© 1999 Robert A. Freitas Jr. All Rights Reserved.

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 ~ T^1/2 (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 E_v ~ 25 zJ and the constant k_v ~ 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 h_a 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 (h_a varies ~20% during the menstrual cycle).^3575-3577 Human synovial fluid becomes increasingly incompressible at higher pressures, with h_a ~ 10 kg/m-sec at 'g ~ 0.1 sec^-1 (e.g., knee flexion during very slow walking) declining to h_a ~ 0.1 kg/m-sec at 'g ~ 10 sec^-1 (very fast running), and to h_a ~ 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 h_a ~ 1 kg/m-sec at low shear rates near 'g ~ 0.1-1 sec^-1, falling to h_a ~ 0.01 kg/m-sec at high shear rates near 'g ~ 100-1000 sec^-1.³⁶² Even ice is a viscoelastic material, with h_a ~ 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 E_v ~ 110 zJ as determined experimentally for pure ice,¹⁶⁰⁹ and taking k_v ~ 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