Parameterization of Data

Each of these curves has a maximum number of parameters associated with the features displayed. Curve A is a single valued Newtonian fluid. The single feature is the value of the Newtonian viscosity, h. Curve B displays two features, the value at two points connected by a line in log-log space or the power-law decay slope and prefactor. Curve B is a power-law fluid that displays shear thinning behavior due to orientation of polymer chains at high rates of strain. This is a type of a structured or dynamically structuring fluid and is typical of polymer melts at high rates of strain: h(dv_x/dy) = m (dv_x/dy)^n-1. Curve C is a typical curve for a polymer melt that displays a plateau at low rates of strain, the zero shear rate viscosity, h₀, and shear thinning behavior at high rates of strain. The curve is described by three parameters, two points in the power-law regime and one in the Newtonian regime or the zero-shear rate viscosity, h₀, and the power-law parameters m and n from above. The Ellis model describes this fluid by h(dv_x/dy) = h₀ [1 + (l(dv_x/dy)²)]^(n-1)/2. There are many other functions that can describe curve C with three parameters one example being the Carreau Model (See Bird Armstrong and Hassager, Dynamics of Polymeric Fluids for many other examples). You should be able to look at each of these curves and to state the maximum number of parameters that should be used to describe the data as well as to list the features in the curve that are associated with each parameter. For example, if curve C displayed only a small part of the Newtonian plateau at low rate of strain, then you could say that the value from a fit using the Ellis model would be expected to have a large uncertainty. It is critical to have a good feel for the features associated with model parameters when estimating the inherent error in values you report or read.

In Curve A the data can be modeled with a maximum of one parameter if the actual number is not of interest (i.e. if the single value is normalized) or two parameters if both the mass and number are of importance since it is single valued or a mono-disperse distribution. In Curve B the data shows symmetry about a mean value and one might expect that if the data resulted from a random process that a Gaussian function could describe the curve. Even if a Gaussian does not fit the curve, the symmetry of the curve indicates that the maximum number of parameters would be 3 if the actual number is of interest, or 2 if the curve is normalized (area set to 1). The two parameters could be two values on the curve such as M_n, and M_w or could equally be the mean, m and standard deviation, s. Any two parameters that functional describe the curve can be mathematically related to other parameters. For example, M_w and M_n can be calculated from the standard deviation and mean, the mean being the number average mass and the standard deviation being given by: