Time-Resolved Rheometry

## TIME-RESOLVED RHEOMETRY

$\alpha$ A stable sample is usually considered as prerequisite for a meaningful rheological experiment. This excludes materials with changing structure such as polymers during gelation, phase transition, decomposition, polymerization, etc. We developed a time-resolved rheometry (TRR) technique which enables characterization of the dynamic properties of such transient polymers. One would like to describe each of the structural states with its own relaxation modulus G(t) or its own spectrum H(). If the rate of mutation (structural change) is small, intermediate can be treated as constant during the taking of individual data points, i.e., the material is considered 'quasi-stable'.

 Figure 1: storage modulus G' and loss modulus G" for a sample which crosslinks with increasing reaction time. Curves are shown for three frequencies. The instant at gelation is marked with a vertical line (at about t=1200 s). For a more complete characterization of the material, data is also needed at different frequencies.

### Characteristic Time Scales

Three time constants govern TRR, one time constant for the experiment and two for the material:
• The duration of the rheological experiment defines the experimental time t. Steady conditions in a dynamic mechanical experiment are reached after about one period .  Figure 2: Characteristic time of dynamic mechanical experiment .
• The material time is the time constant of the relaxation mode which dominates the rheometrical experiment. For steady conditions (steady shear viscosity, equilbrium modulus), this is the longest relaxation time max. In a dynamic mechanical experiment, the stress response is dominated by relaxation processes with a time constant near 1/.  Figure 3: Fading memory of a material as described by the relaxation modulus. In this schematic, the relaxation is demonstrated by a single exponential.
• The mutation time mu is the characteristic time constant for the change in the material. The change is measured indirectly through the property of interest, g (e.g. G', G").  Figure 4: Schematic of the change of the measured property g. The mutation time is the inverse of the rate of change of property g.
The three time constants can be combined into two dimensionless groups:
the Deborah number

and the mutation number which estimates the change during the experiment

One can form a third group

which immediately decides on the feasibility of an experiment:
• if /mu<<1 the material is quasi-stable and TRR is possible
• if /mu > 1 the sample changes during the relaxation process and no reliable data can be taken, e.g. a fast changing sample at low frequencies

### Experimental Technique

Cyclic Frequency Sweeps (CFS)
An appropriate frequency window (e.g. 0.1 - 100 rad/s) is scanned repeatedly while the sample structure is changing. This results in the following type of response:

 Figure 5: Schematic of data (here: loss tangent) during a cyclic frequency sweep experiment. The frequency window (1 - 3) is scanned repeatedly. Each data point represents a different state of the material.

### Data Analysis

We developed a software package (GELPRO) that performs the data conversion from the time domain to the frequency domain. Examples of a polybutadiene during chemical crosslinking are shown below.

 Figure 6: Evolution of dynamic moduli during crosslinking at the three different frequencies 1, 10, and 100 rad/s. Open symbols: G', filled symbols: G". Figure 7: Dynamic moduli at three different stages of crosslinking as interpolated with GELPRO. Open symbols: G', filled symbols: G". The lowest data set (squares) represents the material in the liquid state, the one in the center (diamonds) corresponds to the gel point, and the top data set (triangles) is the material in the solid state.

The interactive computer program allows easy graphical representation of the dynamic moduli, the loss tangent, the rate of change of the dynamic moduli and several other material functions as functions of time and frequency. The mutation number can be evaluated for each data point. To ensure data reliability Nmu should be less than about 0.1.

 Figure 8: Evolution of loss tangent during crosslinking. The gel point can be determined from the crossover (at around t=2300 s). Each curve is measured at constant frequency (1, 10, 100 rad/s) Figure 9: Loss angle as function of frequency at three different material states. Top curve: before gel point, middle: very near gel point, bottom: beyond gel point.

In summary, time-resolved rheometry is a very useful experimental tool to study the rheological behavior and the transition kinetics of any kind of material with transient properties.

### References

Mours M, Winter HH (1994) Time-Resolved Rheometry. Rheol Acta 33:385-397
Mours M, Winter HH (1995) Viscoelasticity During Heating/Cooling Scans. Ind Eng hem Res 34:3217-3222

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