Rheological Phenomena and Structure Formation in Multiphase
Polyme
(work together with M. Holz)
3.1.1 Carbon black compounds and the conductivity leap
The most outstanding property of carbon black compounds is of course
their electrical conductivity, which today may be as much as 30 S/cm (laboratory
produced) [71]. Much more interesting than considering conductivity values
in isolation, however, is a an examination of conductivity per se.
For example, given a steady increase in the concentration of carbon black,
the increase in the conductivity of the compound is not linear; instead,
a moderate increase is followed at a certain carbon black concentration
by a sudden jump, which is again followed by a moderate increase [1].
This sequence of events was formerly called percolation. The carbon
black concentration necessary for the conductivity leap is known as the
critical volume concentration 
.
Since the mechanical properties deteriorate drastically as the carbon black
concentration rises, carbon black compounds are usually only produced with
concentrations slightly above .
A knowledge of and its dependent
relationships is thus of fundamental importance when seeking to achieve
an optimal relationship between conductivity and mechanical properties.
To this end, however, it is necessary to know what is behind the conductivity
leap. Only then is targeted optimisation of the compounds possible. A first
explanation is provided by the percolation theory.
3.1.2 Equilibrium thermodynamics of multiphase polymer systems
As a rule systems are described in terms of equilibrium thermodynamics.
The main proposition of this approach is: equilibrium conditions are the
most stable conditions. A system that is not in equilibrium will seek to
attain this state. Equilibrium is characterised by 
G=0
in non-closed systems, and the tendency of closed systems to move towards
equilibrium can be described by a state function, that of entropy (system
tries to reach maximum entropy).
These main propositions are related to a wide variety of system types,
which also include multiphase systems. One example here is a saturated
solution consisting of two phases (solution and undissolved solid). In
equilibrium, i.e. at saturation, the chemical potential of the substance
dissolved in the solution is equal to the chemical potential of the undissolved
solid.
How can polymer dispersions or dispersions in general be described with
these propositions? Is the most stable state the dispersed state, or do
the separation tendencies represent an attempt to reach a more stable state?
Does this not fail to take account of the time factor, with which the dispersions
could be described as quasi-stationary, i.e. the attainment of equilibrium
is merely inhibited? – These questions make it clear that some systems
should not be described simply by the sweeping generalisation of equilibrium
thermodynamics.
3.1.2.1 Percolation theory
The percolation theory is a statistical/geometrical approach to explaining
the shape of the conductivity curve in carbon black compounds. The theory
does not explicitly embody aspects of thermodynamics, though these are
implicit in it.
The percolation theory proceeds from a statistical distribution of the
carbon black particles, which corresponds to a maximum of entropy. It thus
denies any interaction between matrix and carbon black particles.
As the concentration of carbon black increases, the carbon black particles
inevitably become closer together, and at
they are finally sufficiently close together or even touching, with the
result that conductivity bridges form. If this approximation took place
from all six spatial directions, one could expect
to be reached at a carbon black concentration of around 52 vol.% [72a].
If one restricts the approximation to 2-4 spatial directions,lies
around 27 vol.%. In fact
is found to lie, even for comparatively poor carbon blacks, in the region
of 20 vol.% and even lower. This considerable difference may be due
to the apparently incorrect assumption that the carbon black particles
are spherical in shape. This suspicion is confirmed by a look at the morphology
of carbon black. The smallest structural unit is crystalline, with a graphitic
structure, and these form globular primary particles with a diameter of
10 - 100 nm. Agglomeration of these primary particles gives rise to larger
(up to 1 µm), more or less hollow secondary agglomerations with varying
structure. In view of this state of affairs a model was devised according
to which the primary particles with a diameter of approximately 30 nm
congregate to form highly structured, preferably linear, branched secondary
agglomerates and thereby explain the formation of conductivity bridges
at lower carbon black concentrations.
The introduction of a "universal interfacial energy constant"
must be regarded as a first step away from a purely geometrical view. This
model envisages that percolation takes place in all systems at a universally
valid threshold value of the interfacial energy g*
([69] and the literature cited therein). Unfortunately this theory does
not offer any more profound fundamental information about interactions
at the interfaces. A theory that deals with these interfacial phenomena
is described in
Sidekick 2
Processing of Electrically Conductive Polymers and Compounds
3.1.2.2 Discussion and viability of the percolation
theory
In this sub-section the viability of the percolation theory will be
discussed in connection with experimental findings. These experimental
findings are the result of several years' work by WESSLINGet al. and are
documented in [72] and [73].
as a function of the polymer
matrix
- decreases with:
|
lower mean molar mass (--> lower viscosity) larger proportion
of crystalline regions decreasing polarity (poor wetting) |
- If a polymer blend, in other words a two-phase system is used, two
conductivity leaps can be observed, and consequently there are two values
of (this applies only to interpenetrating
networks, i.e. where the individual phase displays continuous links).
|
It will thus be seen that there is a strong relationship between the
critical volume concentrationand
the properties of the matrix. But was it not one of the basic premises
of the percolation theory that no interactions took place between matrix
and carbon black particles? How else could one explain a random distribution?
A possible, strictly mechanical, explanation would be that the high
viscosity of the matrix partially destroys the urgently needed structure
of the carbon black particles.
as a function of carbon black
characteristics
The strong dependence on the specific surface area and the structure
(expressed as the empty volume, measured by DBP absorption [2])
of the carbon black is shown clearly by fig. 3.1 [72a]. Empirical
quantitative approaches to clarifying this dependence already exist, such
as JANZEN's percolation formula (quoted in [72a])
 |
(3.1) |
| |
| where |
c |
is the density of the carbon black |
| and |
v |
is the DBP absorption. |
Thus the formula contains only "geometrical" factors. No reference
is made to any influence of the specific surface area as an interface.
Any suspected dependences on v, i.e the density of the carbon black,
remain obscure. Or is v perhaps (in relation to the preceding sub-section)
also dependent on the matrix via the form of the secondary agglomerates?
Changes in density with increasing carbon black concentration
Looking at the curve of the density of carbon black compounds against
increasing carbon black concentration, we find that the density initially
shows a linear increase with the carbon black concentration, until at
(determined beforehand) it stagnates or even decreases slightly, after
which the expected increase is resumed. This phenomenon has also been observed
when using a pigment as dispersed phase. As suspected, a polymer blend
then exhibits two interruptions in the density increase.
With the percolation theory one cannot even begin to explain these observations,
which were very surprising at the time. The percolation theory makes no
provision for such a drastic change in structure that would explain a density
effect.
Shift of as a result of
tempering
In tempering, the specimen is rolled at a temperature of approx. 40°C
above the softening point and then cooled slowly after the relevant deformation.
As a result the conductivity of the same specimen is higher than before.
Working on the basis that tempering means supplying energy to the system
and thereby ensuring greater mobility, it appears inconsistent that the
carbon black particles tend to form conductivity threads (for conductivity
is higher after tempering) instead of taking advantage of the mobility
to achieve a better distribution, which would mean a deterioration in conductivity.
It therefore seems completely absurd, from the point of view of the percolation
theory, that tempered specimens display higher conductivity with a consequent
decline in c, when the percolation theory would lead one to expect a rise
in .
CO2 adsorption on carbon black compounds
In studies of CO2 adsorption on carbon black compounds with
different concentrations of carbon black a sudden increase in adsorption
at was observed (studies by TANIOKA
et al., described in [72]), an observation that cannot be explained with
the aid of the percolation theory.
Pyrolysis residues
In pyrolysis, a specimen is ignited for half an hour at 900°C in
the presence of nitrogen. Under these conditions the polymer matrix decomposes
and only the carbon black component remains. This method, with subsequent
ignition in the presence of O2, originates from quality control
and is used to check carbon black concentrations.
Analyses of pyrolysis residues yielded a surprising result. The residues
after pyrolysis were higher (up to 80%) [3] than
was to be expected on the basis of the carbon black concentration. Even
extremely precise execution of the pyrolyses, with a matrix reference,
brought no change. Moreover the residue did not consist of an unstructured
"heap" of carbon, but scanning electron micrographs revealed
a flat layered, wavelike or spongy structure of the residue. This suggests
that some of the matrix has altered its properties so much that this part
was not completely decomposed by pyrolysis, and also that a structure must
have formed in the compound that remained intact during pyrolysis. These
two phenomena cannot however be explained by the percolation theory, since
it rejects both the formation of orderly structures and any change in matrix
properties.
Studies with the scanning electron microscope (SEM)
During studies of specimens with a carbon black concentration less than
or in the region of
the following photographs were taken which reveal a number of interesting
facts.
Fig. 3.2 shows a specimen with a carbon black concentration of
0.5 vol.%, i.e. less than . In
it there are clearly visible individual spherical carbon black particles
which are present in the polymer matrix in isolated and distributed form.
This finding would agree with the percolation theory, although the shape
of the carbon black particles is more similar to a sphere than to a linear
structure.
Fig. 5.3 in sidekick 5 shows a specimen with a carbon
black concentration of 6 Vol.%. It will be seen that isolated particles
and branched chains exist alongside one another. A marked structure has
formed, although free matrix is still available that would permit isolated
distribution. Another important discovery, although not evident from the
photos, is no less interesting. The carbon black concentration found in
the photos is appreciably greater than one would expect on the basis of
the original concentration in the compound. Thus the carbon black particles
seem to concentrate for preference in planes, forming a seam-like structure.
– The percolation theory does not even begin to explain these experimental
results.
Résumé
An examination of all experimental results reveals that the percolation
theory displays distinct deficits when it comes to explaining the results
observed. This shortcoming led to the development of a new theory that
in addition to totally different processes responsible for the occurrence
of conductivity also postulates a fundamentally different view of the systems:
multiphase polymer systems are in a state of thermodynamic inequilibrium.
3.1.2.3 Significance of viscosity
In connection with this new approach, the change in viscosity with increasing
filler concentration was also investigated. The viscosity ultimately determines
the quantity of work Wdisp that has to be input from outside
to achieve a certain dispersion. Moreover the exponential increase in viscosity
may result in a compound becoming impossible to process. In 1982 we worked
out for PVC polymer blends a first approximate formula that, like all others,
describes a linear relationship for the natural logarithm of the resulting
viscosity plotted against free matrix as a percentage of volume:
|
(3.2) |
| |
| where |
 p |
is the free matrix as a percentage of volume |
| |
 |
is the viscosity of the matrix polymer |
This introduced the quantity a as a thermodynamic interaction
factor. All interactions are combined in a without differentiating
between them. The purpose of this formula is to determine empirical values
for a so as to be able to make first forecasts about the
increase in viscosity. A second step then consists in establishing a relationship
between the individual dispersions and their values of a,
in order to identify regular patterns and critical factors.
First values of a proved to be similar in magnitude to
the logarithmic values of the viscosities. This relationship led to the
introduction of apparent viscosity 
for the dispersed phase and to the inclusion of the adsorbed shell when
determining free matrix as a percentage of volume:
 |
(3.3) |
| where |
m |
is the effective free matrix as a percentage of volume |
| |
d |
is the pure dispersed particles as a percentage of volume |
| |
a |
is the matrix combined as adsorbed shell as a percentage of volume
|
| and |
 |
(3.4) |
| where |
 ads |
where ads is the viscosity of the adsorbed shell |
| |
 12e |
excess interface energy (e.g. the energy needed to wet a certain
area of filler with matrix) |
| |
12 |
is the equilibrium interface energy (resulting from filler-matrix cohesion-adhesion).
|
[
TOC |
next
]
