A WATER TREEING MODEL
Jean-Pierre Crine
Consultant
Candiac, QC, Canada
Jpcrine@ieee.org
Jinder Jow
Dow Chemical
Somerset, NJ, USA
IEEE Trans. DEI, vol. 12, pp. 801-8, 2005
Abstract
A new model for water treeing is introduced. It assumes that when the field-induced stress applied on
nanocavities filled with a liquid is larger than the yield strength of the polymer, bonds will be broken and
the nanocavity will expand. The growth of the water trees is enhanced by the fatigue induced by the
alternating electric field . The diffusion of the liquid is also a parameter affecting the water-tree length. A
simple equation relating the water-tree length with field, time, frequency and the nature of the solution is
presented. A very good agreement between theory and experiments for a wide variety of results
obtained with LDPE tested under various fields, frequencies and ionic solutions is observed. The model
predicts the growth of water trees under dc fields after very long times or after many polarity reversals.
Some aspects of the model requiring further refinements or experimental data have been pointed out.
Introduction
Since the first paper on water trees was published nearly thirty years ago, much research has been
devoted to the phenomenon (see Refs. [1-28] and references therein) but there is not yet a
comprehensive model, although several approaches have been proposed (see Refs. [1-7] for a review
of the theories). Many parameters affect the growth phase: the field frequency and value, the
temperature, the polymer morphology, the ions nature and concentration, among many others [1-7]. In
actual cables, water trees are generated during the very slow initiation phase at asperities or interfaces
(semicon-insulation, impurities, voids, etc.) where the local field could be extremely high. It is also well
known that frequency is one of the main accelerating factors in the electrical aging of polymers, and this
is particularly evident in accelerated water-treeing tests (AWTT). This has led to accelerated watertreeing tests where the samples have all sort of surface defects (conical water-needle electrodes,
scratched surface, etc.) exposed to ionic solutions, whose nature and concentration may enhance
water trees initiation. The fact that no tree (or very few) is grown under dc voltage [12,25] and that
frequency accelerates the process strongly suggests that fatigue plays the key role in water treeing
[4,19]. Suzuki et al. [21] and Crine and Jow [22,26] have shown that the water-tree length varies linearly
with the number of field cycles, i.e. the time x frequency, provided all other experimental conditions are
kept constant (Fig. 1). We have also shown [26] that the influence of field is possibly much less strong
than often assumed and the best evidence for that comes from the fact that water trees grown under
constant and moderate fields in Rogowski cells are of nearly the same length than those grown after
the same number of cycles but under very high fields at the tip of water needle electrodes. There is a
need for a comprehensive model describing the main features of water tree growth under various
experimental conditions.
Fig. 1- Water-tree length as a function of the number of cycles measured in PE soaked in NaCl
(different concentrations) and under widely varying ac fields (50 Hz to 30 kHz). Ashcraft cells at room
temperature. The symbols refer to the following field range (in kV/mm): <100; 200<>100;
300<>200; 300<>500; >500; details can be found in [26].
Over the last twenty years many speculations have been offered to explain water treeing and, to
simplify, there are now two main schools of thought. One claims that water droplets are deformed by
the mechanical forces induced by the electric field. This causes the local stresses to exceed the bond
energies of molecular chains, which eventually leads to formation of cavities and of a water tree. Bond
rupture at the tip of deformed water droplets creates additional cavities and thus facilitates the growth of
a water tree. Tanaka et al. [8], Patsch [9] and Sletbak [11] have developed theoretical models and
Filippini et al. [12, 14,16], Bulinski et al. [18] and Crine et al. [19] have provided experimental support to
this approach. The other takes a different approach, in which chemical reactions and especially
oxidation, involving water, ionic impurities and PE, are thought to induce local chain degradation [1,56,17]. Water ingress into the insulation introduces impurities which further induce growth of the
degraded area. This process is assisted by thermal bond breaking either due to high cable operating
temperatures or local heating induced by an electric field at defects. Ross et al. [5,17] have proposed
that among all oxidation by-products, carboxylate ions play the key role in water tree initiation and
growth. Fan and Yoshimura [20] have recently shown that the chemical reduction potential of metal
ions could also be a crucial parameter. We have shown elsewhere [4,24] that oxidation, as such,
should not be a main source of concern since it is a consequence and not the principal cause of bonds
breaking. This paper is devoted to the introduction of a new model based on the influence of fatigueinduced bonds breaking, on the stress induced by the applied electric field and on the elastic properties
of the polymer. The present model is still in development and the needs for some further improvements
are discussed. The validation of the proposed theory is based on water-tree growth experiments
performed at room temperature in low-density polyethylene (LDPE) samples subjected to various ionic
solutions, fields and frequencies in Ashcraft cells (water needle electrodes). Application of the theory to
crosslinked polyethylene (XLPE) or to other polymers will be dealt with in subsequent publications.
The Model: Basic assumptions
The unambiguous relation shown in Fig. 1 between water-tree length and the number N of field cycles
is a clear evidence that one main parameter of any water-treeing model should be the number of
cycles, i.e. bonds breaking is a fatigue-like process. Let us assumes that some of the very tiny “voids”
composing the free volume originally present in PE are filled by a liquid. Assuming spherical voids, the
typical free volume in a 45-50% crystalline PE at room temperature would be vo~ 3x10-28 m3 (see
Appendix A). When a stress (induced by the electric field F) is applied on them, they will tend to
coalesce together to form nanocavities. Eventually, some weak bonds will be broken by the stressinduced field and the nanocavities will grow further to form “cavities”. Their size varies widely depending
on the experimental conditions and reported values are ranging between 10 nm [27] and some m [28].
These nanocavities are more or less interconnected and when filled with water, they form a water tree.
For each field cycle, a certain number n of nanocavities are then formed and this number will depend
on the applied stress (i.e. ½ ’o F2 ) and on the elastic limit of the material. A polymer with a high yield
strength Y will be more difficult to deform and then would grow smaller trees than a polymer with a low
yield strength, as PE. Obviously, a high stress, i.e. high field, should generate more defects than a low
stress. The criterion for water tree growth is that the energy dissipated in the liquid-filled cavities should
be larger than the elastic energy stored in the treed volume. This occurs when the pressure exerted
after N field cycles on a cavity (of volume equal to n vo ) filled with a liquid is larger than the yield
strength of the polymer multiplied by the water-tree volume, the tree grow, i.e. for
N ½ ’o n vo F2  Y x water-tree volume
(1)
where ’ and o are the dielectric constant of the liquid and the free space permittivity, respectively; F is
the field and Y is the yield strength (~1.5x107 N/m2 for PE at 22oC). For the sake of simplicity, we have
assumed that the water-tree volume would be a sphere of diameter L, which is the water-tree length;
the assumed volume is then (4/3)(L/2)3 ~ L3/2. This is a gross simplification but it is reasonably close
to reality and taking different shapes will not induce very different results, especially at this early stage
of model development. For example, assuming a conical shape for the water tree with the length of the
cone equal to the water-tree length and with the cone diameter equal to half the length, the treed
volume is approximately equal to L3/4. Therefore, the shape of the total treed volume has a limited
influence on the calculations. Assuming a sphere, Eq. 1 becomes
N ’o n vo F2 ~ Y L3
(2)
Thus, Eq. 2 predicts a cubic root relationship between L and N, as indeed observed experimentally.
The number of nanocavities filled with water or any other liquid will obviously depend on the presence
of water and we may expect that water diffuses through the polymer at a constant rate. Lets then
assume that n varies with time according to n = no t½ , where no is a constant typical of the material and
its morphology, of the nature of the diffusing liquid and possibly of the temperature. Substituting into Eq.
2 gives
L  (N ’o no t½ vo F2 / Y)1/3
(3)
Equation 3 predicts a relatively small influence of field on water-tree length, as indeed observed in Fig.
1. It is shown later in more detail that it describes extremely well experimental data. An interesting
feature of Eq. 3 is to predict that water trees can grow under dc fields but in that case N = 1 and it would
take an extremely long time to have a measurable tree; more detail later. In Eq. 3, the only unknowns
are no and ’ (although the dielectric constant of water is well known and it could be easily measured for
other liquids) and therefore its validity could be proven if plots of L vs. (No t½ vo F2 / Y)1/3 would yield
straight lines. For that purpose, we have considered water-trees grown at 22oC at the tip of water
needle electrodes in LDPE samples maintained in various solutions (but mostly NaCl).
Preliminary Validation of the Model
Results obtained with water needle electrodes (also known as Ashcraft cell) offer the great advantage
that the field due to a voltage V applied at the tip of the conical depression filled with an ionic solution
can be easily calculated from
F = 2 V / r ln (1 + (4 l /r))
(4)
where r is the radius at the point and l is the distance between the point and the ground. The
field values thus deduced appeared sometimes fairly high (and very close to the breakdown field
for PE) and it could possibly be more appropriate to use the following equation recently used by
Cisse et al. [29]
F = V c / rd [ln (4 c /r)1/2 – 1]
(5)
where c is the distance from the top of the sample to the tip of the electrode and d is the sample
length. The problem is that almost no one has reported their c and d values, whereas l values
were available. The field calculated with Eqs. 4 and 5 for one case where all details were
published indicates that Eq. 5 yields lower (~ 20%) field values than Eq. 4. It remains to verify if
this would apply to other results. The only impact of a lower field would be to increase the value
of no. As already noted, it is still an early stage of development and a 20% variation in the value of
one parameter is not dramatic.
Fig. 2- Experimental results of Yoshimura et al. [10] plotted as a function of (No t½ vo F2 / Y)1/3 .
In Fig. 2 we have replotted the results obtained in distilled water by Yoshimura et al. [10] as a function
of (No t½ vo F2 / Y)1/3 . Obviously, the expected straight line is observed, whatever the frequency or the
field. The slope gives ’ no = 1.15x106 and, assuming 80 as dielectric constant for water, a no value of
1.44x104. Therefore the nanocavity volume is 3x10-28 x 1.44x104 = 4.3x10-24 m3, or a diameter of 20.5
nm. This is within the range observed by Kalkner [27] but a complete validation would require a
statistical analysis of the cavity sizes compared to the no value deduced from the slope of L vs. (No t½
vo F2 / Y)1/3 . Nevertheless, the main features of the model describe some experimental results. Figure 3
shows similar good agreement between theory and experiment for tap water as electrolyte [20]. In Fig.
4, we have calculated the water -tree length grown in deionized water by Fan et Yoshimura [20] using
Eq. 3 and the ’no value deduced from a L vs. (No t½ vo F2 / Y)1/3 plot.
Fig. 3- Experimental results of Fan and Yoshimura [20] obtained in tap water under various fields
(constant time) or under constant field (different times)plotted as a function of (No t½ vo F2 / Y)1/3 .
Fig. 4- Experimental [20] vs. calculated (Eq. 3) water-tree length grown in deionized water.
Let us now look at data obtained with NaCl, the most popular ionic solution. Figure 5 shows the good
agreement between theory and experiment for the results of Filippini and Meyer [14]. In Fig. 6, we
compare the experimental results of Matey et al. [15] with the calculations based on Eq. 3 and the ’no
value deduced from a L vs. (No t½ vo F2 / Y)1/3 plot. The agreement is quite good regarding the usual
enormous scatter in water treeing data.
Fig. 5- Experimental results of Filippini and Meyer [14] plotted as a function of (No t½ vo F2 / Y)1/3 .
Solution: 0.1 N NaCl, F= 203 kV/mm (Eq. 4), f = 1.5 kHz.
Fig. 6- Calculated (Eq. 3) water-tree length grown in 0.1 N NaCl [15].
Figure 7 shows the straight line expected from Eq. 3 when Nitta’s results [8] are plotted as L vs. (No t½
vo F2 / Y)1/3 . Finally, Fig. 8 shows the same good agreement for the results of Suzuki et al. [21]. For the
cases where a high frequency was superposed on the 50 Hz voltage, we have taken as number of field
cycles the number of cycles due to the high frequency stress since we consider that the main
degradation parameter is the fatigue associated with the polarity reversals. Even in that complicated
situation with multiple stresses of different frequencies the model gives a very good description of the
actual results. Interestingly enough, one single ’no value seems to fit all results, as expected. Let us
now examine more closely the field dependence of water-tree growth by comparing data and
predictions made from Eq. 3.
Fig. 7- Experimental results of Nitta [8] plotted as a function of (No t½ vo F2 / Y)1/3 . Solution: 20% NaCl,
F= 205 kV/mm (Eq. 4), f = 50 Hz and 1kHz.
Fig. 8- Experimental results of Suzuki et al. [21
Fields are ranging between 52 and 277 kV/mm (from Eq. 4).
o t½ vo F2 / Y)1/3 .
Influence of Field on Water-Tree Length
The F2/3 dependence of L predicted by Eq. 3 is clearly evident whatever the solution, fields and
frequencies in Figs. 9-11 where we have compared the water-tree lengths to the experimental
values. The ’no value was deduced from L vs. (No t½ vo F2 / Y)1/3 plots. The field values shown in
Fig. 9 were calculated using Eq. 4 and, as already discussed, they seem fairly high but this does not
affect the validity of Eq. 3.
Fig. 9- Comparison between experimental [10] and calculated (Eq. 3) water-tree lengths under
various fields using distilled water as electrolyte. The ’no value was deduced from the slope in
Fig. 2. Frequency : 3 kHz.
Fig. 10- Comparison between experimental [20] and calculated (Eq. 3) water-tree lengths under
various fields using tap water as electrolyte. Frequency : 60 Hz.
The modest field dependence depicted in Figs. 9-11 explains why most accelerated water
treeing data does not appear to be very sensitive to field, as was shown in Fig. 1. This does not
mean that field is not important but it simply means that a relatively low voltage applied at the tip
of a small protrusion may generate a very high field, which will initiate a water tree. Once started,
the water tree growth is not extremely accelerated by a high field and the number of field cycles
and time appear to be more significant factors.
Fig. 11- Comparison between experimental [14] and calculated (Eq. 3) water-tree lengths under
various fields using 0.1N NaCl as electrolyte. The ’no value was deduced from the slope in Fig. 5.
Frequency : 1.5 kHz.
Influence of Ionic Concentration
Another parameter affecting water treeing is the nature and concentration of the ionic solution
[2,4]. From Figs. 12-14, it appears that the influence of the ionic solution will be taken into
account through the ’no parameters, i.e. from the slopes in L vs. (No t½ vo F2 / Y)1/3 plots. Figure
12 shows the results obtained with two widely different NaCl solutions for samples tested under
similar conditions. Obviously, the 1N solution gives larger water trees than the 0.01N solution. We
may expect that the dielectric constant would be different and it seems likely that no was also
different but it has to be verified by further work.
Fig. 12- Water tree lengths [23] as a function of (N o t½ vo F2 / Y)1/3 for two different NaCl
concentrations and under different fields (the other experimental conditions are similar).
In Fig. 13, we have plotted results obtained by several authors [8,13,16,17,20] with all sorts of
NaCl solutions. Quite clearly, the slope, i.e. ’no, increases with the ionic concentration but in a
non linear fashion : above 0.5 N, adding more ions does not seem to change the water-tree
length dramatically. It is well known that the solution conductivity increases with its concentration
[10] but they are limited data on the influence of concentration on dielectric constant. It is also
well possible that depending on the nature and concentration of the solution, chemical or
electrochemical reactions could affect the value of no. Further support to this possibility is shown
in Fig. 14 where we have plotted the water-tree lengths measured by Fan et Yoshimura [20]
under similar conditions but with different solutions. It seems very likely that electrochemical
reactions play a role when the solution with a large negative reduction activation energy (Gred for
AgNO3 = -7.6x10-4 J/mole) gives the largest trees and when the smallest trees are associated
with solutions having positive reduction energy (Gred for NaCl = 2.6x105 and for KCl = 2.8x105
J/mole) [4]. Much more work remains to be done to identify the reactions and also to relate them
to the parameter no. However, our current model has a parameter to take these effects into
consideration and presently it is the only adjustable one.
Fig. 13- Water tree lengths [8,13,16,17,20] as a function of (N o t½ vo F2 / Y)1/3 for widely
different NaCl concentrations and all sort of different experimental conditions.
Fig. 14- Water tree lengths [20
o t½ vo F2 / Y)1/3 for several different ionic
solutions; concentrations : 0.13 N (the other experimental conditions are the same). The results
obtained with AgNO3 () refer to the right hand scale.
Discussion and Conclusion
Our new model therefore seems able to describe fairly well all sort of conditions of water tree
growth in LDPE. An interesting feature of Eq. 3 is the possibility to grow trees under dc voltage. In
that case, N = 1 and obviously this will result in very short trees unless extremely long times are
used. As an example, we have shown in Fig. 15 the calculated water-tree length under dc for the
sample material and experimental conditions used by Matey et al. [15] who performed their test
under a 1.5 kHz ac field. After 400h, the dc tree would have been ~100 times shorter than the ac
tree, which may explain why water trees have been rarely reported under dc [12,25]. However, in
presence of an highly corrosive solution, Anelli et al. [25] have measured sizable water trees.
Note that Zeller had suggested that the dc component of the stress has a small contribution for
solutions of low conductivity (i.e. low ’ ); increasing the conductivity may lead to a non negligible
contribution to the degradation process [3]. If instead of one single voltage application (i.e. N =1 ),
the dc voltage is shutt on and off several times, the number of cycles will increase accordingly. In
the case shown in Fig. 15, 800 voltage interruptions (i.e. one every 30 mn) would have led to a
~2 m water tree after 400 h. This remains to be verified and this would be a good test of the
validity of our contention that fatigue is a major factor in water-tree growth.
Fig. 15- Calculated water-tree length under dc field for one voltage application (line; right hand
scale) compared to experimental data obtained under 1.5 kHz (see Fig. 5).
Another important point to be clarified is the influence of the nature and concentration of the ionic
solution on ’no. The dielectric constant of the solution can be easily measured and the value of n o
could be determined from the ratio of the volume of the nanocavities composing the water tree to
the original free volume (i.e. ~3x10-28 m3 at 22oC). Finally, the influence of the nature and
morphology of the polymer on the parameters in Eq. 3 are of great practical interest and should
deserve further study.
To summarize, we have introduced a simple water treeing model based on the assumption that
nanocavities are formed and grow when the pressure exerted by the field on the liquid exceeds
the elastic limits of the polymer. The growth depends on the number of field cycles and on the
liquid diffusion time. The influence of field is moderate, in agreement with experimental data and
water trees could be grown under dc voltage, especially after very long times. The nature of the
polymer and of the ionic solution appear to influence the size of these original nanocavities but it
remains to be verified by some experimental work.
Appendix A : free volume determination in PE
Polymers are not tightly packed materials and contains a so-called free volume, that is some empty
space between the molecular chains. One technique often used to measure the free volume is positron
annihilation spectroscopy [30-33]. The recombination of an electron and of a positron induces a fast
energetic photon emission whose lifetime is a measure of the size of the free volume. Figure A 1 shows
the free volume calculated from the positron recombination in PE samples of various crystallinity
[32,33]. Assuming that most LDPE samples used for water treeing tests have a crystallinity in the 4550% range, we may conclude that the average volume of the “ free voids” in these samples is ~3x10-28
m3.
average "free volum e" (m ³)
4E-28
3.5E-28
3E-28
2.5E-28
2E-28
1.5E-28
1E-28
40
50
60
70
80
90
X%
Fig. A 1- Dependence of the free volume in PE at 22oC on crystallinity calculated (see Refs. 30-31)
from experimental data obtained by [32,33].
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