Grupp et al. Nature Communications · Grupp et al. Nature Communications Supplementary Information...

Grupp et al. Nature Communications

Supplementary Information 1

SUPPLEMENTARY INFORMATION

5°

Cu spheretilted glass plate

range ofacceptance

a b

Supplementary Figure S1 | Schematic of selection procedure for monocrystalline Cu

spheres. (a) side view. (b) top view.

a b

Supplementary Figure S2 | Electron and X-ray images of sphere and marker holes.

(a) Scanning electron microscopy close-up of a pair of holes drilled into in a copper sphere.

(b) X-ray tomogram of a copper sphere (d = 155 µm) and (inset) magnification of marker hole.



a b

Supplementary Figure S3 | Packing of spheres. (a) Exemplary tomographic reconstruction of

sample I after sintering. (b) Scanning electron microscopy image of sintered spheres after final

sintering stage showing sinter necks.



a

Supplementary Figure S4 | Analysis of grey value lines between particle centres. (a)

Principal illustration of particles and their contact partners. (b) Example for grey value course and

its derivative from a particle centre to beyond its surface in contact areas = blue line in (a). (c)

same as (b) for a particle without contact partner = red line in (a). Broken blue lines in (b) and (c)

mark position of particle radius.

.



Fig. S5.

Supplementary Figure S5 | Different stages of particle identification. (a) 2D slice of a 3D data

set, raw data. (b) Approximated particles identified with the image analysis program. (c) Slice

showing both the original and idealized spheres. (d) Slice showing idealized spheres after

binarisation.

a b

c d



Supplementary Figure S6 | Detection procedure of FIB-drilled marker holes. (a-c), Sinusoidal

projections (2D projections) of a single sphere. (d-f), Schematic illustration. (a,d), Detection of the

sphere surface. Black areas are contact areas. (b,e), Surface smoothed by a median filter. (c,f),

Determination of grey values in a spherical shell below the surface to separate the marker position.



a b

Supplementary Figure S7 | Alignment procedure. Schematic view of the alignment carried out

to exclude global angular displacements of the specimen from one sinter step to the other. Red full

circles: spheres before a sinter step, blue broken circles: spheres after sinter a step. Centre

positions and contact partners are given (a), before, and (b), after the alignment procedure, which

in this example involves a clockwise rotation by 7°. In the real calculations this correction is carried

out in 3D. The angles shown are exaggerated in this sketch to make the procedure clearer.



Supplementary Table S1 | Some details of the tomography experiments on samples I and II.

Sample no. I II beamline used ID15a BAMLine type of experiment in-situ ex-situ sinter program 20°C – 1085°C @ 10 K/min

continuously. (sample inserted at 20°C. First measurement started at 148°C)

pre-sintering 650°C / 10 min 750°C, 850°C, 950°C, 1050°C / 10 min 1050°C / 1 hour heating rate: 10 K/min

Atmosphere 96% He + 4% H2 90% N2 + 10% H2

no. of marked particles included 300 250

no. of marked particles used for analysis

180 170 for θ 20 for α

no. and dimension of holes drilled (length×width×depth) [µm3]

one, 5 x 5 x 8 two, 8 x 8 x 12

quartz capillary used? yes only for pre-sintering, after this, no

type of beam white monochromatic E = 50 keV

number of projections / angle covered

850 / 180° 2500 / 180°

time needed for one tomogram 85 s image acquisition + 125 s measurement overhead

6 h

field of view [mm2] 2 x 2 1.3 x 1.1

pixel size on scintillator [µm] 1.4 0.8

imaging resolution [µm] 2.1 1.8

no. of tomograms acquired 26 6



Supplementary Methods

Production of spherical mono-crystalline particles

Spherical monocrystalline copper particles with radii between 80 and 100 µm were produced by

the Sauerwald process35. This involves the following steps:

(i) heating up to 1130°C in a dry hydrogen atmosphere small quantities of pure CuO

powder embedded in alumina powder and thus reducing the CuO to Cu. Due to surface

tension the copper metal forms near-spherical droplets,

(ii) cooling down very slowly to obtain monocrystalline, approximately spherical particles,

(iii) separating the Cu spheres from the alumina powder.

Apart from employing alumina powder to separate the copper droplets, the same manufacturing

process has also been used by other authors to prepare powders.36,37

Spherical particles were selected by rolling the particles down a slightly tilted (≈5°) glass plate

several times. Spherical particles rolled down in a nearly straight line, whereas particles that

deviated from this shape did not roll at all or rolled to the side, see Supplementary Fig. S1.

Focused ion beam drilling of microscopic marker holes with Ga ions (30 kV, 10 nA) was carried out

in a ZEISS CrossBeam 1540 EsB. Ion drilling took about 1 min for each hole. The spheres were

embedded in electrically conducting resin (see Fig. 1a). After drilling, the resin was removed by a

solvent. Either one or two holes were drilled into each sphere. In the latter case, the angle between

the particle centre and the two holes was 20° on average, see Supplementary Fig. S2a.

The drilled holes with dimensions of either 5x5x8 µm³ or 8x8x12 µm³ cover only 0.02 % or 0.05 %,

respectively, of the surface area of a sphere with a radius of 100 µm. The analysis showed that

due to the small size of the FIB markers almost none of the holes overlapped with a contact area

during the sinter process. The effect of FIB markers on the sintering behaviour can therefore be

neglected.



Tomography and sintering

Sample I (in capillary): The high-energy beamline ID15a of the European Synchrotron Radiation

Facility (ESRF, France) was used in the white beam mode, which allowed us to acquire entire sets

of 850 radiograms (corresponding to one tomogram) in-situ in 85 seconds. Taking into account

additional 125 s needed for the measurement overhead (camera readout, sample repositioning

and acquisition of flat-field image) the temperature difference between subsequent tomograms was

35 K (210 s @ 10 K/min). A designated high-temperature furnace was provided by the ESRF38-41.

The furnace was calibrated by measuring the melting points of the pure metals copper and zinc.

The temperature error can be estimated to ≤2°C.

Sample II (free sintering): Tomography was carried out at the imaging beamline BAMLine of the

synchrotron radiation facility BESSY-II (Berlin) with monochromatic X-rays. Spatial resolution was

better than at ID15a, but data acquisition took much longer, which is why the measurements were

carried out ex-situ. The furnace used in this case was calibrated by measuring the melting points of

the pure elements Cu, Al and Zn.

The parameters describing the two measurements are summarised in Supplementary Table S1.

The software PyHST was used for tomographic reconstruction of the radiographic data sets by

standard filtered backprojection42,43. A typical tomogram is shown in Supplementary Fig. S3a, an

SEM image of the sintered particles after the final sintering step in Supplementary Fig. S3b.

Supplementary Fig. S2b exemplifies how a marker hole appears in the X-ray tomogram. Obviously,

this particular marker hole was close to the resolution limit since it appears rather jagged in this

tomogram.



Detection of the sphere coordinates, FIB-drilled marker holes and rotation/rolling angles

Image analysis took place in three steps:

1. Identification of individual particles in each tomogram: Determination of centre positions

and radii, identification of marker holes and their positions.

2. Analysis of neighbourhoods of each particle: Identification of contact partners, calculation of

coordination vectors.

3. Analysis of changes between different tomograms, i.e. between sinter steps: Calculation of

rolling and rotation angles.

Step 1: First, the custom-made software44,45 searched for uniform spherical areas containing

volume pixels (voxels) above a given minimum grey value to detect the individual copper spheres

and to obtain their approximate centre positions and radii. Then, from the grey values along lines

emanating from the estimated centre of each given sphere, the surface position in a given direction

was deduced from the peak of the derivative of the grey values, see Supplementary Fig. S4c. Over

10,000 directions were analysed for each particle, with an angular mesh distance of ≈2°. Surface

points that were contact areas, marker holes or artefacts in the tomograms, see Supplementary

Fig. S4b, were identified by analysing the derivative of the grey values at the particle surface and

were then excluded from surface fitting. By fitting the detected surface points to an ideal sphere,

the exact coordinates of the particle centres could be determined, see Supplementary Fig. S5.

To detect the positions of the marker holes it was necessary to calculate the surface of the

respective sphere even more precisely, namely with an angular mesh distance of just ≈1° in the

vicinity of the marker holes. This step was already based on the refined particle centre positions,

see Supplementary Fig. S6a,d. After this, the surface was recalculated applying a median filter and

therefore smoothened to detect possible obscured surface points in the vicinity of the contact

areas, artefacts or the marker hole itself, see Supplementary Fig. S6b,e. Finally, a grey value map

was determined in a shell located 2 to 3 volume pixels below the smoothened surface. This grey



value map was then used to distinguish the FIB-drilled marker holes from other contact areas or

artefacts, see Supplementary Fig.S6 c,f. This procedure allowed us to determine the exact

positions of the marker holes. For the illustration of results, a sinusoidal (equal to the cylindrical

Mercator projection known from geodesy) projection was chosen. Using such a projection it is

possible to display the entire surface of a spherical particle 2-dimensionally.

Step 2: Using the data obtained in step 1 it was possible to determine the coordination number

(number of interparticle contacts per sphere) by analysing the grey value lines between

neighbouring particles along lines such as the ones shown in Supplementary Fig. S4. For each

copper sphere, the vectors to the centres to the neighbouring contact partners were determined

and stored in a lookup table.

Step 3: The data obtained separately for each tomogram was now compared between tomograms

representing different sinter stages. Particle rolling was determined from the changes of the

particle centre positions of the contact partners, while intrinsic rotations of each particle were

derived from the movements of the marker holes. The developed software allowed for tracking of

all particles throughout sintering.

Tomographic data sets corresponding to different sinter steps usually showed a small global

translational and/or angular offset. While the translation offset could be easily eliminated, the

angular displacement would add artefacts to the measured rolling and rotation angles without an

appropriate correction. As visualised in Supplementary Fig. S7, groups of particles including a

given central particle and all its contacting neighbours were defined. The configurations before and

after an annealing step were matched as far as possible, by rotating the shell of neighbours such

that the sum of all rolling angles θi was minimal. This minimised angle divided by the number of

contact partners was the locally averaged rolling angle θ of the central particle with respect to all its

interparticle contacts. By then averaging over all particles, the global average θ given in Fig. 3 and

Fig. 4 (main paper) was obtained.

The rotation angles α (see Fig. 2b of main paper) of the marker positions from one sinter step to

another were determined after the corrections for rotational displacements mentioned above had



been carried out. First, the angular rotation of each line connecting a FIB hole and the

corresponding particle centre during a sinter step was determined. The values were then averaged

to a global rotation value α . Rotations around the axis defined by the lines between the particle

centre and the marker could not be detected in this way. Such rotations around a special axis were

only detectable using two marker holes as realized in sample II. This is why the analysis for sample

I possessing only one FIB marker is expected to yield values below the true angle α . A measured

rotation only conforms to the real value if the rotation is around the axis between the particle centre

and the FIB hole. If this axis and the rotation axis form an angle β>0°, a weight factor of cos(β)

applies, since only the projection of the rotation into the plane perpendicular to the axis is detected.

For statistical orientations, the measured average value based on one hole would be 2/π ≈ 0.64

times the true value. This correction has not been factored into the results presented in this paper.

Supplementary Methods 5 gives an analysis of the statistical errors of the sphere coordinates and

radii. It suggests that the statistical error is small enough to rule out that statistical artefacts

determine the rolling and rotation angles. Possible systematic errors can be assessed by

considering tomography measurements that were carried out before the heating stage and in late

stages of cooling where no thermal effects due to sintering are expected. The corresponding

analysis confirms that the measured angles are then actually constant. Moreover, previous

sintering experiments based on particles not containing FIB-drilled markers38,45 showed that the

determination of the rolling angles (which are much smaller than the rotation angles) is robust and

reproducible. One experiment, comparing two samples exposed to both continuous and step

heating programs as applied for samples I and II in this paper, respectively, showed that the

differences in temperature cycles have little effect on the measured angles46.

It is worth emphasising that by using photogrammetric image analysis it was possible to identify the

particles and the marker positions without binarisation, i.e. conversion to Boolean data. Image

analysis based on Boolean images proved less efficient since the sphere edges were blurred over

several voxels (volume pixels) and the sphere position in a given direction depended on the value

of the binarisation threshold chosen. The appropriate choice of valid thresholds was further



complicated by beam intensity variations during SCT measurements. By basing our analysis on

grey values, such arbitrary choices and data losses could be largely avoided and the particle

centre positions determined with a precision of about 0.1 voxel owing to the vast number of surface

pixels analysed for each sphere. Only after all processing had been finished, Boolean images were

created for better graphical representation, see Supplementary Fig. S5d.

Torque and shear stress

The total torque Msphere was calculated in a model system using particles with a = 100 µm radius

and a sinter neck diameter of x = 10 µm. An average torque Mcontact was allocated to each contact.

For more than one contact partner the total torque affects every single contact. The torque for each

contact is:

N

Ad

dE

M contact

⋅= θ

(S1)

The value for the derivative dE/dθ was taken from Wolf et al.46,47. A defines the contact area and N

the coordination number of a particle. A random direction was allocated to each Mcontact. For a

simulation of the torque Msphere as well as of the shear stress, more than 700000 combinations of

measured particle positions, positions of their particular contact partners and crystallographic

orientations were used. The geometrical information came from measured tomograms.

The tangential force F that a contact partner exerts on a sphere is given by geometrical

considerations. Based on the calculated tangential forces of all contact partners acting on a

sphere, a total torque Msphere applying at the centre of a sphere can be determined using:

FaM ×= (S2)

The average shear stress τ per contact is then:

NaA

M sphere

⋅⋅=τ (S3)



This analysis leads to Fig. 5 of the main paper.

Error analysis for sphere fitting procedure

The determination of a sphere centre (x0, y0, z0) starts with fitting a sphere of radius r to the n

measured surface points (x, y, z). The function representing the sphere

( ) ( ) ( ) 022

0

2

0

2

0 =−−+−+− rzzyyxx (S4)

is differentiated with respect to the centre coordinates and the radius, x0, y0, z0 and r, and a 4×n

matrix B is obtained which contains values for all surface points, indexed 1 … n:

( ) ( ) ( )

( ) ( ) ( )

⋅−−⋅−−⋅−−⋅−

⋅−−⋅−−⋅−−⋅−

=

nnnn rzzyyxx

rzzyyxx

B

2222

:

:

2222

000

1010101

(S5)

A 1×n column vector C is defined by:

( ) ( ) ( )( )

( ) ( ) ( )( )

−−+−+−−

−−+−+−−

=

22

0

2

0

2

0

2

1

2

01

2

01

2

01

:

:

nnnn rzzyyxx

rzzyyxx

C (S6)

By calculating the 4×4 matrix N

BBNT ⋅= (S7)

the sphere function can be fitted to the real measured values by determining the differences to x0,

y0, z0 and r in the (finite, but very small) 1x4 matrix dx:

CBNdxT ⋅⋅= −1

(S8)

The weight factor w of the error is calculated using an auxiliary matrix V:

CdxAV −⋅= (S9)



4

)( 0,0

−

⋅=

n

VVw

T

(S10)

The errors of the centre coordinates (x,y,z) and the radius r are then given by:

0,01−=∆ Nwx (S11)

1,11−=∆ Nwy (S12)

2,21−=∆ Nwz (S13)

3,31−=∆ Nwr (S14)

The determined error of the centre coordinates is about 0.075 voxel. Using the error propagation, a

rotation angle error of approximately 0.04° was determined44.



Supplementary References

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Gleichgewicht befindlichen Oberflächen. (Über synthetische Metallkörper. VII.). Z.

Elektrochem. Angew. P. 39, 750-753 (1933).

36. Herrmann, G., Gleiter, H. & Baro, G. Investigation of low-energy grain-boundaries in metals

by a sintering technique. Acta Metall. 24, 353-359 (1976).

37. Mykura, H. Grain-boundary energy and the rotation and translation of Cu spheres during

sintering onto a substrate. Acta Metall. 27, 243-249 (1979).

38. Kieback, B., Nöthe, M., Banhart, J. & Grupp, R. Investigation of sintering processes by

tomography. Mater. Sci. For. 638-642, 2511-2516 (2010).

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metal powder compacts during sintering. Nucl. Instr. Meth. Phys. Res. B 200, 287-294

(2003).

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sintering by X-ray synchrotron microtomography. Acta Mater. 52, 977-984 (2004).

41. Vagnon, A., Lame, O., Bouvard, D. Di Michiel, M., Bellet, B. & Kapelski, G. Deformation of

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