If an external stress is exerted on a fluid, the latter will flow to a degree that is determined by internal friction forces derived from internal molecular interactions and by the magnitude of the external stress applied. The measure of resistance to flow is defined as the viscosity of the fluid. The coefficient of viscosity, , historically termed the absolute viscosity, is defined by Newton's law of viscous flow:

	=	= shear stress, defined as the applied stress that causes successive parallel layers of a material body to move in their own planes relative to each other
	=	= shear rate, defined as the rate of change of shear strain with time

It should be noted that although stress and strain are often used interchangeably, the terms are not used interchangeably in rheology: stress is synonymous with force per unit area or a system of forces per unit area, and strain refers to deformation (i.e., a change in shape or size). A fluid is said to exhibit Newtonian flow if the viscosity is a constant that is independent of the shear rate or applied shear stress. Because viscosity depends on temperature, the temperature of the substance being measured should be controlled to within ±0.1.

In general, Newton's law is obeyed by liquids of low molecular weight and by solutions of low molecular weight solutes in low molecular weight solvents. However, liquids of high molecular weight, solutions that contain high molecular weight solutes and colloidal dispersions (e.g., suspensions and emulsions) typically do not obey Newton's law of viscous flow and are referred to as non-Newtonian fluids.

In the International System of Units (SI), the unit of viscosity is the pascal-second (Pa·s), which is equivalent to 1 kg·m1·s1 or 10 Poise in the centimeter-gram-second (cgs) system. The viscosities of most Newtonian fluids are measured in millipascal-seconds (1 Pa·s = 1000 mPa·s; 1 mPa·s = 1 centipoise). Kinematic viscosity, , relates the viscous force to the inertial force by relating the Newtonian viscosity of a fluid to its density, , at the same temperature:

The units of kinematic viscosity in SI units are m2·s1 or, more commonly, mm2·s1. [Note—1 mm2·s1 = 1 centistoke (cS or cSt). Stokes or centistokes are the units in the cgs system for kinematic viscosity. ]

Newton's law of viscous deformation or flow, described in the section on Newtonian viscosity, describes the relationship between an applied stress, , and the resultant flow with a rate of shear, or velocity gradient, of . Disperse systems (i.e., suspensions or emulsions) or fluids that contain macromolecular components usually do not obey Newton's law. These exceptions to Newtonian behavior are the subject of this section. The categorization of such exceptions into different flow behaviors such as shear-thinning (pseudoplastic), shear-thickening (dilatant), Bingham, etc., is not rigid. Non-Newtonian fluids may exhibit different rheological behavior depending on shear rate, shear stress, and temperature. Time-dependent rheological behavior, such as rheopexy or thixotropy, is not considered in this chapter.

The rheological behavior of a non-Newtonian fluid is evident when shear stress is plotted as a function of shear rate. The shear stress–shear rate relationship for Newtonian compositions is linear and passes through the origin, whereas the shear stress–shear rate relationship for non-Newtonian fluids is nonlinear and may not pass through the origin, or it is linear but does not pass through the origin (Bingham body). Rheograms, or flow curves, of some typical non-Newtonian fluids are contrasted with those of Newtonian fluids in Figure 3.

All the rheograms in Figure 3 can be described by the Herschel–Bulkley equation:

	=	= shear stress, in Pa
K	=	= consistency coefficient
	=	= shear rate, in s1
n	=	= the flow behavior index
0	=	= yield stress, in Pa

For Newtonian fluids and Bingham plastics, K is designated as the viscosity () and plastic viscosity (pl), respectively. Because the viscosity of Newtonian fluids and the plastic viscosity of Bingham plastics are independent of the shear stress or shear rate, the flow behavior index, n, is equal to 1 for Newtonian fluids and Bingham plastics. For shear-thinning fluids, 0 < n < 1, and for shear-thickening fluids, 1 < n < . Shear thinning can be observed in certain complex solutions such as aqueous hydrocolloid solutions. Dilatant behavior is generally exhibited by concentrated suspensions of nonaggregating monodisperse solid particles such as mixtures of corn starch and water. At very high viscosities or very low shear stresses, Bingham plastics exhibit a yield stress, 0, a shear stress below which no apparent deformation occurs (Figure 3). Other non-Newtonian fluids do not appear to exhibit a yield stress, 0, because flow occurs even at very low shear stresses. Thus, for non-Bingham bodies, in the absence of 0 the Herschel–Bulkley equation reduces to the power law, i.e., the Ostwald–de Waele equation:

Many pharmaceutical macromolecular materials exhibit non-Newtonian behavior in solution. Although the viscosity of Newtonian fluids is a constant, i.e., = / so that is independent of shear rate or shear stress, there is no singular value of viscosity for non-Newtonian fluids because the viscosity of non-Newtonian fluids varies with shear rate or shear stress. Accordingly, applying the Newtonian concept of viscosity to non-Newtonian fluids is inappropriate. The viscosity value observed for a non-Newtonian fluid is an apparent viscosity, app, and is a function of the conditions of measurement. As the shear rate or shear stress changes, the apparent viscosity is likely to change as well. Figure 4 depicts the dependence of app on the shear rate for an aqueous solution of a typical hydrocolloid.

Figure 4 depicts apparent viscosities, app, ranging from high values at very low rates of shear to low values at very high rates of shear. The curve's asymptotes—corresponding to the zero-shear viscosity (0) and the infinite-shear viscosity ()—together with the linear log–log region at intermediate rates of shear, are characteristic of the shear-thinning behavior of these materials.

During the course of manufacturing, in-process materials that exhibit non-Newtonian behavior may be subjected to wide ranges of shear rates and shear stresses. Fluid dynamic conditions during drug product storage and administration and release of the active constituent(s) could broaden those ranges even further. For materials that exhibit non-Newtonian behavior in a solution or dispersion, the rheological profile (rheogram or flow curve), as described in Figure 3, gives a better functional characterization than a single viscosity measurement of the apparent viscosity at a single shear rate.

Determination of the rheological profiles of non-Newtonian solutions or dispersions requires the use of a rheometer or a viscometer that is capable of measuring rheological behavior over the range of shear rates and/or shear stresses encompassing the conditions that the excipient or product is likely to encounter. The exact mathematical characterization of deformation and flow of a test substance requires the precise delineation of sample dimensions, the forces exerted, and the resulting deformation or flow. Instruments that enable a thorough, geometrically accurate analysis of deformation and flow are termed absolute rheometers. Such absolute rheometers are preferred for measurements of non-Newtonian fluids so that results can be compared from instrument to instrument for corresponding ranges of shear stresses and shear rates. When flow or deformation is not well defined or is indeterminate, the resulting measurements are likely to correspond, at best, to approximations of the true rheological behavior of the sample. Rheometers that can provide useful information about material behavior, but do not enable the true rheological character of the material to be determined, are known as relative rheometers. Such instruments are subject to geometry-dependent limitations. Results obtained with relative rheometers generally are not comparable to those obtained with absolute rheometers or other differently configured relative rheometers. Relative rheometers often are called viscometers. However, the term viscometer should be reserved for a subset of rheometers that measure only steady shear flow and the corresponding Newtonian viscosity.

Temperature control during the course of measurement is critical to the rheological evaluation of non-Newtonian fluids. An Arrhenius-type equation has often been invoked to characterize Newtonian and non-Newtonian flow properties as a function of temperature:

	=	= Newtonian or non-Newtonian viscosity at some absolute temperature T
ref	=	= Newtonian or non-Newtonian viscosity at some reference temperature Tref
A	=	= constant
E0	=	= energy for viscous flow
R	=	= universal gas constant

Energies for viscous flow and the constant A are determined from experimental data. Energy dissipation by viscous flow is a function of measuring system geometry, temperature control, shear rate, and shear stress, but increased variability is observed in particular at high shear rates and for fluids that exhibit high apparent viscosity. Sample temperatures of non-Newtonian fluids should be controlled to ±0.1, as is the case for Newtonian viscosity measurements.