# Giant segregation transition as origin of liquid metal embrittlement
in the Fe-Zn system

Reza Darvishi Kamachali1 Corresponding author<EMAIL_ADDRESS>(Reza
Darvishi Kamachali) Theophilus Wallis1 Yuki Ikeda1 Ujjal Saikia2 Ali Ahmadian2
Christian H. Liebscher2 Tilmann Hickel1,2 Robert Maaß1,3 1Federal Institute
for Materials Research and Testing (BAM), Unter den Eichen 87, 12205 Berlin,
Germany 2Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-Straße 1,
40237 Düsseldorf, Germany  3University of Illinois at Urbana-Champaign,
Urbana, 61801 Illinois, USA

###### Abstract

A giant Zn segregation transition is revealed using CALPHAD-integrated
density-based modelling of segregation into Fe grain boundaries (GBs). The
results show that above a threshold of only a few atomic percent Zn in the
alloy, a substantial amount of up to 60 at.% Zn can segregate to the GB. We
found that the amount of segregation abruptly increases with decreasing
temperature, while the Zn content in the alloy required for triggering the
segregation transition decreases. Direct evidence of the Zn segregation
transition is obtained using high-resolution scanning transmission electron
microscopy. Base on the model, we trace the origin of the segregation
transition back to the low cohesive energy of Zn and a miscibility gap in Fe-
Zn GB, arising from the magnetic ordering effect, which is confirmed by ab-
initio calculations. We also show that the massive Zn segregation resulting
from the segregation transition greatly assists with liquid wetting and
reduces the work of separation along the GB. The current predictions suggest
that control over Zn segregation, by both alloy design and optimizing the
galvanization and welding processes, may offer preventive strategies against
liquid metal embrittlement.

††preprint: arxiv

Liquid Metal Embrittlement (LME) is a major safety concern in structural
engineering. For Zn-coated steels, LME leads to severe limitations in
manufacturing and related designs in the automotive industry [1, 2]. Extensive
research has been devoted to understanding the underlying mechanisms of LME
(see for instance [3, 4, 5] and references therein). Embedded in several
analyses, wetting of grain boundaries (GBs) has been proposed as a leading
mechanism in promoting crack initiation and propagation in Zn welding regions
[6, 7]. The extent and significance of wetting are argued to depend on the
solubility of the embrittling metal in the base alloy [8]. Another mechanism
repeatedly discussed for LME is stress-assisted diffusion, promoting the
formation of a Zn-penetration-zone and thus cracking [9, 10]. Recently, it has
also been proposed that LME can be driven by the formation of brittle Zn-rich
$\Gamma$ precipitates that further create stress heterogeneities in the
uncracked GBs [11].

As a common point of departure in various mechanisms, the interaction between
Zn and Fe atoms at GBs is of fundamental importance: In particular, one can
ask if, prior to wetting or GB precipitation, the segregation of Zn could play
a critical role on a much smaller scale? Indeed, careful investigations
revealed that Zn-rich precipitates form along the GB far ahead of the crack
tip [11, 12]. Furthermore, atom probe tomography has revealed significant
elemental segregation along GBs [13]. These observations brace Zn segregation
as a precursor for LME, weakening the GB and accelerating interfacial wetting
and precipitation prior to crack initiation. Even for the mechanism of stress-
assisted diffusion, the enlargement of the so-called Zn-penetration-zone
should be studied in conjunction with the GBs’ segregation response. For these
reasons and because of its potential for preventive design against LME, it is
highly desirable to explore and understand Zn segregation and related GB phase
behavior.

In the bulk Fe-Zn phase diagram, the BCC solid solution shows a miscibility
gap over a significant range of temperatures and compositions, see Fig. S1 in
the Supplementary Material (SM). Although often neglected in the usual Fe-Zn
phase diagram (due to the assumption of an equilibrium formation of the Zn-
rich $\Gamma$ phase), this miscibility gap has important implications for the
Zn segregation behavior: In fact, the presence of such a miscibility gap can
lead to a segregation transition phenomenon as demonstrated in other alloy
systems [14, 15, 16]. Furthermore, due to the small cohesive energy of Zn, the
driving force for any Zn segregation is expected to be high, i.e., the energy
of an Fe GB can be dramatically reduced with Zn segregation. To capture these
two major features, density-based Gibbs free energy formalism for describing
GBs is considered [14]:

$\displaystyle G(T,\rho,X_{Zn})$
$\displaystyle=X_{Fe}G_{Fe}(T,\rho)+X_{Zn}G_{Zn}(T,\rho)$
$\displaystyle+{\rho^{2}}\Delta H^{B}\left(T,X_{Zn}\right)-T\Delta
S^{B}\left(T,X_{Zn}\right)$ (1)

with
$G_{i}(T,\rho)={\rho^{2}}E_{i}^{B}+{{\rho}}\left(G_{i}^{B}(T)-E_{i}^{B}\right)$.
Here, the dimensionless atomic density $\rho$ defines the parent bulk phase
when $\rho=1$ or a GB when $\rho<1$.

Figure 1:  Grain boundary (GB) phase diagram. (a) The GB miscibility gap
(circles) computed utilizing the CALPHAD-integrated density-based method.
Thermo-Calc TCFE11 database was used. The miscibility gap of the parent
$\alpha$ (BCC) bulk phase is shown for comparison in the background. The
superscripts B and GB refer to bulk and grain boundary compositions,
respectively. (b) GB segregation isotherms computed for various temperatures
using CALPHAD-integrated density-based method. The abrupt jump in each
isotherm indicates the segregation transition. (c) The complete density-based
GB phase diagram is obtained, combining the results in panels (a) and (b). At
each given temperature, the transition line (squares) marks the critical alloy
(bulk) composition at which the segregation transition occurs. Massive Zn
segregation transition is predicted for alloy compositions and temperatures
changing from the left to the right side of the transition line. The
segregation transition at $T=623$ K is marked, as an example, to link the
results in panels (b) and (c).

All other terms are bulk thermodynamic properties available from CALPHAD
databases: The Gibbs free energy $G_{i}^{B}$ and potential energy $E_{i}^{B}$
of the pure $i$ substance, the mixing enthalpy $\Delta H^{B}$ and the mixing
entropy $\Delta S^{B}$ —the superscript $B$ denotes bulk properties throughout
this paper. Compared to its original formulation [14], in this study, we omit
the gradient energy contributions in the free energy and, instead, focus on
the temperature and composition dependence of the Zn segregation phenomenon.

The density-based approach has been successfully applied for microstructure
and alloy design in a broad range of materials [17, 18, 19, 20, 21]. The key
strength of this method is its natural capacity to be integrated with CALPHAD
framework and atomistic simulations, providing in-depth multi-scale insights
into the phase behavior of GBs. We study a prototypical GB with $\rho=0.92$
motivated by atomistic simulations of BBC-Fe GBs. For the calculations, we use
the Thermo-Calc TCFE11 database. The computational details and input
parameters are given in Appendix A and B in the SM.

Figure 1(a) shows the computed GB miscibility gap (solid lines with circles).
Compared to the parent bulk phase (red curve in the background), the GB
miscibility gap slightly shrinks and shifts, however, it still covers a broad
range of temperatures and compositions. Applying the equal chemical potential
condition [14], one can study the coexistence of the GB and bulk. This gives
the segregation isotherms shown in Fig. 1(b). The isotherms reveal a
segregation transition, i.e., an abrupt increase (jump) in the GB Zn
segregation with increasing Zn content in the bulk solid solution. This
corresponds to the miscibility gap of the GB. We found that the amount of Zn
in the GB massively increases during the segregation transition, markedly
exceeding the previously observed large segregation transitions in iron alloys
[15]. For instance, at $T=623$ K, about 60 at.% Zn segregate to the GB if only
4 at.% of Zn are added to BCC-Fe.

The critical alloy compositions, corresponding to the segregation transitions,
can be obtained from segregation isotherms, i.e., the solid black line with
squares on the left corner in Fig. 1(c), referred to as the transition line.
With this critical information, the GB phase diagram is complete: Figure 1(c)
illustrates that for alloy compositions and temperatures to the left side of
the transition line, low segregation of Zn to the GB is expected. In contrast,
shifting to the right side of the transition line induces massive Zn
segregation, in which the GB composition dramatically increases, crossing its
own miscibility gap. This corresponds to the high-segregation branch in the
segregation isotherms, Fig. 1(b). The results in Figs. 1(b) and (c) show that
the magnitude of the segregation transition (the height of the jump) and the
degree of segregation significantly increase with decreasing the temperature.
Such low temperatures and temperature decreases can occur during the cooling
stages in galvanization and welding processes.

The origin of the miscibility gap in the Fe-Zn bulk system has been attributed
to a magnetic ordering effect, i.e., the presence of different magnetic states
[22, 23]. Using the density-based thermodynamic approach, here we show that
this miscibility gap of the bulk is inherited by the GB and plays a central
role in governing the segregation transition. To verify the existence of such
a magnetic miscibility gap for GBs, we have performed DFT calculations of Zn
segregation in a $\Sigma$5 GB, with and without the consideration of magnetic
disorder. The computational details are given in Appendix C in the SM. Figure
2 reveals that the strong segregation for Zn to GBs discussed in previous DFT
calculations [24, 25] is only present in the ferromagnetic case. Without
magnetic ordering (paramagnetic state), the segregation of Zn only reduces the
GB energy up to a coverage of 50 %, i.e., the energy curve is convex. In the
presence of a magnetic ordering (ferromagnetic state), the GB shows a concave
energy curve, indicating that the Zn-segregated GB becomes thermodynamically
unstable. This demonstrates that the miscibility gap due to the magnetic
ordering effect is inherited by the GB; thus, above a certain composition, the
GB prefers to decompose into Zn-rich and Zn-poor segments.

Figure 2:  GB energy from DFT calculations. The energy of a $\Sigma$5
[100](013) GB is computed for various levels of Zn coverage (C: coverage),
with and without the magnetic ordering effect. The black curves connect the GB
energies for the end-members with no and full coverage. In the paramagnetic
state, Zn favorably dissolves in the GB (convex) while in the ferromagnetic
state, the GB shows a miscibility gap (concave).  Figure 3:  GB segregation
transition and related wetting. (a) High-angle annular dark-field (HAADF)
imaging in the scanning transmission electron microscope (STEM) showing a
$\Sigma 5$ GB (scale bar 7 nm). Both grains are oriented along the [001] axis.
The corresponding elemental Zn map and line profile along the GB, obtained by
STEM energy dispersive spectroscopy (STEM-EDS), reveal interfacial spinodal
decomposition with localized Zn-rich patches, which is the signature of a
segregation transition. (b) Schematic sequence of segregation and segregation
transition, assisting the formation of the liquid phase in the GB. (c) Bright-
field scanning transmission micrograph (scale bar 1 $\mu$m) evidencing Zn-rich
wetting along a GB after a spot welding process. Further experimental details
are given in Appendix D in the SM.

In fact, recent studies have revealed that such chemical decomposition at the
GB is direct evidence of a segregation transition [15, 20]. To examine this
for the Fe-Zn system, we performed high-resolution transmission electron
microscopy of a $\Sigma 5$ GB in a Zn-coated BCC-Fe bicrystal. Figure 3(a)
shows a HAADF scanning transmission micrograph and the corresponding STEM-EDS
elemental Zn map and line profile for the GB. Experimental details are given
in Appendix D in SM. We found that with only 5 at.% Zn in the bulk, the Zn
strongly segregates to the GB and decomposes into rich and poor patches. The
existence of the Zn patches along the GB reveals an interfacial phase
decomposition, as previously demonstrated [15, 20], and confirms the existence
of the segregation transition in the Fe-Zn system. The concentration of Zn
patches reaches 40 at.% which is in the range of predicted values in Fig.
1(b). We note that an additional effect of gradient energy terms is neglected
in the current calculations that can be responsible for the spread of Zn in
the grain boundary region in Fig. 3(b). Co-segregation with other elements is
also evidenced to reduce the Zn segregation at the GB [26]. The cross-
validations from the DFT calculations and the experiment confirm the density-
based modeling predictions of the segregation transition in the Fe-Zn system.

Given the complex temperature profiles experienced during the processing of
Zn-coated steels [27], the alloy compositions and experienced temperatures
fall easily beyond the transition line in the GB phase diagram (Fig. 1(c)).
Once the Zn supply into the alloy is available via diffusion into the grain
interior, the GB segregation transition can be triggered. This also can happen
during the cooling stage, as the required amount of Zn in the grain to trigger
the segregation transition is low and further decreases with decreasing the
temperature. The experimental results shown in Fig. 3(a) reveal such
segregation transition occurring during the cooling after a heat treatment
process. Interestingly, our calculations show that the Zn segregation becomes
much stronger with decreasing the temperature.

The significant amount of Zn in the GB, originating from the segregation
transition, can dramatically affect the course of GB wetting, as schematically
shown in Fig. 3(b). Experimentally, we observed this in the welding process of
a steel specimen, giving the thick Zn solution layer shown in Fig. 3(c).
Although the segregation might not be the sole origin of wetting (e.g.
stresses may play a significant role), the segregation transition is expected
to dramatically boost it. To understand this influence, we studied the barrier
for the formation of the liquid phase from a Zn-enriched GB, $\Delta
G=G^{L}-G^{GB}$. Here the free energies of the liquid solution
$G^{L}(T,X_{Zn})$ and the segregated GB, $G^{GB}=G(\rho=0.92,T,X_{Zn})$ are
obtained from CALPHAD database TCFE11 and through Eq. (1), respectively.

Figure 4:  Segregation-induced liquid formation at the GB. (a) Free energies
of GB (thin curves) and liquid Fe-Zn solution (thick curves) for different
temperatures. The intersections of two free energy curves are also marked with
triangles. For GB compositions beyond the intersection point, the liquid phase
becomes stable, i.e., $\Delta G<0$. GB compositions for right before and after
the segregation transition are marked on the GB free energy curves with
unfilled and filled circles, respectively. (b) $\Delta G$ as a function of
temperature are shown for the two sets of GB compositions marked in panel (a).
The same markers are used as in panel (a). The arrow in panel (b) indicates
that the massive segregation transition dramatically decreases the barrier for
the formation of the liquid phase from segregated GBs.

Figure 4(a) shows the free energies of the GB and the liquid solution for
various temperatures. The intersections of the two free energies corresponding
to $\Delta G=0$ are marked (triangles). On the GB free energies, we also
marked the GB compositions at the segregation transition, and the low and high
values at the jump (circles). We computed $\Delta G$ values for these two sets
of GB compositions to specifically reveal the effect of segregation
transition. Figure 4(b) reveals that the barrier for the formation of the
liquid phase significantly increases with a decrease in the temperature. The
results show that both compositional and energetic barriers for the formation
of the liquid phase are greatly reduced by the Zn segregation transition.
Further applying the density-based model for intermetallic phases, we found
that the segregation transition enables the formation of the Zn-rich $\Gamma$
phase ahead of the crack tip. This is thoroughly discussed in [12].

Prior to the liquid formation, the massive Zn segregation can also have great
significance for promoting an easy crack initiation by weakening the GB. We
studied the work of separation of a $\Sigma$5 GB as a function of Zn coverage,
using DFT calculations. The results in Fig. 5 show that the segregation of Zn
dramatically reduces the GB’s work of separation. These confirm the results
from previous works [24, 25, 26]. We found that in the ferromagnetic state,
the GB exhibits a slightly reduced work of separation and weaker dependence on
the Zn content. Nevertheless, the overall reduction in the work of separation
is significant. In fact, for 50% Zn coverage, the work of separation for the
GB reduces almost by half. These results suggest that once the Zn segregation
transition occurs, GB weakens dramatically. Such a weakened GB ahead of a
crack tip is highly prone to opening, thus promoting embrittlement in the
material.

Figure 5:  GB’s work of separation. The work of separation of a $\Sigma$5
[100](013) GB for various levels of coverage with Zn, with and without
magnetic ordering effect.

To summarize, we reveal and explain a giant segregation transition occurring
during Zn segregation into Fe GBs using CALPHAD-integrated density-based
modelling. A central insight of the model is that it shows the GB can have a
large magnetic miscibility gap, which is also confirmed by DFT calculations.
The segregation transition is evidenced by our HAADF-STEM experimental
measurements, revealing an interfacial phase decomposition (formation of Zn-
rich patches) along GB. Our results shed light on the origin and mechanisms of
LME in Zn-coated steels and its strong dependence on temperature and alloy
composition. In particular, we found that the amount of segregation
dramatically increases with decreasing the temperature, while the required
amount of Zn in the alloy to trigger the segregation transition decreases.
These emphasize the significance of the cooling stage during processing. We
show that the segregation transition markedly assists with the formation of an
interfacial liquid phase and weakens the GB by reducing the work of
separation. The strong temperature and composition dependence of the Zn
segregation implies that LME in the Fe-Zn system should be preventable via
alloy and processing design strategies that suppress the revealed segregation
transition. Follow-up density-based phase-field simulations and guided
experiments are planned to explore the detailed influence of the alloying and
processing parameters.

## References

  * Sierlinger and Gruber [2017] R. Sierlinger and M. Gruber, A cracking good story about liquid metal embrittlement during spot welding of advanced high strength steels, White Paper; voestalpine Stahl GmbH: Linz, Austria (2017).
  * Bhattacharya [2018] D. Bhattacharya, Liquid metal embrittlement during resistance spot welding of zn-coated high-strength steels, Materials Science and Technology 34, 1809 (2018).
  * Joseph  _et al._ [1999] B. Joseph, M. Picat, and F. Barbier, Liquid metal embrittlement: A state-of-the-art appraisal, The European Physical Journal-Applied Physics 5, 19 (1999).
  * Nilsson and Hojna [2018] K. Nilsson and A. Hojna, Overview of mechanisms & models for liquid metal embrittlement and future directions, EU Science Hub-European Commission 450 (2018).
  * Razmpoosh  _et al._ [2021a] M. Razmpoosh, C. DiGiovanni, Y. Zhou, and E. Biro, Pathway to understand liquid metal embrittlement (lme) in fe-zn couple: From fundamentals toward application, Progress in Materials Science 121, 100798 (2021a).
  * Wolski  _et al._ [2001] K. Wolski, V. Laporte, N. Marie, and M. Biscondi, About the importance of nanometer-thick intergranular penetration in the analysis of liquid metal embrittlement, Interface Science 9, 183 (2001).
  * Bhogireddy [2016] V. S. P. K. Bhogireddy,  _Liquid metal induced grain boundary embrittlement_ , Ph.D. thesis, Ruhr-University Bochum (2016).
  * Shunk and Warke [1974] F. A. Shunk and W. R. Warke, Specificity as an aspect of liquid metal embrittlement, Scripta Metallurgica 8, 519 (1974).
  * Ling  _et al._ [2020] Z. Ling, M. Wang, L. Kong, and K. Chen, Towards an explanation of liquid metal embrittlement cracking in resistance spot welding of dissimilar steels, Materials & Design 195, 109055 (2020).
  * DiGiovanni  _et al._ [2021] C. DiGiovanni, A. G. Kalashami, E. Biro, and N. Zhou, Liquid metal embrittlement transport mechanism in the fe/zn system: Stress-assisted diffusion, Materialia 18, 101153 (2021).
  * Ikeda  _et al._ [2022] Y. Ikeda, R. Yuan, A. Chakraborty, H. Ghassemi-Armaki, J. Zuo, and R. Maaß, Early stages of liquid-metal embrittlement in an advanced high-strength steel, Materials Today Advances 13, 100196 (2022).
  * Ikeda  _et al._ [2023] Y. Ikeda, H. Ni, A. Chakraborty, H. Ghassemi-Armaki, J. Zuo, R. Darvishi Kamachali, and R. Maaß, Segregation-induced grain-boundary precipitation during early stages of liquid-metal embrittlement of an advanced high-strength steel, Acta Materialia , 119243 (2023).
  * Razmpoosh  _et al._ [2021b] M. Razmpoosh, B. Langelier, E. Marzbanrad, H. Zurob, N. Zhou, and E. Biro, Atomic-scale investigation of liquid-metal-embrittlement crack-path: Revealing mechanism and role of grain boundary chemistry, Acta Materialia 204, 116519 (2021b).
  * Darvishi Kamachali [2020] R. Darvishi Kamachali, A model for grain boundary thermodynamics, RSC Advances 10, 26728 (2020).
  * Darvishi Kamachali  _et al._ [2020] R. Darvishi Kamachali, A. Kwiatkowski da Silva, E. McEniry, D. Ponge, B. Gault, J. Neugebauer, and D. Raabe, Segregation-assisted spinodal and transient spinodal phase separation at grain boundaries, npj Computational Materials 6, 191 (2020).
  * Wang and Darvishi Kamachali [2021a] L. Wang and R. Darvishi Kamachali, Density-based grain boundary phase diagrams: Application to Fe-Mn-Cr, Fe-Mn-Ni, Fe-Mn-Co, Fe-Cr-Ni and Fe-Cr-Co alloy systems, Acta Materialia , 116668 (2021a).
  * Li  _et al._ [2020] L. Li, R. Darvishi Kamachali, Z. Li, and Z. Zhang, Grain boundary energy effect on grain boundary segregation in an equiatomic high-entropy alloy, Physical Review Materials 4, 053603 (2020).
  * Wang and Darvishi Kamachali [2021b] L. Wang and R. Darvishi Kamachali, Incorporating elasticity into CALPHAD-informed density-based grain boundary phase diagrams reveals segregation transition in Al-Cu and Al-Cu-Mg alloys, Computational Materials Science 199, 110717 (2021b).
  * Zhou  _et al._ [2021] X. Zhou, R. Darvishi Kamachali, B. L. Boyce, B. G. Clark, D. Raabe, and G. B. Thompson, Spinodal decomposition in nanocrystalline alloys, Acta Materialia 215, 117054 (2021).
  * Wallis and Darvishi Kamachali [2023] T. Wallis and R. Darvishi Kamachali, Grain boundary structural variations amplify segregation transition and stabilize co-existing spinodal interfacial phases, Acta Materialia 242, 118446 (2023).
  * Wang and Darvishi Kamachali [2023] L. Wang and R. Darvishi Kamachali, Calphad integrated grain boundary co-segregation design: Towards safe high-entropy alloys, Journal of Alloys and Compounds 933, 167717 (2023).
  * Reumont  _et al._ [2000] G. Reumont, P. Perrot, J. Fiorani, and J. Hertz, Thermodynamic assessment of the fe-zn system, Journal of Phase Equilibria 21, 371 (2000).
  * Su  _et al._ [2001] X. Su, N.-Y. Tang, and J. M. Toguri, Thermodynamic evaluation of the fe–zn system, Journal of Alloys and Compounds 325, 129 (2001).
  * Bauer  _et al._ [2015] K.-D. Bauer, M. Todorova, K. Hingerl, and J. Neugebauer, A first principles investigation of zinc induced embrittlement at grain boundaries in bcc iron, Acta Materialia 90, 69 (2015).
  * Scheiber  _et al._ [2020] D. Scheiber, K. Prabitz, L. Romaner, and W. Ecker, The influence of alloying on zn liquid metal embrittlement in steels, Acta Materialia 195, 750 (2020).
  * Ahmadian  _et al._ [2023] A. Ahmadian, D. Scheiber, X. Zhou, B. Gault, L. Romaner, R. Darvishi Kamachali, W. Ecker, G. Dehm, and C. H. Liebscher, Interstitial segregation has the potential to mitigate liquid metal embrittlement in iron, Advanced Materials , 2211796 (2023).
  * Hou  _et al._ [2007] Z. Hou, I.-S. Kim, Y. Wang, C. Li, and C. Chen, Finite element analysis for the mechanical features of resistance spot welding process, Journal of Materials Processing Technology 185, 160 (2007).

## Acknowledgments

RDK acknowledges the financial support from the German research foundation
(DFG) within the project _DA 1655/3-1_ and the Heisenberg program’s project
_DA 1655/2-1_. US and TH acknowledge the DFG support within the transfer
project T07 of the _SFB761_ “Stahl - ab initio”. CHL acknowledges the DFG
support within the project _LI 2133/7-1._ YI gratefully acknowledges the
Takenaka Overseas Scholarship Foundation for their kind support of his
graduate work. This research was carried out in part at the electron
microscopy center at BAM.

## Author contribution

RDK conceived the idea and collaborations, conducted the thermodynamic
calculations, supervised grain boundary density calculations and wrote the
manuscript. TW computed the grain boundary density from atomistic simulations.
US conducted the ab initio calculations. AA conducted the experiments on the
bicrystal specimen. YI conducted the experiments on the welding specimen. TH
supervised the ab initio calculations and revised the manuscript. CHL
supervised the experiments on the bicrystal specimen and revised the
manuscript. RM supervised the experiments on the welding specimen and revised
the manuscript. All authors reviewed and edited the manuscript. RDK, CHL, TH
and RM secured funding.

## Supplementary material

Methods and additional discussions are presented in the Supplementary Material
(SM) attached.

## Data availability

Further data and codes of this study are available from the corresponding
author.

## Competing financial interests

The authors declare no competing financial interests.

in 1,…,9 See pages \x of SM_.pdf