When any substance moves across a surface, the motion is impeded by friction. If the substance is a liquid, this friction generates the effect called viscosity. Flow in a liquid generates a velocity gradient which produces a kind of deformation called shear. Newton showed that the frictional force f between the layers in a liquid is proportional to the area A of the layers and to the velocity gradient between them; that is,
where is the coefficient of viscosity, or simply the viscosity; f/A = F, the shear stress; and the shear gradient, or shear rate. If is a constant, the fluid is called Newtonian; if is a function of F or G, the solution is called non-Newtonian. The addition of macromolecules to a solvent with viscosity , yields a solution of higher viscosity, . This can be thought to result from increased friction between adjacent unimolecular liquid planes caused by the fact the macromolecules are larger than the solvent molecules and hence extend through several of these hypothetical planes. The viscosity increase is a function of several parameters of the molecule, including the volume of the solution that is occupied, the ratio of length to width of the molecule (the axial ratio or the ratio of the axes of the smallest ellipsoid of revolution in which the molecules could fit), and the rigidity of the molecule. For globular proteins, the principal effect is through molecular volume and this is simply related to molecular weight.
The change in viscosity is usually expressed as a ratio, , called the relative viscosity, . Einstein showed that is a function of both the size and the shape of the macromolecule and derived the equation:
in which a is a shape-dependent constant (a = 5/2 for spheres), is the fraction of the solution volume occupied by the molecules, and b is a second shape-dependent constant. This equation can be rewritten in terms of the concentration, c, of the macromolecules by defining V as the specific volume of one molecule, so that , to give
Viscosity is frequently expressed as the specific viscosity, , which is the fractional change in viscosity produced by adding the solute, that is:
Neither nor can be simply related to molecular parameters (i.e., shape and volume) because of intermolecular interactions (e.g., collision, entanglement). To avoid this problem, a situation at very low (i.e., zero) concentration is considered. To do this, the intrinsic viscosity is defined as:
which depends only on the shape-dependent constant, a, and the specific volume, V. Operationally, this means that is determined by measuring at several concentrations, plotting versus c and extrapolating to c = 0. The intrinsic viscosity is directly related to molecular weight and reveals the volume occupied by individual molecules in the case of very dilute polymer solutions (Morris, 1984). The above plot called the Huggins plot often gives a straight line, the intercept of which is .
where k' is a dimensionless constant, called the Huggins' constant. Kraemers' plot also yields as the intercept,
where k" is also Kraemers' constant.
The Huggins and Kraemer constants (k' and k") are related as shown below:
As the concentration, c, of a polymer solution is increased, a stage is reached at which the individual polymers are forced to interpenetrate one another. The concentration at which this occurs is known as (Morris, 1992). Below c*, individual polymers are free to move through the solvent with little mutual interference, viscosity is virtually independent of shear rate ( known as the 'Newtonian' behaviour). Above c*, where chains can move only by the much more difficult process of "wriggling" (reptation) through neighbouring chains, viscosity becomes higly dependent on shear rate.
If the log of specific viscosity is plotted against the log of , a dimensionless product which is called the 'coil overlap parameter' provides an index of the total volume occupied by a polymer, a line with two gradients emerges. It has been observed empirically, that for a wide range of random coil polysaccharides the log of hsp varies approximately linearly with the log of over the viscosity range , with a slope of about 1.4 (Morris et al., 1981). At higher values of hsp, however, the concentration dependence changes suddenly to a slope of about 3.3, because a point is reached where the individual coils start to entangle and overlap .