result, viscosity is different in the various phases. From flow behavior, the viscosity of E. coli protoplasm was estimated as ~1000 kg/m-sec; from measured diffusion rates of sucrose, dextran, and b-galactosidase, the apparent viscosity was 3-4 x 10-3 kg/m-sec.1407 Cytoplasm is inhomogeneous and anisotropic at many levels of organization.

Last updated on 21 February 2003