Density - Conductivity Measurements
in Green-State Compacts
Research Team: Reinhold Ludwig
(508) 831 5315 [email protected]
Georg Leuenberger (508) 831 5238 [email protected]
Focus Group:
Pierre Blanchard, Federal-Mogul
[email protected]
Dave Kasputis, Hoeganaes
[email protected]
Richard Scott, Nichols Portland
[email protected]
Ian Donaldson, PresMet
[email protected]
Paul Bishop, GKN Sinter Metals
[email protected]
Michael Krehl, Sinterstahl
[email protected]
Marcus Eismann, Osterwalder
1
2
Project Goals
•
Establish correlation between density and electric conductivity of green-state
P/M compacts with the ultimate aim of detecting density variations through
electric conductivity measurements.
•
Develop an algorithmic approach to infer density distributions from voltage
measurements recorded over the compact’s surface.
•
Develop instrumentation that monitors density distributions.
Achievements in this Work Period
•
Confirmed inverse behavior of density – conductivity relationship for highdensity, lubricated parts.
3
•
Explained inverse behavior of density – conductivity relationship for highdensity, lubricated parts physically.
•
Modeled density – conductivity relationship for all densities and lubricant
mixtures.
•
Procured controlled samples exhibiting large density variations.
•
Demonstrated that first measurements on these samples confirm the feasibility
of the derivation of the density distribution from voltage measurements.
•
Introduced a first proposal for instrumentation approach
Next Steps
•
Proof of concept for “simple” parts.
•
Proposal for sensor for complex parts.
•
Formulate an inverse algorithm whereby the voltage recording can predict
changes in density.
2
Density – Conductivity Measurements in Green-State
Compacts
1
Introduction
Building on the work in detecting surface-breaking and near-subsurface crack detection, the
current program focuses on the detection of density variation within a green state P/M part [1,2].
The research approach attempts to exploit the macroscopic evaluation of potentially very small
differences in conductivity compared to the microscopic exploitation of large variations in
conductivity between the metal compact and its microscopically small defects such as air pockets.
The following milestones have been identified:
•
Development of a correlation between sample density and electric conductivity
•
Formulation of a generic mathematical voltage-current-conductivity model
•
Conversion of the recorded voltages to conductivity predictions via an inverse algorithm
During the course of our density measurement program for green state PM parts, an initial
effort was made to link the density of the part to a specific conductivity, consistent with our first
milestone. This conductivity – density relationship was later to be used in the reconstruction of
the density map of the part. The outcome of the conductivity – density measurements, however,
differed significantly from the expected results. For non-lubricated parts, a linear relationship
between the nominal density and the conductivity could be established. For lubricated parts
however, not only the relationship was nonlinear, but it exhibited saturation and even an inversion
at high densities [3]. This behavior is depicted in Figure 1, where we see the conductivity –
density relationship for various lubricant mixtures over a wide range of nominal densities.
Initial attempts to explain the observed inversion focused on process effects. Especially
lamination, lubricant migration and increased oxidation through higher temperatures were taken
into account. However it became clear that none of these effects could be responsible for the
inversion behavior. As a consequence, further considerations were given to the theory of
conductivity of mixtures and answers were found that seemed to explain the initially puzzling
effects through physical laws. The following paragraphs will explain the inversion as a physical
phenomenon due to induced charges and polarization effects.
Having met the requirements for the first milestone, the next step targeted the development of
sensors and algorithms suitable for the measurement of density. New green state P/M parts were
obtained which exhibited a density gradient. These parts were measured in an initial attempt to
reconstruct the density distribution throughout the samples. The results of these measurements are
presented in the second part of this report.
3
15000
40000
Conductivity [S/m]
Conductivity [S/m]
50000
30000
20000
10000
5000
10000
0
0
6
6.5
7
6
2
7
Density [g/cm3]
Density [g/cm3]
Figure 1:
6.5
Conductivity – density relationship of green state PM parts with no lubricant,
0.375% and 0.75% AWX lubricant respectively (left); same curves without the
non-lubricated parts, illustrating the inversion of the relationship at high
densities (right).
Non-Conducting Particles in Conducting Medium
The problem of determining the electrical conductivity and the dielectric properties are
closely related. Both cases result in almost identical Laplace-type equations, where the potential
must be continuous across the interface of adjacent regions with different material parameters.
The other required boundary condition arises from the continuity of the electrical current density
J in the conductive case and the displacement D in the dielectric case. Since the electric field E
plays the same role in both cases, the averaging equations for the conductivity and the dielectric
constant become identical.
Starting from the Clausius-Mossotti equation for inhomogeneous media [5], we can, within a
part, derive an expression for the effective dielectric constant, and hereby for the conductivity
Assuming that the non-conducting particles are evenly distributed within the sample volume and
subjected to a constant external field of E0, we find
εEeff − ε 0 E0 =
4π
4π
εP =
nLE eff
3
3
(1)
where L represents the depolarization factor.
Considering a medium consisting of two (or more) constituents with conductivities σ 1 and
σ 2 and volume fractions f 1 and f 2 on a completely symmetrical basis leads to Bruggeman’s
symmetric theory [5,6]. This theory calculates the conductivity of a random mixture of spherical
4
particles of two constituents, which together completely fill the media. Generalizing the equation
for all three dimensions d=1,2,3, the conductivity of the medium σ m is given in the equation
f1
(σ 1 − σ m )
(σ 2 − σ m )
+ f2
=0
(σ 1 + (d − 1)σ m )
(σ 2 + (d − 1)σ m )
(2)
Using the case of a non-conducting dispersion material (σ 1 = 0 ) in a highly conductive host
medium with σ 2 = σ h , Eq.(2) can be solved for the conductivity of the resulting mixture as

df 
f 

σ m = σ h 1 −
 = σ h 1 −  .
fc 
 d − 1

(3)
Here fc denotes the critical insulator volume fraction at which the conductor - insulator transition
occurs. In three dimensions fc is 2/3.
The conductivity of a medium, where the dispersion with conductivity σ 1 consists of an
effectively infinite size range of spheres, each of which remains coated at all volume fractions
with the host medium of conductivity σ m , can be calculated using Bruggeman’s asymmetric
theory [6,8]. Using the more general approach of oriented ellipsoids instead of spheres, the
equation can be written as [11]
(σ m − σ d )1 / L
σm
=
(1 − f )1 / L (σ m − σ h )1 / L
σh
,
(4)
with L denoting the depolarization factor of the oriented ellipsoids in the direction of the current
flow. When the dispersion can be regarded as an insulator compared to the conductivity of the
host medium ( σ 1 = 0 ), equation (4) becomes
1
σ m = σ h (1 − f )1− L .
(5)
Combining the two theories into a semi-phenomenological effective medium equation
developed by McLachlan [9] results in one equation that allows to handle media whose
morphologies are those of the symmetric and asymmetric media of Bruggeman or lie in between
these two extremes. This generalized effective medium equation is given by
(
1/ t
σ1
)
(
(1 − f c ) σ 2 − σ m
−σ m
+
f
f
1/ t
1/ t
1/ t
σ2 + c σm
+ c σm
1 − fc
1− fc
1/ t
f σ1
1/ t
1/ t
1/ t
)= 0 .
(6)
with t = f c /(1 − L) for oriented ellipsoids and fc being the volume fraction at which the
conductor-insulator transition occurs. Again assuming σ 1 = 0 , we find

f 
σ m = σ h 1 − 
fc 

5
t
(7)
for the conductivity of the mixture. Although this equation can be used for any mixture and
morphology type, the value for fc is not readily available. Furthermore we can assume that in our
case all the lubricant particles will be surrounded by the metal powder, since the volume fraction
of the lubricant is so low. Hence, we may as well use Bruggeman’s equation for asymmetric
media as given in (5).
Modeling the conductivity of a green state P/M compact was accomplished by calculating the
volume fraction of air and lubricant at each density. Using Eq. (5), the conductivity for nonlubricated parts was calculated with the conductivity iron as the base material. In a next step, the
resulting conductivity was used as the background conductivity in the calculation of the lubricated
parts, resulting in an overall equation of
1
1
σ PM = σ Fe (1 − f air )1− Lair (1 − f lub )1− Llub .
(8)
Using these parameters to simulate the conductivity – density relationship over a wide
density range does not show the results we obtained during the experiments. Although the volume
fractions for the air and the lubricant account for a reduction in the overall conductivity, the
relationship is still linear. This is explained by the fact that the volume fraction of the lubricant
stays constant through the compaction process. The increased amounts of non-conducting
lubricant particles per volume at high densities only results in a lower slope in the still linear
relationship and cannot explain the inversion behavior.
3
Depolarization Effect
So far the lubricant particles and the air inclusions within the pressed part were regarded as
perfect spheres. As seen above, the linear increase of low-conducting particles per volume with a
linear increase in density leads to a linear increasing conductivity, where the amount of lubricant
in the mixture determines the slope of the relationship. Instead of regarding the depolarizing
particles as constant spheres, we now change this viewpoint and take the geometrical deformation
of the lubricant into account.
The depolarization factor, which enters equation (6), depends heavily on the geometry of the
inspected particle. Let us consider an ellipsoidal piece of uniform dielectricity, which is bounded
by a surface defined by
2
2
 x  y z
f ( x, y , z ) =   +   +  
a b c
2
(9)
In the presence of a uniform external field E0, the interior field is given by the superposition
of the external field and the depolarization field E1, which is produced by a surface charge. The
depolarization potential at any point within the ellipsoid is given by the surface integral
Φ (r ' ) = ∫
P ⋅n
dS ,
r − r'
6
(10)
where P = χE is the polarization of the dielectric. Since the depolarization field inside the
ellipsoid is uniform [10], we can determine it at any point. The center of the ellipsoid is the
obvious point to choose. Here the potential is given by
E1 (r ' ) = E1 (0) = − ∫
 P x Py y P z 
P ⋅n
rdS = − ∫  x2 + 2 + z2  .
3
r
b
c 
a
(11)
The depolarization factors, which are usually defined by
E1i = − Li Pi ,
(12)
can now be calculated by rewriting (11) in polar coordinates [3]
2π
2π
1
cos 2 ϑ
∂ϑ
L = 2 ∫ ∂ϕ ∫ sin ϑ
sin 2 ϑ
sin 2 ϑ 2
cos 2 ϑ
c 0
2
2
0
cos ϕ +
sin ϕ +
cos ϕ
a2
b2
c2
(13)
In a spheroidal geometry ( a = b ≠ c ) this integral can easily be evaluated. If the external field
is applied along the z-Axis, the depolarization factor for the oblate spheroid results in
LZ =
4π
e2
2


1 − 1 − e sin −1 e ,


e


(14)
2
2
where e = 1 − (c / a ) refers to the ellipticity of the rotated ellipse with c denoting the short
half-axis, and a the long half-axis of the oblate spheroid.
As depicted in Figure 2, the depolarization factor clearly changes in a non-linear fashion.
Starting from a perfect sphere with Lz=1/3, a linear change of the dimension in z-direction results
in a non-linear change of the depolarization factor and approaches 1, when the extension in zdirection goes to 0.
7
Depolarization Factor for Oblate Spheroid
0.9
Depolarization factor L
0.8
0.7
0.6
0.5
0.4
0.3
0
0.1
0.2
0.3
0.4
.5
Ellipticity e = SQRT(1-c
Figure 2:
0.6
.7
0.8
.9
1
a )
Change of depolarization factor of the oblate spheroid as the geometry changes
from a sphere to a flat disk. The external field is assumed in the direction of the
short half axis of the spheroid.
If we again use (6) as the basis of the calculation of the conductivity of our PM part, the
depolarization factor is no longer constant. Increasing the density of the parts will result in a
deformation of the lubricant particle. Since the lubricant itself is incompressible, the volume of
each lubricant particle is assumed constant. Further assuming that the spheroid character of the
particle is conserved, any compression in z-direction results in a reduction of the short half axis c
and in an increase of the longer half axis a. The sharp increase of the depolarization factor at high
densities in turn leads to a highly increased depolarization effect. Therefore the voltages recorded
over a given length, while injecting a constant DC current, start to increase instead of decrease.
This, however, leads to the observed decrease in conductivity at high densities. Figure 3 shows
the result of the simulations, where the conductivity was calculated for non-lubricated parts and
for three different lubricant mixtures. The inversion can clearly be observed.
8
10
x 10
6
onductivity vs. Density
No Lub
9.5
9
0.375%AWX
Conductivity [S/m]
8.5
8
0.5%AWX
7.5
7
0.75%AWX
6.5
6
6000
Figure 3:
4
6200
6400
600
6800
Density [g/cm3]
7000
7200
7400
600
Simulation of conductivity vs. density behavior for green state PM parts with
different amounts of lubricants
Summarizing the Conductivity-Density Relationship
The above simulations show a very good agreement between theory and actual
measurements. The measurement results for various parts together with the simulated data show
that the basic relationship is understood. Several different parameters are still available to adjust
and fine tune the simulation results. Different possibilities include the degree of deformation of
the lubricant particles, the density at which lubricant deformation ensues, the rate of deformation
with increasing density, and so on. In the same sense, one could also argue that the air bubbles
within the part will be deformed to ellipsoidal shape too, before they are actually forced out of the
compressed powder. Such a deformation would increase the depolarizing effect of these air
bubbles much in the same way as the deformation of the lubricant.
These parameters are all likely to change with each base material, lubricant, and even
lubricant amount. The exact relationship is therefore difficult to predict. But the presented theory
sufficiently explains the reasons for the inversion of the conductivity – density relationship at
high densities for lubricated PM parts.
9
5
Parts and Measurement Setup for Density-Reconstruction
A new set of green state P/M parts was needed to measure density variations. Contrary to the
previous parts (which were wide and flat to achieve a density distribution that was as uniform as
possible) the new parts needed to be long and slim. Samples of aspect ratios of about 4:1
(length:diameter) would inherently exhibit significant changes in density from top to bottom, even
more so when pressed in a single punch process.
We were able to obtain two sets of parts, of which especially the set provided by GKN
Worcester proved to be exactly formed in the required aspect ratio. The parts, shown in Figure 4,
were 2.75” to 3.25” inches long and all had a radius of 0.75”. Using the same powder mixtures
that were used to press the flat disks in the first part of the project, the green samples had the
following properties:
•
iron 1000B as base material,
•
four different sets of parts with no lubricant (die wall lubricated), 0.3% AWX,
0.5% AWX and 0.75% AWX respectively,
•
each set with 7 to 8 parts of different pressures, ranging from 25 tsi to 55 tsi
•
single punch pressing (to increase density gradient)
Figure 4:
Green state P/M samples with length : width ratio of 4:1, used for measurement
of density gradient.
The above parts were contacted in the same way as the previous samples, injecting a direct
current of 1A from top to bottom and recording the voltages along the surface. But this time, we
did not take an integral voltage measurement over the entire length of the part, effectively
averaging all possible density variations, but rather measured over small slices of 0.25” each (see
Figure 5 for the basic measurement arrangement.
10
Aluminum rod
I
Figure 5:
V
V
V
V
V
V
σ1 σ2 σ3 σ4 σ5 σ6
I
Measurement setup for the measurement of density variations over the length of
a green state P/M rod. A controlled DC current is injected and the voltage on the
surface is recorded on several slices of 0.25” thickness each.
In such a setup, the conductivity, and hence the density, are considered constant over the
measured length of 0.25”. While this is obviously only an approximation (since it effectively
averages the density over the measured length), it provides the means for a straight forward
reconstruction of the conductivity and the corresponding density. Since the constant current I is
forced through each of the slices and is homogeneous throughout the volume, the conductivity
can be calculated from the measured voltage and the geometrical measures of the measured length
L and the cross-section A as
σi =
6
I L
,
Vi A
(15)
Measurement Results
The results of the voltage measurement on the surface of the long, thin green state P/M rods
show a distinct change in conductivity over the length of the part. Even with the limited resolution
of 0.25” for our first measurements, a conductivity and hence a density distribution is apparent.
Furthermore, comparing the results from the set of non-lubricated parts and the parts with 0.75%
AWX, the voltage measurements indicate the difference in the conductivity – density relationship.
While the density distribution in both sets is expected to be similar, the shape of the voltage curve
over the length of each part is different. Figure 6 shows the monotonically increasing voltages
measured for non-lubricated parts.
In the lubricated parts on the other hand, we find a very slow increase, if not even a decrease
of the voltage near the top of the part (where we find the highest green state densities), with the
expected monotonic increase following for the lower part. This again reflects the conductivity –
density relationship in lubricated parts, which exhibits a maximum conductivity between densities
of 6.8 and 7.0 g/cm. The voltages recorded for the parts lubricated with 0.75% AWX is depicted
in Figure 7.
11
Voltage Measurements - No Lubricant
9
8
7
V [mV]
6
25 tsi
5
32.9 tsi
4
55 tsi
3
2
1
0
0
0.5
1
1.5
2
2.5
3
l [in]
Figure 6:
Voltage measurements on slices of non-lubricated green state samples of various
initial densities.
Voltage Measurements - 0.75% AWX
9
8
Voltage [mV]
7
6
25 tsi
5
33.6 tsi
4
40.8 tsi
3
55 tsi
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
position [in]
Figure 7:
Voltage measurements on slices of green state samples of various initial
densities, lubricated with 0.75% AWX.
Together with the previously recorded relationship between the conductivity of pressed
powder and its density, we can now try to reconstruct the density distribution over the length of
the parts. This reconstruction is especially straight forward in the case of non-lubricated parts,
12
where the relationship is linear. Here every conductivity can be directly mapped into a
corresponding density. The in this way reconstructed density distribution indicates that the
densities at the bottom of the parts are the same, regardless of the pressure used. The calculated
densities at the top, where the press contacts the powder, correspond to the amount of pressure
used. The reconstructed densities for four different pressures are shown in Figure 8. Figure 9
provides the same data as presented in Figure 8, but the high and low pressure parts have been
separated to allow rescaling of the axis. This way we recognize the same inherent behavior of the
density distribution in high and low pressure parts.
Density Reconstruction - No Lubricant
8.00
Density [g/cm3]
7.50
25 tsi
7.00
32.9 tsi
41.2 tsi
6.50
55 tsi
6.00
5.50
0
0.5
1
1.5
2
2.5
3
position [in]
Figure 8:
Reconstructed density distribution over the length of the non-lubricated green
state P/M rods. Highest densities are found at the top, lowest at the bottom of the
parts (single punch pressed).
The densities for the lubricated parts can be reconstructed in much the same way, with the
only additional requirement that the algorithm must be somewhat intelligent to discriminate
between densities above and below the inversion point. Such an algorithm not only takes into
account the voltage reading for the inspected segment, but also the readings from the neighboring
segments in order to determine the general direction of the density gradient in the vicinity. This
algorithm is currently being developed and will be presented soon.
13
Density Reconstruction - No Lubricant
7.75
Density [g/cm3]
7.50
7.25
7.00
41.2 tsi
6.75
55 tsi
6.50
6.25
6.00
5.75
0
0.5
1
1.5
2
2.5
3
position [in]
6.35
Density [g/cm3]
6.25
6.15
25 tsi
6.05
32.9 tsi
5.95
5.85
5.75
0
0.5
1
1.5
2
2.5
3
position [in]
Figure 9:
7
Reconstructed density distribution over the length of the non-lubricated green
state P/M rods (same as in Figure 8). The separation of the high and low
pressure parts into separate diagrams allows to rescaling of the axis and the
demonstration of the same inherent density distribution in the four different
parts.
Conclusions
The theory connecting the base material, lubricant, lubricant concentration and nominal
density to its electric conductivity are now well understood. These findings allow us to deduce
equations calculating the density-conductivity relationship at all points. They will form the basis
for a reconstruction algorithm.
Additionally, a first set of measurements was conducted on parts exhibiting a density
gradient. The results of these measurements clearly show the feasibility of a partial reconstruction
of the density map throughout the part from voltage measurements on the surface. Especially in
the simple case of non-lubricated parts with their linear conductivity – density relationship, the
14
inverse algorithm and a density reconstruction were implemented with promising results. Further
work is required to develop suitable algorithms for all powder mixtures. In a future step,
appropriate sensors and refined algorithms must be developed to account for more complex part
shapes.
References
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of green-state powder metal compacts using the electrical-resistivity method,” Measurement
Science and Technology, Vol. 11, pp. 157 – 166, 2000.
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Resistivity Test Of Green-State Metal Powder Compacts,” in Review of Progress in
Quantitative NDE, Vol. 17B, pp. 1462 - 1470, Plenum Press, 1998.
3 . R. Ludwig and G. Leuenberger, “Electrical Measurements of Density in Green-State
Compacts,” PMRC Meeting Spring 2001, Worcester, MA, April 2001.
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Solid State Phys., vol. 18, 1985, pp. 1891 - 1897
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Compacted Mixtures of Polymeric and Metallic Powders”, in Journal of Appl. Physics, vol.
42, 1971, pp. 614 - 618
15