and viscosity 
If one follows rheology from the description of the ideal parent substance through the introduction of non-Newtonian fluids to the rheology of suspensions, one gets the feeling of moving deeper and deeper into a "jungle" of assumptions, exceptions, approximated formulae and empirically determined constants. For this reason this sub-section confines itself to introducing the principal quantities and describing the relationships that are important for the rheological measurements performed.
Shear rate sr, shear stress and viscosity
By moving a plate with an area A=1xb over the surface of a liquid one obtains a simple shear current, whereby there is
a constant shear stress and the speed of the liquid wx(y) increases with the height coordinate y (fig. 3.3).
| Shear stress: |  | (3.5) |
| Shear rate: |  | (3.6) |
| Dynamic viscosity: |  | (3.7) |
Newtonian fluids
Newtonian fluids are characterised by pure linear viscous behaviour. When a load is applied they display a linear change in shear over time, and there is a linear relationship between shear rate and stress, i.e. dynamic viscosity is independent of shear rate. When the load is removed, the shear remains completely preserved.
Since eq.(3.7) yields a constant for Newtonian fluids, these form the basis for calculating the apparent viscosity 
in
a second step correction is applied to calculate the true viscosity of the system in question.
Pseudoplastic behaviour
Polymers exhibit a decrease in viscosity with increasing shear rate. This behaviour is known as pseudoplastic behaviour, and the result is that eq.(3.7) does not yield a constant. Fig. 3.4 shows the different shape of the viscosity curves.
The pseudoplastic behaviour of polymers can be explained by the fact that in response to shear the thread-like molecules that are initially present in coils become uncoiled and an orientation takes place which results in a reduction in friction and hence in viscosity.
Other important features
The viscosity of polymers depends on their mean molar mass and on the breadth of distribution of the mean molar masses.
A temperature increase of 1°C can result in a reduction in viscosity of up to 10%.
If the shear rate is too high, the laminar flow may give way to turbulent flow [4]. Typical shear rates occurring during
processing are [74]:
| 1 - 10 | 1/s | compression moulding |
| 10 - 102 | 1/s | calendaring |
| 102 - 103 | 1/s | extrusion |
| 103 - 104 | 1/s | injection moulding |
| 104 - 105 | 1/s | spinning |
As mentioned earlier, calculations from measurements are always made on the basis of Newtonian behaviour. This
results in apparent quantities which have to be corrected. These corrections are necessary because of the reduced wall
adhesion of many polymers (especially with a large filler content), and to this end the Rabinowitsch correction is
applied to the shear rate. Also, owing to pressure losses on entry into the capillaries (when measuring with a capillary
viscometer), the Bagley correction is applied to the shear stress .
The correction methods are explained in detail in sub-section 3.2.3.1.2
For the purpose of these descriptions only certain data have been selected that characterise the substances used to the
extent that is of interest for the measurements performed. For further data reference should be made to the
manufacturers listed under the trade names, from whom detailed data sheets are obtainable.
| Trade Name / Distributing Company | Chemical Characterisation | Remarks |
| Matrix Polymers: | ||
| Pebax 2533 SA 00 Atochem Density: | Polyether block amide PA: polyamide segment | Thermoplastic elastomer Use: For modifying polyamides for pump membranes, sports shoe soles, hoses etc. |
| PS N 2000 Shell Density: | Polystyrene | Low-melting standard polystyreneOwing to good flow
properties mainly used in injection moulding Use: Packaging, disposable items |
| Fillers: | ||
| Printex XE-2 (beads) Degussa Density: N2 Specific surface area2 950 m2/g (BET) | Elementary carbon
Average particle size3 30 nm | Very high conductivity
DBP oil absorption4 380 cm3/100g |
| Black Pearls 880 Monarch Cabot Density: N2 Specific surface area2 220 m2/g (BET) | Elementary carbon
Average particle size3 16 nm | Readily dispersible carbon black
DBP oil absorption4 105 cm3/100g |
| Titanium dioxide RL 90 Titafrance (Rhône Poulenc) Density: | 95% TiO2 Crystal form: rutile Oil absorption value5: 17 | Pigment for (white) colouring of plastics |
The choice of specimens sought to cover the following three areas:
To this end eight series of concentrations were prepared and measured. Filler concentrations were specified in mass percent and followed the series: 0.5 - 1 - 2 - 3 - 5 - 7 - 10 - 15 - 20 - 25 - 30 - 35 - 40 - 45 - 50 - 55 mass %, [5]
whereby the concentration series for the individual systems were broken off at different points (viscosity too great or no longer possible to produce specimen).
The individual systems:
1 Pebax 2533 SA 00 with Printex XE-2
Pebax as matrix remains elastic even at high filler concentrations and is characterised by good wetting of the carbon black.
Printex XE-2 with its very good conductivity possess a large specific surface area. Concentration series up to 20 mass % carbon black.
2 Pebax 2533 SA 00 with Printex XE-2 and wax
Matrix and filler as in 1, but with a wax. The dosage is in each case 20% (of the weight) of the carbon black content. It should be noted that the additive component is incorporated at the expense of the matrix component. This system was intended to indicate what influence the additive has on viscosity or on the shape of the curves. Concentration series up to 20 mass % carbon black.
3 PS N2000 with Printex XE-2
PS N2000 as a very brittle matrix with poorer wetting properties. Concentration series up to 20 mass % carbon black.
4 PS N2000 with Printex XE-2 and a wax
System analogous to 2. Concentration series up to 20 mass % carbon black.
5 Pebax 2533 SA 00 with Black Pearls 880 Monarch and wax
BP 880 as carbon black colourant with much smaller specific surface area. Constant admixture of 2 mass % wax as external lubricant to make preparation of the specimen easier. Concentration series up to 40 mass % carbon black.
6 PS N2000 with Black Pearls 880 Monarch
System analogous to 5, but free from additives. Concentration series up to 25 mass % carbon black.
7 PS N2000 with Black Pearls Monarch and wax
System analogous to 4 and 6, but with different additive dosage to prevent expulsion of the additive at high concentrations. The additive concentration was halved, i.e. only 10 mass % of carbon black component. Concentration series up to 25 mass % carbon black.
8 PS N2000 with Titanium dioxide RL 90
Titanium RL 90 as filler with varied chemical character and changed density. Owing to the much greater density it was necessary to change the concentration accordingly. For this reason and owing to the poorer processability (marked adhesion to machine parts), only one concentration series was prepared with TiO2 as filler. Concentration series up to 55 mass % TiO2.
To permit subsequent comparison of the measurement curves, and because volume contents are used in the theoretical discussions, the mass percentage figures were recalculated as volume percentage figures with the aid of the density of the individual components. The calculated figures are only approximations, as the densities of the individual components fluctuate.
3.2.2.3 Preparing the specimens
Individual batches of 1 kg were weighed out. An exception was made only in the case of series 5, where the 2 mass % of wax was in each case added to the 1000g. Weighing in was performed separately for matrix and for filler with additive. The components were then added in the laboratory kneader in such a way that the powdered components were deposited on a layer of matrix polymer and this was then covered with the remaining matrix polymer. Only then was the plunger lowered sufficiently to slightly compress the mixture. In the subsequent kneading process the upper layer of matrix formed a kind of seal, with the result that losses were only to be expected at the seals of the paddle shaft. By careful "dosing" of the plunger pressure it was possible to limit such losses to an acceptable level. To melt the specimen, the housing temperature of the kneader (not the material temperature) was increased to approx. 100°C. After melting, the plunger was removed from the specimen, the temperature reduced to approx. 25°C and the "lump" further homogenised. Here it was necessary to divide the Pebax specimens, as the sticky, readily flowing mixture clung to the kneader paddles, preventing adequate mixing. After homogenisation for approximately equal periods, it was possible to remove the PS specimens from the kneader as a single lump, whereas in the case of the Pebax specimens the mixtures could only be partially removed. Following this, depending on the state of the kneader, either the next higher mixture in the series was kneaded or a mixture of ABS was kneaded in between for cleaning purposes (obligatory in the case of Pebax). The still hot lumps were cut into smaller "cubes" with a side of approx. 2-3 cm using an impact cutter. After cooling, the specimens were cooled further in a deep freeze, before being ground the next day in a mill. As a result the specimens now took the form of an irregular granulate with a more or less large proportion of finer matter. The specimens were then transferred to plastic cans and stored.
Measurements performed
The main focus of the measurement programme was the rheological measurements with the high-pressure capillary
viscometer. In addition a number of measurements were performed with the aim of permitting statements about the
dispersion state (pyrolysis, microscopic investigations) and determining 
(conductivity measurements).
The aim was to investigate the curve of the measuring points on a graph of the natural logarithm of dynamic viscosity against the volume percentage of the filler, in order to confirm the predicted linearity or obtain evidence of structural influences.
To this end the dynamic viscosity of the various specimens was measured with a high-pressure capillary viscometer at various shear rates.
A large number of trial measurements were performed to determine suitable measuring conditions before work started on the measurement series itself.
Temperature
The temperature was specified as 180°C. Although 190°C is the most widely used temperature, the Pebax concentrates have such a marked tendency to stick that a lower temperature seemed more appropriate. On the other hand an even lower temperature causes the viscosity, especially of the PS concentrates, to rise so far that high concentrations can no longer be measured at higher plunger speeds. The chosen temperature therefore represents a compromise.
Quantity of specimen
The quantity of specimen that was placed in the measurement channel had no influence on the precision of the measurements. For this reason there was no need to weigh out the quantities of specimen used.
Moisture
As some polymers react very sensitively to moisture, one Pebax and one PS concentrate were a) dried for 4 days at 80°C and b) stored for 4 days in a closed container together with a beaker of water. Subsequent measurement compared with the untreated specimen revealed the following results:
PS concentrate: All three specimens were within the range of measuring accuracy.
Pebax concentrate: The moist specimen displayed up to 20% lower viscosity at low shear rates, whereas the dried and the untreated specimen were within the range of measuring accuracy.
Thus the Pebax concentrates are much more susceptible to hydrolysis than the PS concentrates, but given suitable storage of the specimens the water uptake is so slight that it is negligible. The state of the specimens must always be examined before measurement, however!
Components
It is in any case important to ensure that the matrix polymers in particular are taken from a single consignment, or preferably from a single sack. If this is not possible, the matrix polymer must first be subjected to a kind of receiving inspection, since differences in viscosity of up to 35% were found in the case of Pebax. There would seem to be no need for additional mixing of the substances used, as the fluctuations are large-scale ones.
Die geometry
The existing round-section dies all vary in the ratio l/d, i.e. length of capillary to diameter. As a basic principle it may be said that as the diameter decreases it is possible to measure smaller differences in concentration owing to the larger force differences. As the length increases, the variance of the measured values decreases. But combining the two reduces the maximum measurable concentration. The choice of capillaries is therefore made on the basis of the criteria: concentration range to be measured, concentration differences to be measured, and desired shear rates.
It will therefore be necessary to perform targeted test measurements with various selected specimens and different dies, and then select those that appear most suitable.
For the specimens to be measured here, the 30/1 die (length 30 mm, diameter 1 mm) appeared to be the best compromise.
Threshold value
By inputting a threshold value it is possible to influence the force constancy at which the measurement is to be registered. To this end one inputs a tolerance in percent and specifies via the number of comparison intervals (1-100) how many measured values must be within the tolerance for the force to count as constant and be registered. For example, if the tolerance entered is 1% and the number of intervals is 15, the computer takes measurements from the force transducer at regular intervals and checks whether the value lies within the tolerance. If the force stabilises around a certain value, then the 15th value that lies within the tolerance of the stabilised value is registered as the current force. This also means, however, that the force registered may differ from the actual force by up to 1%.
For the measurements performed here, the tolerance was specified as 1% and the value was accepted after 15 comparison intervals. Increasing the number of intervals or reducing the tolerance makes the measurements excessively lengthy because the specimens are very inhomogeneous, which results in sharp fluctuations in force at low plunger speeds and hence low volume flows. For this reason it was decided to perform an extra series of measurements rather than spend a long time over a single measurement series with virtually no resulting increase in accuracy.
Plunger speeds
The plunger speeds were selected such that the corrected shear rates were subsequently as close as possible to the shear rates used for interpolation (see sub-section 3.2.3.1.3).
The plunger speeds were 0.01 - 0.05 - 0.1 - 0.5 - 1 - 2 mm/s (it is not possible to say anything about the quantitative extent of the fluctuations in speed), and the corrected shear rates are therefore approximately 12, 75, 200, 750, 1500 and 3000 s–1. Values were interpolated to 20, 50, 100, 500 and 1000 s–1.
In the sequence of measurements, all speeds were run in succession starting with the slowest plunger speed. As soon as the force was registered as constant, the speed was increased. In other words it was necessary to load the test channel with enough material to be able to run through all the speeds before the channel was empty.
Melting time
The melting time, at 3 minutes, was kept very short, but is quite sufficient when one considers that thorough filling of the test channel takes so long that the granulate put in first has already had long enough to heat up.
Force transducer
Since the measuring error of the force transducer is 0.5% of the final value of its nominal range, smaller concentrations were measured with the 500 bar transducer, while the 1000 bar transducer was used for higher concentrations.
Guide values: for PS N2000 with Printex XE-2 0.5-5 mass % – 35 - 500 bar 7-15 mass % – 50 - >1000 bar. Apparatus settings
As a result of the preliminary tests, the following apparatus settings were used for performing the measurements:
| – Die geometry: | Round-section die of 30 mm length and 1 mm diameter |
| – Temperature: | 180°C (0.3°C fluctuation) in all three heating zones |
| – Threshold value: | Tolerance 1%, 15 comparison intervals |
| – Force transducer: | 500 bar force transducer as basic version and 1000 bar transducer for pressures >500 bar. |
| – Melting time: | 3 minutes |
| – Plunger speed: | 0.01 - 0.05 - 0.1 - 0.5 - 1 - 2 mm/s |
Method
At the start of every measuring day the measuring channel and the region of the force transducer were each thoroughly cleaned (the material, which was still solid and only slightly warm, was easier to remove than in the hot state). Then the cavity behind the "pressure hole" was filled with HD-PE, because this does not decompose even when exposed to temperature for long periods. Next the die and the force transducer were screwed into the test chamber (allow to heat up before final tightening!). This was followed by zero adjustment of the force transducer.
The first measurements for each day were discarded, but otherwise the specimens were measured one after the other with increasing concentration. No cleaning was undertaken between the individual specimens; only the first two or three measurements were discarded. In the first measuring sequence for a complete concentration series, each specimen was measured four times. Given a time of 12 to 20 minutes per measurement (including melting time), this meant that three to four specimens could be measured in a day. In the repeat measurements a complete concentration series was measured in one day to compensate for any fluctuations from one day to another. The disadvantage of this was that it was only possible to measure each individual specimen two to three times. Each concentration series was measured a total of three times.
During the measurements the display indicated the temperatures of the three heating zones, the current plunger speed and the force. If the force rose above 500 bar, the highest plunger speed (2 mm/s) was deleted. If the force at the next higher concentration rose higher than 500 bar (at 1 mm/s), the 100 bar force transducer was installed and zero adjusted. After each measurement the viscosities measured were plotted against the shear rates on a plotter, using a log-log scale, and any conspicuous mavericks were immediately deleted.
At the end of a measuring day the equipment was always run once with pure matrix polymer in order to clear the capillaries of highly viscous concentrate residues. After this the die and the force transducer were removed. The only item cleaned while hot was the membrane of the force transducer.
On the basis of the geometrical conditions and the force registered, the internal computer calculated in accordance with
the following equations the values which were then printed out (nomenclature and units as in [75]):
| Volumetric flow rate Q in cm3/s | ||
 | (3.8) | |
A plunger area =  | (3.9) | |
| s plunger travel (cm) | ||
| t time (s) | ||
| v plunger speed (cm/s) | ||
| Ds plunger diameter (1.2 cm) | ||
| Apparent shear rate D in s–1 | ||
 | (3.10) | |
| R radius of capillaries (cm) | ||
Apparent shear stress  in N/m2 | ||
 | (3.11) | |
| p melt pressure (bar) | ||
| L length of capillary (cm) | ||
| d diameter of capillary (cm) | ||
Apparent viscosity  in Pa x s | ||
 | (3.12) |
Corrections
The calculations performed must be seen as no more than approximations, since they are based on the assumption of Newtonian behaviour, something from which polymer dispersions are far removed. For this reason the instrument offers two correction methods: the Bagley method for determining the true shear stress and the Rabinowitsch method for correcting the shear rate.
As a result of the partial correction of shear rates the values for dynamic viscosity are obtained at completely different shear rates, which means that a comparison of viscosities from different series of measurements is problematic, or at worst pointless. For this reason a subsequent analytical procedure was developed that makes it possible to compare the dynamic viscosity figures by interpolation to "straight" shear rates. To this end, all dynamic viscosities measured (in a given day) and the associated shear rates were input into a computer. The computer then plotted viscosity against shear rate on a log-log scale. After eliminating mavericks, a regression line was drawn through these points. Along this curve the viscosities were read off and written down at sr = 20 - 50 - 100 - 500 1000 s–1. All further analysis was then based on these interpolated values.
This method of analysis yielded values that can be directly compared with one another without any graphic plotting. There is no denying, however, that this method involves a degree of uncertainty. The position of the regression curve is subject to an unquantifiable error, and a reading error also exists in interpolation, as was shown by several independent analyses. The method has however proved valuable despite its error sources, as the values obtained are extremely convenient to use. Fig. 3.5 shows a printout of the computer-aided interpolation.
The cross hairs visible in the printout are controlled by the mouse. The sensitivity of this control is limited by the smallest step size permitted by the program. As a result it is not possible to read off the coordinates to any desired degree of accuracy. This can be seen from the fact that the shear rate as the x value can only be approximated to within 0.21 s–1 of the desired 50 s–1, and it also means that the viscosity read off is only accurate to within approx. 1.5%. However, since this accuracy is within acceptable limits it was decided to dispense with subsequent correction.
The pyrolyses performed provide, as mentioned in sub-section 3.1.2.2, information about the ratio of the mass of the dispersed particles to the mass of the adsorbed matrix. Here the shape of the curve actually obtained compared with the expected curve can give an indication of the dispersion state. The results must however be regarded with a certain caution, as the pyrolysis method is not quantitative with regard to the non-decomposed adsorbed polymer layer. Accordingly the pyrolysis results are not used for the analysis proper.
Method
Each of three specimens is weighed into two porcelain crucibles. The six crucibles are placed in a glass dish and put in the unheated part of the approx. 1 m long pyrolysis tube (fig. 3.6). After the tube has been flushed with nitrogen for a quarter of an hour, the dish is pushed into the hot part of the tube which is at a temperature of 900°C. After half an hour the dish is removed into the unheated part to cool off. When it has cooled in the tube for a quarter of an hour, the dish is removed and the flow of nitrogen is turned off. On cooling to approximately room temperature, the specimens are weighed again. For control purposes this is followed by ignition in air and renewed weighing.
Analysis
For the analysis, the ignition residue was subtracted from the pyrolysis residue. The pyrolysis residue was then
calculated as a percentage of the initial amount weighed out. The percentage of mass accounted for by the dispersed
phase (only dispersed particles) was deducted from this percentage. The remaining difference was expressed as a
percentage of the amount of dispersed particles weighed out.
| Example: | PS N 2000 with 10 mass % Printex XE-2 | ||
| Amount weighed out | 3.3494 g | ||
| Pyrolysis residue | 0.4154 g | or 12.40% of amount weighed out | |
| Ignition residue | 0.0063 g | or 0.19% of amount weighed out | |
| Corrected pyrolysis residue 12.21% | |||
| Subtracting concentration shows that 2.21% of the amount weighed out was not decomposed | |||
| In terms of the 10% carbon black concentration this means a pyrolysis excess of 22.1%. | |||
The conductivity measurement was only intended to determine , in the specimens with Printex XE-2. Since the
values do not represent specific conductivities, and since they are only of secondary importance for the actual analysis
of the work, only the fundamentals of the method of measurement are outlined here.
A rolled sheet [6] of each specimen was made from granulate. This rolled sheet was then compression moulded to form a plate. The surface resistance of the pressed plate was measured with a ring electrode (DIN 53 482). The measured values represent the resistance for the measurement setup used, and cannot be converted into specific resistance values. This is of no importance in the determination of c, however.
= 1.01 g/cm3 
cm/cm