(3.14) 3.4 Discussion
Consequences for an application-oriented view
The relationship between logarithmised dynamic viscosity and increasing filler concentration may be regarded as being – with sufficient accuracy for practical purposes – linear. This means that eq.(3.2), which describes a linear relationship, can be used for calculations intended to permit forecasts about viscosity.
It remains to be clarified, however, how compositions of a more complex structure can be predicted. Strictly speaking, the statement above applies only to the specimens measured. Ist universal validity would of course have to be tested before real mixtures can be the object of such studies. If the statement were to be confirmed in this field too, this would open up promising opportunities for the future. With only a small number of carefully prepared specimens it would be possible to construct something like a calibrated curve against which one could not only measure current production batches (quality control), but also determine maximum contents with an eye to machine loadings.
In all this euphoria, however, it is important to stress that the statement above makes no claim to universal validity. From the eight formulations measured, one cannot draw conclusions about all other formulations. Nevertheless the linear relationship remains as quite an accurate approximation.
Looking at the correlation coefficients for the regression lines, one finds a – for rheological measurements on polymers – high degree of correlation. Naturally one has to bear in mind that the logarithmisation of viscosity involves a certain smoothing. On the other hand a look at the unconverted values reveals variations of 3-7%. These figures may seem high, but for polymer melts they are within normal limits. The good correlation can thus be taken as a measure of the linearity and as support for the statement above.
Theoretical consideration
The aim of the measurements carried out was not only to determine the trend of logarithmised dynamic viscosity in response to increasing filler content, but also, with regard to the new theory, to look for indications of structural influences.
If one assesses the results with an eye to the new theory, one can find a number of interesting interrelationships, though it has to be said that these are largely speculative and are intended more to stimulate new ideas.
Looking at the graphs in sub-section 3.3.2 one is struck by two conspicuous features in addition to the expected effects. Firstly, there is the crossing-over of the two Pebax lines in the first block of specimens (with Printex XE-2). Apparently waxes do not act, only as a lubricant. Although the intercept is smaller than for the Pebax specimen without wax, which suggests a lubricant effect, the gradient is steeper. On the basis of eq.(3.2) this corresponds to a higher interaction factor (as can be seen in sub-section 3.3.4). Since the intensity of the interaction is presumably not altered by the wax, however, the only possibility left for increasing the factor is an increase in the "active" surface area. This would mean an improvement in dispersion, thereby identifying the wax as a dispersion aid. This is not new, and one could have found it out from the product description, but it does support the idea that the "size" of the surface is also crucial for viscosity. Furthermore, in the conductivity figures we find that this fact is expressed in a higher conductivity of the Pebax specimen with wax.
The second striking point is the shape of curve 5, in other words Pebax with BP 880 and 2% wax. Here the extreme gradient, which is due to the large proportion of wax in the lower concentration range, is less interesting than the fact that the line actually crosses line 6 at low shear rates. One possible explanation might be found in the good wetting properties of Pebax. The good wetting would leave less free matrix available. The question which then arises is why this effect is not seen in the specimens with Printex XE-2. It may be that the reason should be sought not so much in line 5, but rather in the PS specimens which apparently react so drastically to the change in the type of carbon black. But this would take us into the realms of wild speculation, and the discussion will therefore be broken off at this point to turn our attention to another very interesting point.
A closer look at the trends of the "straight lines" in sub-section 3.3.1 reveals a deviation that is common to all the curves. To illustrate this, let us take two curves that display this behaviour particularly clearly and that are suitable from the point of view of shear rate. When considering the shear rates plotted here, it must be remembered that at sr = 1000s–1 the flow is in the critical range for the transition to turbulent flow, and also that low shear rates result in very low volumetric flow rates during measurement and are therefore subject to sizeable variations. For this reason sr = 500s–1 was chosen as a suitable shear rate. The specimens chosen were Pebax with Loxiol and PS N 2000, in each case with Printex XE-2. The specimens with Printex XE-2 are especially suitable because even small differences in concentration lead to measurable changes in viscosity and the readings therefore yield a particularly clear picture.
Fig. 3.24: PS N 2000 with Printex XE-2
It will be seen from the graph that the points in the lower concentration range can be joined up much better by a curve. The last point, which represents the highest concentration, is distinctly out of line with this curve, however. This fact appears particularly interesting, since while the percolation theory might possibly have explained a curve instead of a straight line, it certainly has no explanation to offer for a more or less sudden change in curvature of the kind indicated here.
The curvature can be emphasised by altering the scales so that the last point disappears and the curve is stretched slightly in the direction of the y-axis. This version is shown in the next graph.
Fig. 3.25: PS N 2000 with Printex XE-2, modified scale
Having demonstrated the curvature so clearly, we are left with the question of whether this shape is surprising or whether there is already some hint of it in the new theory.
One answer to this question might lie in further development of eq.(3.2). Eq.(3.3) on its own apparently also describes a linear relationship. If one however inserts eq.(3.4) in eq.(3.3), the result is a highly promising but problematical relationship:
(3.14)
The important point here is the dependence of the resulting viscosity on 12e, whereby 12eA12 represents the work
that has to be input to wet the area A12 with matrix. If work has to be input, it follows that a resistance has to be
overcome. This resistance is the resistance offered by a unit of volume V with the viscosity 
when a shear rate sr is
applied to make it flow.
It is thus necessary to input the work
[37] (3.15)
And this closes the circle, since according to [37]
(3.16)
Thus eq.(3.14) is a kind of function within a function that escalates itself (if increases, so does 
12e and so in turn
does ...).
Another attempt at explanation might be made in terms of the structures that are formed or in the process of forming. But since the further course of the gradient after this supposed salient point is not yet known with certainty, any consideration of structures at the present time would lead too far into the realm of speculation.
For this reason the discussion will be broken off here.
As evidence that the curve discussed is no exception, it is worth considering another graph, and the reader is also recommended to make a critical scrutiny of the graphs in sub-section 3.3.1 from this point of view as well.
Fig. 3.26: Pebax 2533 with Printex XE-2 and wax
Summary
If one looks at the relationship between logarithmised dynamic viscosities and increasing volume percentage of the disperse phase, it is possible with sufficient accuracy to assume a linear relationship for practically oriented applications. For more exact fundamental research it is necessary to suppose a curve, the course and dependent relationships of which have yet to be clarified. Further studies in this direction are undoubtedly the priority goal for those seeking to clarify the structural relationships and their effects.
FIGURES
Fig. 3.1: Diagram: percolation curves of carbon.black with various size specific surface areas and various size empty volumes [72a].
Fig. 3.2: SEM-picture of a carbon-black compound with a carbon- black concentration of 5% [17d].
Fig. 3.3:Schematic representation of an apparature to measure the shear rate [76].
Fig. 3.4: Viscosity curves [76].
Fig. 3.5: printout of a computer aided interpolation [76].
Fig. 3.6: Schematic representation of the pyrolization apparatures [76].
Results of measurements and corrections. Dynamic Viscosity in slope of the Volume-portion-dispersed phase [76]:
Fig. 3.7: Pebax 2533 with Printex XE-2 and wax
Fig. 3.8: Pebax 2533 with Printex XE-2
Fig. 3.9: PS N 2000 with Printex XE-2 and wax
Fig. 3.10: PS N 2000 with Printex XE-2
Fig. 3.11: Pebax 2533 with Black Pearls 880 Monarch and wax
Fig. 3.12: PS N 2000 with Black Pearls 880 Monarch and wax
Fig. 3.13: PS N 2000 with Black Pearls 880 Monarch
Fig. 3.14: PS N 2000 with titanium dioxide RL 90
Results of measurements and corrections : Dynamic Viscosity in slope of the Volume Portion Filler [76]:
Fig. 3.15:
Fig. 3.16:
Fig. 3.17:
Fig. 3.18:
Fig. 3.19:
Results of measurements and corrections: pyrolytic measurements [76]:
Fig. 3.20: PS N 2000 with carbon black
Fig. 3.21: PS N 2000 with titanium dioxide RL 90
Fig. 3.22: Pebax 2533 with carbon black
Fig. 3.23: Carbon black concentrates with Printex XE-2. Conductivity vs. Carbon-Black Portion [76].
Dynamic viscosity compared to the volume portion dispersed phase [76]:
Fig. 3.24: PS N 2000 with Printex XE-2
Fig. 3.25: PS N 2000 with Printex XE-2, modified scale
Fig. 3.26: Pebax 2533 with Printex XE-2 and wax
[1] The conductivity curves of a number of samples are shown in a figure in sub-section 3.3.3.2
[2] See footnote 3 on page 7.
[3] 80% is the result of the following calculation: pyrolysis residue minus the carbon black concentration in the compound gives the quantity which is then calculated as a percentage of the carbon black content. Thus if a 1% (mass%) carbon black compound has a residue of 1.8%, this means an 80% excess.
[4] Identifiable by "melt fracture" (roughened, often fish-scale-like surface)
[5] Weighing precision 1/10 g
[6] Term describing the layer of plastic that wraps around the roller.
[7] It was only possible to analyse one series of measurements, as the wax migrated to the surface, thereby distorting the viscosities in the subsequent measurements. Cf. sub-section 3.4.
