Submitted Manuscript: Confidential
Supplemental Online Material: Percolation and universal scaling in composite
infiltration processing – Detailed Description of Methods and Additional
Remarks
Authors: Alain Léger1*, José-Miguel Molina-Jordá2, Ludger Weber1 and Andreas Mortensen1*
Affiliations:
1
Laboratory of Mechanical Metallurgy, EPFL, Station 12, CH-1015 Lausanne, Switzerland.
2
Instituto Universitario de Materiales de Alicante, Universidad de Alicante, E-03080 Alicante,
Spain.
*Correspondence to: alain.leger@epfl.ch (experiments), andreas.mortensen@epfl.ch
(discussion)
The infiltration apparatus – The pressure infiltration apparatus is sketched in Fig. S1a. It
comprises a water-cooled pressure chamber, with its inner wall padded with insulation and
containing an induction coil. A cylindrical graphite susceptor placed within the induction coil
provides a heated working space 60 mm in diameter and 250 mm high. The machine in its
current configuration is capable of delivering melt temperatures up to 1,500 K controlled to
roughly ± 5 K over several hours. An automated system of pressure control, composed of a set of
electrovalves that regulate the gas inlet and outlet pressures to a precision of 10-3 MPa, enables
controlled pressure ramps in the range between 10-1 and 10-3 MPa/s as well as a fixed constant
pressure P to be held for several hours. The nominal design pressure of the apparatus is 20 MPa;
however, temperature control becomes difficult above P = 12 MPa (due to heat transfer by
conduction and convection through the pressurized argon gas).
A special device enables dynamic tracking of the molten metal surface while the apparatus is
pressurized. This is accomplished using a sensor that is connected via an alumina tube to a
graphite plunger that floats atop the melt within the crucible. Upon pressurization the floater
follows the melt surface as it descends to accompany flow of the metal into the preform; this
allows to measure in situ the volume of liquid metal that has invaded the preform. The local
resolution of the system is on the order of ± 5 µm while the total precision (after calibration) is
around 40-50 μm over a 6 mm displacement of the graphite plunger. The device features a
custom-built optics-based sensor to track the movement of the graphite plunger. The sensor
includes two pressure-resistant windows, one for a light source and the second to enable
tracking, using a digital camera, of a series of white and black stripes lining a vertical rod linked
to the graphite plunger.
Percolation and universal scaling in composite processing
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Figure S1 - Infiltration machine and experimental procedure
(a) Sketch of the optics-based metal level-tracking device fixed on top of the infiltration
apparatus. (b) Sketch of an infiltration experiment, conducted under stepwise increasing pressure
at 1,423 K. (c) Cross-section through a solidified, partly infiltrated, sample (left) and a typical
drainage curve collating data from five separate experiments and thus showing data
reproducibility (right).
Percolation and universal scaling in composite processing
3
A dedicated LabView software was developed to compute the metal level position using image
analysis. A sketch explaining the technique is displayed in Fig. S1b. Movement of the plunger is
captured optically by tracking a rectangular “Region Of Interest” (ROI in LabView; red
rectangle in Fig. S1b), in which the same dark stripe is followed during the whole infiltration
run. In more detail, the procedure is as follows: recorded images of the stripes are first binarized
in real time; a ROI is then placed manually around a selected dark stripe before infiltration starts.
Two horizontal lines are selected; the upper and lower boundary of the dark stripe inside the ROI
(for which the pixel intensity shifts from black (0) to white (255)). This ROI is scanned for any
variations of pixel intensity below an adjustable threshold value (typically 50). The ROI then
follows the selected pixels (typically 5 for the upper and the lower boundary) for as long as
infiltration proceeds. The program then measures their location in real time and the average
value is recorded (corresponding in fact to the middle of the black stripe). The origin (0 ; 0) is
defined as the top left of the image (hence the camera may not be moved during infiltration). The
pressure is then increased to another value level P2, and so on until the plunger no longer moves
when pressure is increased, indicating that all available pores within the preform are now
invaded. At that moment, the preform is considered as fully infiltrated. Since the inner diameter
of the crucible is constant, assuming that the composite is fully infiltrated at that point, the
saturation S is defined as being S = 0 at the first plateau h0 and S = 1 at h3. In-between, discrete
points along the saturation curve are obtained by simple linear interpolation, which holds given
the constant crucible cross-sectional area (Si = (hi – h0)/(h3-h0)).
At such high temperatures the free vertical movement of the plunger can at times be impeded
by sticking or by any oxide that might cover the melt. Another custom-built device was therefore
added to drive both rotary and vibratory motion of the plunger, the former allowing to disengage
the plunger during an experiment if required (this was seldom used with unalloyed copper),
while the latter was used periodically to ensure free plunger vertical movement.
Provided pressurization is sufficiently slow [1] or if a series of constant increasing pressure
plateaus is produced during infiltration, measurements gleaned in one single experiment can be
used to plot the entire drainage curve for the system at hand (Fig. S1c). This apparatus was used
with a less perfected position sensor in earlier studies in our laboratory [1-6].
Materials systems and preform preparation – All infiltration experiments reported here were
conducted at 1,423 K in round-bottom alumina crucibles of inner diameter between 1.6 and 1.8
cm, using 99.95% pure copper (Swissmetal, Dornach, Switzerland) as the molten infiltrating
matrix. Six different reinforcements were studied. These are listed with their main characteristics
in Table 1 and comprise: (i) F320 or F1000 grit alumina powder (crushed powder of irregular
shape) purchased from Treibacher Schleifmittel (Germany), (ii) vapour-grown AA18
“Sumicorundum” alumina powder (of more regular and equiaxed shape, produced by Sumitomo
Chemical in Japan), (iii) spheroidized alumina powder (produced in the US by Gamma
Technology, Valencia, CA), (iv) cylindrical alumina short fibers (Saffil™, produced by ICI in
Runcorn, UK) packed, pressed, and bonded into preforms using small amounts of colloidal silica
and (v) fine-grained microporous graphite (Ringsdorff EK98 from SGL Group in Germany).
Prior to infiltration, cylindrical preforms (usually about 14 mm in diameter and 14 mm in
height) were prepared. The main challenge in preform preparation is to ensure that the ceramic,
and hence the pores, are uniformly distributed, free of channels and free of density gradients (see
Percolation and universal scaling in composite processing
4
below). Preform preparation methods were thus designed as follows, to produce homogeneous
preforms as a function of the reinforcement:
• Preforms of F320 and AA18 particles were produced identically; namely, the powder was
poured in a single step (without tapping or vibrating) into a mould of sintered alumina. This was
followed by partial sintering for 10 hours at 1,773 K, which bonded slightly the particles to one
another, this in turn preventing their relative motion during infiltration.
• Preforms of F1000 and of spherical particles were on the other hand cold isostatically
pressed (CIPped) at 150 MPa for 1 minute, as the resulting interlocking of particles gave
sufficient cohesion to the preform.
• Preforms of Saffil™ and graphite were already bonded, and were hence simply cut and
machined (delicately, to avoid smearing) into the desired cylinders.
The volume fraction of ceramic Vr in the fully infiltrated composites was determined by
densitometry using a Sartorius MC 210P microbalance of sensitivity ± 10 μg knowing the bulk
powder density. This was measured by helium-pycnometry (F320, AA18 and F1000 measured
on Quantachrome Instrument Multipycnometer, Boynton Beach, Florida, USA while for
Saffil™, spherical alumina and graphite preforms, a Micromeritics Accupyc 1330, Norcross,
Atlanta, USA was used); values are given in Table 1. Note that sample-to-sample density
variations within the same ceramic system are one cause for small shifts observed between
drainage curves gathered on several similar samples.
Infiltration under stepwise increasing pressure – At the start of an experiment a porous
preform is first placed at the bottom of an alumina crucible. A small ingot of copper is then
placed atop the preform. Induction heating is started once a stable primary vacuum (around 1.5
Pa) is reached. Upon melting of the metal, the preform becomes surrounded with molten copper;
this seals it from the outside atmosphere. For this reason, when pressurized argon is let into the
apparatus, it drives the metal into open pores of the evacuated preform. To prevent preforms
from floating once the metal is molten, a small piece of dense alumina is wedged horizontally
over each preform, blocking it at the bottom of the crucible by friction against crucible walls.
Drainage curves are measured under slow infiltration conditions, to ensure that the metal
position and the pressure are governed by capillary forces. Preforms were therefore infiltrated
under stepwise increasing pressure, holding each pressure level for a time long enough for the
displacement of the graphite plunger to reach a plateau, or in other words, for the saturation to
stabilize. A series of discrete points along the saturation curve of the sample in question are then
obtained; with a sufficient number of such points the curve can be drawn by interpolation, Fig.
S1c. It is to be noted that the number of pressure steps, as well as the pressurization rate between
each step (as long as the step itself is not too high), do not affect significantly the results. This is
shown by the reproducibility of the drainage curves obtained for each system (note that different
pressure steps were applied from experiment to experiment). After complete or partial
infiltration, the composite is cooled within the apparatus by switching off the heating while
maintaining the applied gas pressure. It can then be retrieved, infiltrated with epoxy, after which
its cross-section (an example of which is given in Fig. S1c) can be used for characterization of
the metal structure (see below).
Experiment with an empty crucible - Figure S2a shows the displacement of the graphite
plunger as the pressure is increased automatically at a rate of 0.003 MPa/s up to 12 MPa over a
crucible containing copper alone (i.e., with no preform). Figure S2b compares the temperature
Percolation and universal scaling in composite processing
5
and the displacement versus time for the same system. The displacement signal is not perfectly
flat (a decrease of 40 µm is recorded from the beginning to the end of the experiment); however,
this difference (which is roughly within experimental uncertainty) is small compared to the
whole displacement for a typical infiltration experiment (usually more than 5 mm, depending on
the preform porosity and the inner diameter of the crucible used). This shows that, up to a
pressure of 12 MPa, there is no significant plunger movement if there is no infiltration.
Starting at 10 MPa, due to the high gas pressure in the chamber, temperature control becomes
more difficult (as seen from the slightly wavy shape of the temperature curve). This in turn can
affect the movement of the plunger, as differential thermal expansion causes slight shifts in its
position if the temperature is not constant. There is, therefore, a somewhat higher uncertainty in
the measured plunger position at P > 10 MPa. For this reason, with graphite preforms (which
necessitated pressures above 10 MPa), a special procedure was used; this is described next.
Figure S2 - Experiment with an empty crucible
(a) Plot of the displacement of the graphite plunger and pressure (pressurization rate 0.003MPa/s)
as a function of time for an experiment in which the crucible contained copper without a
preform. (b) Temperature and displacement for the same experiment. The experiment was
stopped by turning the heating power off (explaining why pressure is maintained while thermal
contraction causes the sensor signal to denote a sudden change).
Microporous graphite
Temperature control becomes difficult above 10 MPa and this poses a problem for microporous
graphite because 10 MPa is not sufficient to fully infiltrate this preform material: note the visible
difference in metal fraction between P = 11 MPa and P = 17 MPa in Fig. S3b-c. Since the solid
volume fraction Vr is high in graphite preforms, the total displacement of the graphite plunger is
also relatively small (usually 1.5 mm compared to roughly 5 mm for F1000 system). This
implies that the slight variations of temperature that can appear above 10 MPa may affect the
precision at which the last portion of the drainage curves is measured.
A slightly different procedure was therefore used for these preforms. The density of nine
partially or fully infiltrated composites was measured by densitometry (using Archimedes’
Percolation and universal scaling in composite processing
6
principle, after sealing open pores along the surface of partially infiltrated composites by
smearing after polishing their surface using 600 grit SiC paper). Metallographic observation and
a lack of motion of the average plunger position indicate that the preform is fully infiltrated
above 17 MPa. With the pressure required for S = 1 thus calibrated, each partially infiltrated
sample was then used to place one individual point of the drainage curve knowing its density.
These points are marked in red on Fig. S3a and give a first estimate of the drainage curve. From
this, it was deduced that S = 0.9 when P = 10 MPa. Knowing this, in turn, one can calibrate
lower-P sensor-displacement data generated at pressures below 10 MPa, to trace a continuous
drainage curve up to S = 0.9. As seen, within experimental error this more precisely traced curve
matches data gathered by densitometry of individual lower-pressure partly infiltrated samples,
Fig. S3a.
Figure S3 - Microporous graphite
(a) Comparison of drainage curves of microporous graphite infiltrated by copper with the
continuous method of this study (black points) and data points produced by interrupted
infiltration of individual samples followed by densitometry (red points; here each point
corresponds to one single infiltration). S = 1 was assumed at P = 17 MPa on the basis of
metallography. (b)&(c), Backscattered electron images of composites infiltrated at 11 MPa and
17 MPa respectively (dark phase: graphite, white phase: copper; note that, due to smearing and
copper detachment during polishing, the fraction copper on these micrographs does not reflect
the true volume fraction in the composite)
Determination of λ and Pb in Eq. (1), Pc and C in Eq. (3) – Fitting the data to the universal
scaling law in Eq. (3) or to the Brooks and Corey relation (Eq. (1)) was done by plotting data in
coordinates that produce a straight line when one of these relations is obeyed. Where a straight
line obtained, its slope and intercept were deduced by linear regression. Note that, in doing so,
results vary somewhat with the saturation range used for the fit: the saturation range used was
therefore kept fixed across systems. The threshold pressure Pc and constant C in Eq. (3) were
Percolation and universal scaling in composite processing
7
thus determined for all experiments by linear regression of data plotted as S(1/0.41) vs P over the
range 0.02 ≤ S(1/0.41) ≤ 0.35 (corresponding to 0.2 ≤ S ≤ 0.65); results are reported in Table 1 of
the main article text. For the Brooks and Corey relation (Eq. 1) plotted in the usual coordinates
of ln(1-S) versus ln(P) (see Fig. 5 of the main article text), we have used the range -1 > ln(1-S) >
-2, corresponding to 0.6 ≤ S ≤ 0.9 and have then averaged results across similar samples.
Resulting values for all systems are given in Table 1 of the main article text.
Metallographic preparation methods - A challenge in metallographic sample preparation for
the visualization and analysis of partially infiltrated structures was to preserve the highly
branched, partly porous, regions of the (fractal) sample while cutting and polishing. This was
accomplished by first drilling a 1 mm hole through the crucible and metal shell surrounding
preforms, right after samples were infiltrated and solidified. Epoxy was then infiltrated through
the hole into remaining pores within the preform, first under vacuum, and then under 5 MPa
pressurized argon. The epoxy was left to set for at least 10 hours. The resulting twice-infiltrated
composite sample was then cut and polished, using additional epoxy pressure-infiltration steps if
needed, to produce clear and sharp images of the metal in the preform, featuring sufficient detail
for the measurement of fractal characteristics of the microstructure (an example is given in Fig.
S1c). Samples were observed in a scanning electron microscope (FEI XLF30), all under
backscattered electron mode to bring out the metal.
Fractal dimension measurement method - We have applied standard fractal dimension
measurement methods [7-10] on two-dimensional (2D) metallographic cuts through frozen
composites at low saturation. To this end we used thresholded images from 2D metallographic
cuts through partly infiltrated samples observed in back-scattered electron mode using a scanning
electron microscope.
Several different software packages (ImageJ, U. S. National Institutes of Health, Bethesda,
Maryland, USA), Fractalyse 2.4.1 (Research centre ThéMA, CNRS-Université de FrancheComté, FRA), Benoit version 1.31 (TruSoft Int’l Inc, St. Petersburg, USA) and different fractal
dimension measurement methods (the box-counting method, the radius mass method and the
correlation function method) were first tested on more than ten 2D fractal structures of known
fractal dimension (such as the Sierpinski carpet or the Menger sponge). Differences between
methods were slight; the method giving overall the most accurate fractal dimension with the
software packages that we used was the box counting method.
In box counting, the fractal dimension D2D comes directly as D in the relation N(h) ~ h-D,
where N(h) is the number of boxes of linear size h necessary to cover a data set of points
distributed in each image. Once suitable parameter values were identified (number of boxes,
their minimal and maximal sizes, the threshold value and the size in pixels of the picture
analyzed), images were taken from a metallographic cut, prepared as described above, through
samples of all six systems explored here for which the infiltration process was stopped at low
saturation (i.e., for P just past Pc), using the most porous (central) portions of the samples at high
magnification. Figure 4 (of the main article text) illustrates the structure and image analysis
process for partially infiltrated AA18 alumina preform. Figure S4 shows typical examples of
binarized pictures used, together with corresponding measured 2D fractal dimension measured
(as well as corresponding metal volume fraction) associated with selected areas (indicated with a
red square) for each system studied. Selection of analysis areas (red squares in Fig. S4) was done
by operator judgment, based on the following criteria: (i) that the square be sufficiently large
Percolation and universal scaling in composite processing
8
(roughly two orders of magnitude wider than the average pore and visually representative) and
(ii) situated far from preform edges (where the fraction metal is higher). The measured fractal
dimension was found to increase gradually with the local volume fraction Vm of metal; this is
illustrated in Fig. 4. This occurs because as Vm increases, the number of boxes of given linear
size falling on metal, N(h), also increases, and does so most markedly at low h, causing the slope
of a plot ln(h) versus ln(N(h)) (giving D2D, the fractal dimension in 2D) to increase slightly with
increasing Vm. For this reason, final measurements of the metal cluster fractal dimension are
reported in the form of ranges instead of single values.
Figure S4 - Fractal metal trees in the various preforms
Thresholded SEM images of partially infiltrated preforms by copper at 1,423 K under stepwise
increasing pressure of (a) F320 at 0.23 MPa, (b) AA18 at 0.46 MPa, (c) SAFFIL at 0.44 MPa, (d)
Spherical alumina at 3.4 MPa, (e) F1000 at 3.7 MPa and (f) graphite at 3.4 MPa. The red squares
(768 x 768 pixels) indicate the area used to calculate the fractal dimension using the box
counting method from imageJ (function: “Fractal box counter”, size of the boxes: 2, 3, 4, 6, 8,
12, 16, 24, 32, 48, 64, 96, 128, 192 pixels).
Measured ranges of 2D fractal dimensions (D2D = D - 1) are reported in Fig. S4, together with
the resulting estimation of the three-dimensional fractal dimension D of the metal cluster at the
lowest saturation present. The low value obtained with the graphite preforms (1.33) may be
related to the combined effects of lower volume fraction and image resolution.
Self-similarity of the metal cluster geometry was also investigated by measuring its fractal
dimension at different magnifications (x29, x50, x80 and x120) for a partially infiltrated F320
sample. Results are given in Fig. S5: D does not vary over the (limited) range that could be
explored, (D2D ≈ 1.70 ± 0.02 at all four magnifications). This is as should be for a fractal
structure.
Percolation and universal scaling in composite processing
9
A potential concern in these measurements is that they might be affected by solidification
shrinkage in the metal. To check for this, a similar partially infiltrated F320 sample was
produced using the eutectic lead-bismuth alloy instead of copper (nominal composition: 44.5
wt.%Pb + 55.5 wt.%Bi; exact composition of the prepared alloy: 44.52 wt.%Pb + 55.48 wt.%Bi).
This alloy is known to have essentially no shrinkage during solidification [11, 12]. Its fractal
dimension was similarly measured. Good correspondence was found between the measured
value (1.69) and that found with copper in F320 (1.68), Fig. S6, suggesting that solidification
shrinkage effects are either absent (because the metal was fed by the pressurized riser above) or
do not affect D significantly (for example because solidification shrinkage does not significantly
redistribute metal within the infiltrated preform).
Figure S5 - Self-similarity of the metal cluster geometry
Four different magnifications (X29, X50, X80 and X120) of thresholded SEM partially
infiltrated F320 images, and resulting 2D fractal dimension measured by the box-counting
method: the resulting fractal dimension (D2D ≈ 1.70) is nearly independent of magnification. The
red squares (768 x 768 pixels) indicate the area used to calculate the fractal dimension using the
Percolation and universal scaling in composite processing
10
box counting method from imageJ (function: “Fractal box counter”, size of the boxes: 2, 3, 4, 6,
8, 12, 16, 24, 32, 48, 64, 96, 128, 192 pixels).
Figure S6 - Fractal cluster of eutectic lead-bismuth alloy
(a) Backscattered electron images of F320 infiltrated by eutectic lead-bismuth alloy at 0.08 MPa
at 773 K, (b) corresponding thresholded SEM image and resulting 2D fractal dimension
measured by the box-counting method. The red square (768 x 768 pixels) indicates the area used
to measure the fractal dimension using the box counting method from imageJ (function: “Fractal
box counter”, size of the boxes: 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192 pixels).
Preform packing and channeling - A challenge we met in exploring the early stages of
infiltration was to achieve homogeneity in particle packing at both macroscopic and microscopic
levels. Initial infiltrations were performed with tapped and/or vibrated F320 powder; however,
the preforms obtained in this manner were never homogeneous in that they always contained
macroscopic cracks, or channels of lower packing density. These then became preferential flow
paths for the metal in initial phases of infiltration, affecting both the drainage curve and metal
cluster formation in early phases of infiltration; Fig. S7 gives an example of metal channels in
the early phases of infiltration with such preforms.
We mention the effect because it was important in experimentation but also because it, too,
highlights the importance of pore size and connectivity statistics in composite processing. Pore
channel formation was eliminated here by careful particle packing, and for some systems by
including a slight sintering step (see Methods). In industrial practice, however, heterogeneity in
pore statistics will often be encountered and will also influence early phases of infiltration, for
example by giving rise to percolation size effects (if channels are sufficiently numerous to
enclose preform regions 40 or less pores in width, then the infiltration threshold pressure of those
regions will be affected; some areas in Fig. S7 below seem to be near this limit) and/or by
creating interesting dual-scale percolation network effects.
Percolation and universal scaling in composite processing
11
Figure S7 - Scanning electron microscopy (back-scattered electron mode) image of a partly
infiltrated preform of spherical alumina powder that was prepared by sequential powder packing,
i.e., by pouring and vibrating layers successively, finishing with a light sintering step (10 hours
at 1,773 K): metal-filled channels (corresponding to preform regions of lower particle packing
fraction, probably formed at the interface between each packed particle layer) and their influence
in the overall process of pore invasion are clearly visible.
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