TimeResolved Rheometry
TIMERESOLVED 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 timeresolved
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 'quasistable'.
 
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 quasistable 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 N_{mu} 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, timeresolved 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) TimeResolved Rheometry.
Rheol Acta 33:385397
Mours M, Winter HH (1995) Viscoelasticity During Heating/Cooling
Scans. Ind Eng
hem Res 34:32173222
IRIS DEVELOPMENT
14 Elm Street
Amherst, MA 010022007
USA
email: IRISrheo@yahoo.com
FAX (413) 549 4129
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