Curve-Fitting the Linear Model Parameters for One Mode

Use the logarithmic plot of the experimental $G\prime$ , $G"$ curves to obtain estimate initial values for the linear model parameters: the polymer viscosity and the relaxation time. In the plot, you find the intersection point of the two curves. Then, you read off the values for $G"$ and the frequency $\omega$ from the x and y axes. By applying the following rules of thumb, you obtain suitable initial values for the curve-fitting algorithm:

Initial relaxation time: ${\lambda }_{init}\approx 1/\omega$

Initial polymer viscosity: ${\mu }_{0,init}\approx 2G"\cdot \lambda$

For the minimum and maximum values of relaxation time $\lambda$ and viscosity $\mu$ , use a rational range, generally starting from one decade smaller for minimum to one decade larger for the maximum.

More exact values for the minimum and maximum values of the polymer relaxation time and viscosity can be calculated using following relationships:

• ${\lambda }_{\min }\approx 1/{\omega }_{\max }$ where ${\omega }_{\max }$ is the highest frequency on the frequency vs GpGpp curves.
• ${\lambda }_{\max }\approx 1/{\omega }_{\min }$ where ${\omega }_{\min }$ is the lowest frequency on the frequency vs GpGpp curves.
• ${\mu }_{\min }\approx \text{min}[({G\prime }_{\max }+{G"}_{\max })/{\omega }_{G\max },({G\prime }_{\min }+{G"}_{\min })/{\omega }_{G\min }]$ where ${\omega }_{G\max }$ and ${\omega }_{G\min }$ are the frequencies for the respective $G\prime$ and $G"$ values.
• ${\mu }_{\max }\approx \text{max}[({G\prime }_{\max }+{G"}_{\max })/{\omega }_{G\max },({G\prime }_{\min }+{G"}_{\min })/{\omega }_{G\min }]$ .

To curve-fit the linear model parameters: