Supporting Information
Nanocavitation around a Crack Tip in a Soft Nanocomposite: a Scanning Microbeam
Small Angle X-ray Scattering Study
Huan Zhang†a*, Arthur K. Scholzb,c, Jordan de Crevoisiera, Daniel Berghezane,
Theyencheri Narayananf, Edward J. Kramerb,c,d*, and Costantino Cretona*
a)
Laboratoire de Sciences et Ingénierie de la Matière Molle, ESPCI Paristech-CNRS-UPMC, 10
rue Vauquelin, 75005, Paris, France
b)
Materials Research Laboratory, University of California Santa Barbara, 93106 CA, U.S.A.
c)
Department of Materials, University of California Santa Barbara, 93106 CA, U.S.A.
d)
Department of Chemical Engineering, University of California Santa Barbara, 83106 CA,
U.S.A.
e)
Michelin, CERL Ladoux, F-63040 Clermont Ferrand, France.
f)
ESRF, 6 rue Jules Horowitz, BP 220, 38043 Grenoble Cedex 9, France.
Table of Contents
Data Reduction............................................................................................................... 3
2D Mapping of Fresh Cracks ......................................................................................... 8
2D Mapping of Fatigued Cracks .................................................................................. 10
Miscellaneous .............................................................................................................. 13
References .................................................................................................................... 14
Data Reduction
1. Data Reduction of the Scattering Intensity. The incident beam intensity IC1 and
the transmitted beam intensity IC2 were measured from two different apparatus so a
correlation factor is defined as: R = (IC1air/IC2air). The in-house data acquisition
system at ID-02 automatically computed most of the data reduction according to
Equation 1.
𝐼img (𝑞) =
𝑓
𝑇m 𝐼𝐶1
(𝐼 − 𝐼dark )]
[(𝐼r − 𝐼dark ) −
𝐼𝐶1𝐴∆Ω𝑒𝑇𝑚
𝑇air 𝐼𝐶1air air
(S1)
Idark: dark current of the detector.
Ir: the raw scattering intensity of the sample
Iair: the scattering intensity of the air
Tm: the transmission of the sample (= IC2/IC1)
Tair: the transmission of the air (= 1/R)
A: exposure area
ΔΩ: solid angle correction
e: detector efficiency
f: calibration factor to the standard sample (water)
Here, Iimg (q) is in absolute unit.
2.1 Effect of Tilting. At the edges of the crack the streak is slightly rotated relative to
the tensile axis and the cylindrical symmetry may not be appropriate. We therefore
evaluated the effect of tilting of the cavitation streak on the calculation of Q. We used
the set of data obtained at ALS under uniaxial loading[1]. We picked out the scattering
pattern at  = 360%, whose principal axes are well aligned with the coordinates. This
scattering pattern was then purposely rotated with respect to the tensile direction to
mimic the case that we have observed around the crack tip. We then calculate Q by
Equation S2 and normalized it to the unrotated value Qu.
Figure S1. The effect of tilting of the streak on the calculation of scattering invariant.
Q/Qu as a function of tilting angle θ for an anisotropic streak pattern ( = 360%, solid
line) and an isotropic pattern ( = 0%, dashed line). Qu is the scattering invariant of
untilted pattern.
We first tested an isotropic scattering pattern ( = 0%, dashed line) to verify our
algorithm. As we expected, its Q/Qu is around 1.0 and irrespective to the tilting angle
θ between 0 to 90o (Figure S1). Q/Qu from an anisotropic pattern ( = 360%, solid line)
varied from 1.0 to 0.995 as θ increased from 0 to 10o, respectively. As θ exceeded 10o,
Q/Qu dropped slowly (only 10% decrease at θ = 45o) and eventually reached 0.75 at θ
= 90o. The extent of the decrease at fixed θ dependents on the anistropicity of the
scattering pattern (therefore on the strain ). In our current work, although some of the
streaks were tilted around the crack tip, the tilting angle was small that the deviation
from the cylindrical symmetry can be neglected and Equation S2 is still valid in this
case.
2.2 Porod Tail and Thickness Correction. The calculation of the scattering invariant
by Equation S2 cannot be carried out to infinite q because of the limitation of the
detector size. Thus a conventional Porod tail was added onto the data, see reference 2
for details. Qimg was then normalized to the value of Q at the reference position
(denoted as Qimg_ref) to get the normalized scattering invariant Q/Q0 used in the main
text.
𝑄img 𝑡ref
𝑄
=
𝑄0 𝑄img_ref 𝑡
(S3)
where t, tref are the thickness near the crack and at reference position, respectively.
The right term in Equation S3 is to correct for the thickness change due to straining of
the sample.
2.2.1 The Superposition Approach. In previous work carried out in uniaxial tension
[1-3]
, the value of tref/t was obtained from the value of R and the ratio of IC2 to IC1
(Equation 4)
𝑅𝐼𝐶2ref
𝑡ref ln ( 𝐼𝐶1ref )
=
𝑅𝐼𝐶2
𝑡
ln (
)
𝐼𝐶1
(S4)
However in this case, because the beam size was so small, a small vibration in the
surroundings could perturb the collimation system and the readings of IC2 were not
stable in the current setup and could not be reliably used to measure thickness. We
hence adopted an alternative method to calculate the value of tref/t other than Equation
S4.
In all our work, only the normalized scattering invariant Q/Q0 is of interest and the
effect of the thickness correction is just to vertically shift Iimg(qx, qyz) relative to the
reference value Iimg_ref(qx, qyz) (Equations S2 and S3). For small and intermediate
strains where no additional scatterers appear, Iimg(qx, qyz) should superpose perfectly
with Iimg_ref(qx, qyz) (The butterfly scattering pattern may disrupt the validity of this
superposition but that occurs at very small q < 0.04 nm-1 and doesn’t affect the
scattering invariant2). When the strain is above the critical cavitation strain there is a
range of strain (the extent of which is discussed below) where this superposition
remains valid in the high q region (see Figure 5 in reference 1) along both meridian
and equator. Q/Q0 can then be calculated without a precise knowledge of IC2 by
superposing Iimg(qx, qyz) relative to the reference value Iimg_ref(qx, qyz).
Figure S2. Illustration of the calculation of Q/Q0 by the superposition in the high q
region
We chose the scattering intensity along the tensile direction x, Iimg(qx) for the
superposition because it is relatively less affected by the strong scattering streak from
cavities (Figure S2). The coefficient C to shift Iimg(qx) relative to Iimg_ref(qx) is
determined by the following equation.
𝐶=
∑ 𝐼img ∗ 𝐼img_ref
2
∑ 𝐼img
(S5)
∆= ∑(𝐼img − 𝐶𝐼img )2
(S6)
This expression is based on a least square regression and only the data with q > qcutoff
were used for calculation. The value of qcutoff was chosen at 0.35 nm-1 in this work.
When the beam was partially or completely off the sample in the crack, Δ was
unreasonably large and therefore these points could be removed. The normalized
scattering invariant Q/Q0 thus is expressed by:
𝐶𝑄img
𝑄
=
𝑄0 𝑄img_ref
(S7)
2.2.2 Experimental Validation
Figure S3. Q/Q0 calculated by two methods for the same data set obtained under
uniaxial loading. Control (dashed line) denotes the method we previously used in
references 1-3. The symbols are results from current method and the number indicate
the value of qcutoff.
To evaluate the systematic error compared to the more accurate method based on
measuring IC2, we used both methods with the same set of data previously obtained
at the Advanced Light Source (ALS) under uniaxial loading for the same material.[1]
As shown in Figure S3, when the strain  was smaller than 300%, Q/Q0 obtained by
Equation S5 to S7 coincided well with that obtained from our previous method
(Equation S3 and S4). As  exceeded 300%, the results deviated from the Control
curve. Because more and more smaller cavities appeared under high stress, the
scattering intensity increased in the whole range of q. However, the superposition
method (Equation S5 to S7) has neglected these smaller cavities in the calculation of
Q/Q0 therefore the current method may have underestimated the void volume fraction
at high strain. Increasing the value of qcutoff did not improve the result significantly.
3. Plotting Q/Q0 in the x-y Plane. For Procedure A, the obtained Q/Q0 values were
directly plotted without further treatment. For Procedure B, the scattered data points
were interpolated onto a mesh grid using Matlab program to obtain matrixed data of
Q/Q0.
4. Calculation of the Void Volume Fraction. void was evaluated based on a
previously developed three-phase model,[1-3]
2
2
𝑄
𝜙SBR 𝜌SBR
+ 𝜙CB 𝜌CB
=1+[
− 1]𝜙void
𝑄0
𝜙SBR 𝜙CB (𝜌SBR − 𝜌CB )2
(S8)
where ρSBR (8.756 × 1010 cm-2) and ρCB (15.26 × 1010 cm-2) are the X-ray scattering
length density (XSLD) of the SBR matrix and of the CB filler particles, respectively.
According to above discussion and Equation S8, the maximum value of void very
close to the crack tip was underestimated up to 20% (Figure S3, relative error)
depending on the local strain.
Notice: Analyzing the voids size and their aspect ratio requires precise data at high q
(> 0.5 nm-1) which was not available with the current setup. Therefore they were not
discussed in this study.
2D Mapping of Fresh Cracks
Figure S4. The scattering images around a fresh crack at εm = 90 %. To fit the size of
the page, the x-axis and the y-axis were rotated.
A strong cavitation streak could be observed around the crack tip and some of the
streaks were tilted, indicating a slight deviation of the orientation along the tensile
axis. When the beam was partially off the same, the cavitation streak was still
observable but there was a drastic drop in the scattering intensity.
Figure S5. Void volume fraction void of a fresh crack at εm = 90 % in the x-y plane.
2D Mapping of Fatigued Cracks
Figure S6. Q/Q0 of a fatigued crack G3000 at εm = 30 % in the x-y plane. The open
symbols are the scattered data from three x scans and three y scans. The colorful
surface plot is the matrixed Q/Q0 by interpolation.
Figure S7. Q/Q0 of a fatigued crack G3000 at εm = 60 % in the x-y plane. The open
symbols are the scattered data from three x scans and three y scans. The colorful
surface plot is the matrixed Q/Q0 by interpolation.
Figure S8. Q/Q0 of a fatigued crack G3000 at εm = 90 % in the x-y plane. The open
symbols are the scattered data from three x scans and three y scans. The colorful
surface plot is the matrixed Q/Q0 by interpolation.
The crack tip moved in respect to its previous position.
Figure S9. Q/Q0 of a fatigued crack G3000 at εm = 60 % in the x-y plane for different
cycles: 1st cycle (cross symbols) and 2nd cycle (open circles).
Figure S10. Q/Q0 of a fatigued crack G700 at εm = 60 % in the x-y plane.
Figure S11. Q/Q0 of a fatigued crack G700 at εm = 90 % in the x-y plane.
The crack tip moved in respect to its previous position.
Figure S12. Typical scattering patterns of G700 at εm = 90 % just in front of the crack
tip.
Miscellaneous
Figure S13. Geometry of the mini pure shear sample.
References
1. Zhang, H.; Scholz, A. K.; de Crevoisier, J.; Vion-Loisel, F.; Besnard, G.;
Hexemer, A.; Brown, H. R.; Kramer, E. J.; Creton, C. Macromolecules 2012, 45,
1529-1543.
2. Zhang, H.; Scholz, A. K.; Vion-Loisel, F.; Merckel, Y.; Brieu, M.; Brown, H.;
Roux, S.; Kramer, E. J.; Creton, C. Macromolecules 2013, 46, 900-913.
3. Zhang, H.; Scholz, A. K.; Merckel, Y.; Brieu, M.; Berghezan, D.; Kramer, E. J.;
Creton, C. J. Polym. Sci. B-Polym. Phys. 2013, 51, 1125-1138.
4. PJ, F., Principles of polymer chemistry. Cornell University Press: 1953.
5
Mzabi, S. (2010). PhD thesis Paris, Université Pierre et Marie Curie: 304p.