Skip to main content

Crystallographic and Morphological Evidence of Solid–Solid Interfacial Energy Anisotropy in the Sn-Zn Eutectic System

Abstract

In this paper, we explore the morphological evolution during two-phase growth in the Sn-Zn eutectic system, which has a particularly low volume fraction of the minority Zn phase. The reason for this choice is its exotic nature, as even with such a low volume fraction, the reported morphology is “broken-lamellar,” in contrast to the usually expected hexagonal arrangement of Zn rods in the Sn matrix. Thus, the main objective of the study is to investigate the reasons behind this phenomenon. We begin by presenting experimental results detailing the morphology and crystallography of the eutectic microstructures under various combinations of thermal gradients and velocities in directional solidification conditions. Based on the crystallography and further specially designed experiments we find that the solid–solid interface between the Sn and Zn crystal is anisotropic. On the basis of the results, we propose a hypothesis that the presence of solid–solid interfacial energy anisotropy leads to the formation of predominantly broken-lamellar structures, even when the minority fraction is significantly low.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

References

  1. W. Kurz, D.J. Fisher: Fundamentals of Solidification, vol. 1: trans tech publications Aedermannsdorf, Switzerland, 1986.

  2. J.A. Dantzig, M. Rappaz, Solidification, EPFL Press, 2009.

  3. G. A. Chadwick: Progress in materials science 1963, vol. 12, pp. 99–182.

    Google Scholar 

  4. J. Hunt, K. Jackson: Trans. Metall. Soc. AIME 1966, vol. 236 (6), pp. 843–852.

    CAS  Google Scholar 

  5. R. Elliott: Eutectic Solidification Processing: Crystalline and Glassy Alloys, Elsevier, 2013.

  6. M. Croker, R. Fidler, R. Smith: Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 1973, vol. 335 (1600), pp. 15–37.

    CAS  Google Scholar 

  7. K. Kassner, C. Misbah: Physical Review A 1991, vol. 44 (10), pp. 6513.

    CAS  Google Scholar 

  8. U. Hecht, L. Gránásy, T. Pusztai, B. Böttger, M. Apel, V. Witusiewicz, L. Ratke, J. De Wilde, L. Froyen, D. Camel, B. Drevet, G. Faivre, S. G. Fries, B. Legendre, S. Rex: Materials Science and Engineering: R: Reports 2004, vol. 46 (1-2), pp. 1–49.

    Google Scholar 

  9. A. Karma, M. Plapp: JOM 2004, vol. 56 (4), pp. 28–32.

    CAS  Google Scholar 

  10. H. Walker, S. Liu, J. Lee, R. Trivedi: Metallurgical and Materials Transactions A 2007, vol. 38 (7), pp. 1417–1425.

    CAS  Google Scholar 

  11. A. Parisi, M. Plapp: Acta Materialia 2008, vol. 56 (6), pp. 1348–1357.

    CAS  Google Scholar 

  12. M. Perrut, S. Akamatsu, S. Bottin-Rousseau, G. Faivre: Physical Review E 2009, vol. 79 (3), pp. 032602.

    Google Scholar 

  13. M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi: Acta Materialia 2009, vol. 57 (4), pp. 941–971.

    CAS  Google Scholar 

  14. S. Akamatsu, M. Plapp: Current Opinion in Solid State and Materials Science 2016, vol. 20 (1), pp. 46–54.

    CAS  Google Scholar 

  15. R. Contieri, C. Rios, M. Zanotello, R. Caram: Materials characterization 2008, vol. 59 (6), pp. 693–699.

    CAS  Google Scholar 

  16. A. Dennstedt, L. Ratke: Transactions of the Indian Institute of Metals 2012, vol. 65 (6), pp. 777–782.

    CAS  Google Scholar 

  17. A. Choudhury: Transactions of the Indian Institute of Metals 2015, vol. 68 (6), pp. 1137–1143.

    CAS  Google Scholar 

  18. J. Hötzer, M. Jainta, P. Steinmetz, B. Nestler, A. Dennstedt, A. Genau, M. Bauer, H. Köstler, U. Rüde: Acta Materialia 2015, vol. 93, pp. 194–204.

    Google Scholar 

  19. J. Hötzer, P. Steinmetz, M. Jainta, S. Schulz, M. Kellner, B. Nestler, A. Genau, A. Dennstedt, M. Bauer, H. Köstler, U. Rüde: Acta Materialia 2016, vol. 106, pp. 249–259.

    Google Scholar 

  20. P. Steinmetz, J. Hötzer, M. Kellner, A. Dennstedt, B. Nestler: Computational Materials Science 2016, vol. 117, pp. 205–214.

    CAS  Google Scholar 

  21. S. Liu, J. Lee, R. Trivedi: Acta Materialia 2011, vol. 59 (8), pp. 3102–3115.

    CAS  Google Scholar 

  22. D. Cooksey, D. Munson, M. Wilkinson, A. Hellawell: Philosophical Magazine 1964, vol. 10 (107), pp. 745–769.

    CAS  Google Scholar 

  23. A. Parisi, M. Plapp: EPL (Europhysics Letters) 2010, vol. 90 (2), pp. 26010.

    Google Scholar 

  24. D. Lewis, J. Warren, W. Boettinger, T. Pusztai, L. Gránásy: Jom 2004, vol. 56 (4), pp. 34–39.

    CAS  Google Scholar 

  25. M. Taylor, R. Fidler, R. Smith: Journal of Crystal Growth 1968, vol. 3, pp. 666–673.

    Google Scholar 

  26. T. Digges, R. Tauber: Metallurgical Transactions 1971, vol. 2 (6), pp. 1683–1689.

    CAS  Google Scholar 

  27. H. Kerr, M. Lewis: Journal of Crystal Growth 1972, vol. 15 (2), pp. 117–125.

    CAS  Google Scholar 

  28. T. Digges Jr, R. Tauber: Journal of Crystal Growth 1971, vol. 8 (1), pp. 132–134.

    CAS  Google Scholar 

  29. M. Savas, L. Clapham, R. Smith: Journal of Materials Science 1990, vol. 25 (2), pp. 909–913.

    CAS  Google Scholar 

  30. H. Kerr, W. Winegard: Canadian Metallurgical Quarterly 1967, vol. 6 (1), pp. 67–70.

    CAS  Google Scholar 

  31. M. Taylor, R. Fidler, R. Smith: Metallurgical Transactions 1971, vol. 2 (7), pp. 1793–1798.

    CAS  Google Scholar 

  32. M. Şahin, E. Çadirli: Journal of Materials Science: Materials in Electronics 2012, vol. 23 (2), pp. 484–492.

    Google Scholar 

  33. J. Bromley, F. Vnuk, R. Smith: Journal of materials science 1983, vol. 18 (10), pp. 3143–3153.

    CAS  Google Scholar 

  34. G. Piatti, G. Pellegrini: Journal of ScieMaterialsnce 1976, vol. 11 (5), pp. 913–924.

    CAS  Google Scholar 

  35. G. Beghi, G. Piatti, K. Street: Journal of Materials Science 1971, vol. 6 (2), pp. 118–125.

    CAS  Google Scholar 

  36. M. Notis, D. Shah, S. Young, C. Graham: IEEE Transactions on Magnetics 1979, vol. 15 (2), pp. 957–966.

    Google Scholar 

  37. M. Sahoo, G. Delamore, R. Smith: Journal of Materials Science 1980, vol. 15 (5), pp. 1097–1103.

    CAS  Google Scholar 

  38. G. Nishimura, R. Fidler, M. Taylor, R. Smith: Canadian Metallurgical Quarterly 1969, vol. 8 (4), pp. 319–322.

    CAS  Google Scholar 

  39. P. Taylor, H. Kerr, W. Winegard: Canadian Metallurgical Quarterly 1964, vol. 3 (3), pp. 235–237.

    CAS  Google Scholar 

  40. D. Jaffrey, G. Chadwick: Metallurgical Transactions 1970, vol. 1 (12), pp. 3389–3396.

    CAS  Google Scholar 

  41. F. Vnuk, M. Sahoo, D. Baragar, R. Smith: Journal of Materials Science 1980, vol. 15 (10), pp. 2573–2583.

    CAS  Google Scholar 

  42. H. Kaya, M. Gündüz, E. Çadirli, O. Uzun: Journal of Materials Science 2004, vol. 39 (21), pp. 6571–6576.

    CAS  Google Scholar 

  43. H. Kaya, E. Çadırlı, M. Gündüz: Journal of materials engineering and performance 2003, vol. 12 (4), pp. 456–469.

    CAS  Google Scholar 

  44. Y. Goto, M. Kurosaki, H. Esaka: Journal of the Japan Institute of Metals 2011, vol. 75 (7), pp. 392–397.

    CAS  Google Scholar 

  45. M. Şahin, F. Karakurt: Physica B: Condensed Matter 2018, vol. 545, pp. 48–54.

    Google Scholar 

  46. K. Sharma, R. Rai: Thermochimica acta 2012, vol. 535, pp. 66–70.

    CAS  Google Scholar 

  47. K. A. Jackson, J. D. Hunt: Transactions of The Metallurgical Society of AIME 1966, vol. 236, pp. 1129–1142.

    CAS  Google Scholar 

  48. B. Caroli, C. Caroli, G. Faivre, and J. Mergy: J. Cryst. Growth, vol. 118 (1992).

  49. A. Valance, C. Misbah, D. Temkin, and K. Kassner: Phys. Rev. E vol. 48, (1993).

  50. S. Bottin-Rousseau, M. Şerefoǧlu, S. Akamatsu, and G. Faivre: IOP Conference Series: Materials Science and Engineering, vol. 27 (1), pp. 012088, (2012).

  51. S. Akamatsu, S. Bottin-Rousseau, M. Şerefoğlu, G. Faivre: Acta Materialia 2012, vol. 60 (6-7), pp. 3199–3205.

    CAS  Google Scholar 

  52. S. Akamatsu, S. Bottin-Rousseau, M. Şerefoğlu, G. Faivre: Acta Materialia 2012, vol. 60 (6-7), pp. 3206–3214.

    CAS  Google Scholar 

  53. S. Ghosh, A. Choudhury, M. Plapp, S. Bottin-Rousseau, G. Faivre, S. Akamatsu: Physical Review E 2015, vol. 91 (2), pp. 022407.

    Google Scholar 

  54. S. Ghosh, M. Plapp: Acta Materialia 2017, vol. 140, pp. 140–148.

    CAS  Google Scholar 

  55. U. Hecht, J. Eiken, S. Akamatsu, S. Bottin-Rousseau: Acta Materialia 2019, vol. 170, pp. 268–277.

    CAS  Google Scholar 

  56. S. Bottin-Rousseau, O. Senninger, G. Faivre, S. Akamatsu: Acta Materialia 2018, vol. 150, pp. 16–24.

    CAS  Google Scholar 

  57. S. Mohagheghi, U. Hecht, S. Bottin-Rousseau, S. Akamatsu, G. Faivre, and M. Şerefoğlu: Effects of interphase boundary anisotropy on the three-phase growth dynamics in the \(\beta \) (in)–in2bi–\(\gamma \) (sn) ternary-eutectic system: In: IOP Conference Series: Materials Science and Engineering: Vol. 529: IOP Publishing, p. 012010, (2019).

  58. R. Elliot, A. Moore: Scripta Metallurgica 1969, vol. 3 (4), pp. 249–251.

    Google Scholar 

  59. P. Pandey, C. Tiwary, K. Chattopadhyay: Journal of Electronic Materials 2016, vol. 45 (10), pp. 5468–5477.

    CAS  Google Scholar 

  60. P. Pandey, C. Tiwary, K. Chattopadhyay: Journal of Electronic Materials 2019, vol. 48 (5), pp. 2660–2669.

    CAS  Google Scholar 

  61. M. Straumanis and N. Brakes: Z. Physik. Chem. B, vol. 38 (1937).

  62. H. Herman: Adv. Mater. Res. Vol. 4: Wiley-Interscience (1970).

  63. B. Cantor, G. Chadwick: Journal of Crystal Growth 1975, vol. 30 (1), pp. 109–112.

    CAS  Google Scholar 

  64. Z. Moser, J. Dutkiewicz, W. Gasior, J. Salawa: Journal of Phase Equilibria 1985, vol. 6 (4), pp. 330–334.

    CAS  Google Scholar 

  65. H. G. Weller, G. Tabor, H. Jasak, C. Fureby: Computers in physics 1998, vol. 12 (6), pp. 620–631.

    Google Scholar 

  66. A. J. Shahani, X. Xiao, P. W. Voorhees: Nature communications 2016, vol. 7, pp. 12953.

    CAS  Google Scholar 

  67. I. Ansara, A.T. Dinsdale, and M.H. Rand: COST 507, Thermochemical Database for Light Metal Alloys, vol. 2, Office for Official Publication of the European Communities, Luxemburg, 1998.

  68. A.J. Schwartz, M. Kumar, B.L. Adams, and D.P. Field: Electron Backscatter Diffraction in Materials Science, Springer, 2000.

  69. D.B. Williams and C.B. Carter: The Transmission Electron Microscope, Springer, 1996.

  70. D. P. Field: Ultramicroscopy 1997, vol. 67 (1-4), pp. 1–9.

    CAS  Google Scholar 

  71. L. G. Ware, D. H. Suzuki, Z. C. Cordero: Journal of Materials Science 2020, vol. 55, pp. 1–12.

    Google Scholar 

  72. A. Passerone, N. Eustathopoulos: Acta Metallurgica 1982, vol. 30 (7), pp. 1349–1356.

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank DST-SERB for funding through the project (DSTO1679). The authors would like to thank AFMM, IISc for characterization facilities. The authors would also like to thank Prof. Mathis Plapp, Prof. Silvère Akamatsu, and Prof. Sabine Bottin-Rousseau for insightful discussions during the course of the work. The authors thank Mr. Ravi Kumar for computing the thermal profile inside the directional solidification apparatus.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abhik Choudhury.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Manuscript submitted May 15, 2020.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 4712 KB)

Appendices

Appendix A. Calculation of Thermal Gradient at the Solid–Liquid Interface

The model of the directional solidification experimental setup is as shown in Figure A1a. We calculate the thermal profiles by solving heat transfer equations in the software OpenFOAM[65], which allows us to determine accurate thermal gradient at the solid–liquid interface. The material parameters reported by References [43, 64] for the alloy are used in the calculation. The computed thermal profile is shown in Figure A1b. The axial temperature profile (Figure A1c) in liquid shows a gradient close to 6 °C/mm.

Fig. A1
figure 18

Modeling using OpenFOAM software, where (a) is directional solidification setup 2D view, (b) is thermal profile, and (c) is axial temperature profile in liquid

Appendix B. Calculation of Jackson–Hunt Parameters for Lamellar and Rod Morphology

Using Jackson and Hunt theory,[47] undercooling vs. spacing relation for lamellar (L) and rod morphology (R) writes,

$$\begin{aligned} \Delta T = K_1V \lambda + K_2/\lambda . \end{aligned}$$
(B1)

For lamellar morphology, \(K_{1}^{L} = {\overline{m}} C_{o} P / f_{\alpha }f_{\beta } D\) and \(K_2^{L}=2{\overline{m}} (\Gamma _{\alpha } \sin \theta _{\alpha }/f_{\alpha }m_{\alpha } + \Gamma _{\beta } \sin \theta _{\beta }/f_{\beta }m_{\beta })\), where \(C_o=\left( C_\alpha - C_\beta \right) \) ( i.e., \(C_o\) is the composition difference between the solid phases at the eutectic temperature), \(f_\alpha \), \(f_\beta \) are the volume fractions of the solid phases, \(1/{\overline{m}} = \left( 1/m_{\alpha } + m_\beta \right) \), \(m_\alpha \), \(m_\beta \) are the liquidus slopes of the solid phases. \(\Gamma _\alpha \) and \(\Gamma _\beta \) are the Gibbs–Thomson coefficients of the solid and the liquid phases while the \(\theta _\alpha \) and \(\theta _\beta \) are angles the \(\alpha \)-liquid and the \(\beta \)-liquid tangents make with the horizontal. P is a function of the volume fraction of one of the phases that reads \(\sum _{n=1}^{\infty } \dfrac{1}{n\pi ^{3}} \sin ^{2}\left( n\pi f_\alpha \right) \). Similarly for rod morphology, \(K_1^{R}={\overline{m}} C_{o} M / f_{\beta } D\), where \(M=\sum _{n=1}^{\infty }\dfrac{1}{\left( \gamma _n\right) ^{3}}\dfrac{J_1^{2}\left( \sqrt{f_\alpha }\gamma _n\right) }{J_0^{2}\left( \gamma _n\right) }\), where \(\gamma _n\) is the \(n\)th zero of the first-order Bessel function \(J_1\), while \(J_o\) is the zeroth-order Bessel function. We calculated \(K_1^{L}\) and \(K_1^{R}\) values as 0.00836 and 0.0018, respectively, using material parameters reported[43,64] for the Sn-Zn alloy. Thereafter using the value of \(\lambda ^{2}V\) (= \(K_2^{L}/K_1^{L}\)) that we find using our experiments, we assessed the value of \(K_2^{L}\) as 0.226. Using the relation \(K_2^{R}/K_2^{L} = \sqrt{f_\alpha }\) from Jackson–Hunt theory, we calculated \(K_2^{R}\) value as 0.0678. This allows us to determine the \(\Delta T\) vs. spacing \(\lambda \) relationship for both morphologies.

Appendix C. Calculation of Interface Shape and Undercooling

Assumptions are

  • Directional solidification conditions with gradient G and velocity V;

  • Composition profile derived for a planar solidification interface; and

  • All other Fourier coefficients are derived from the planar solution except the boundary layer composition \(B_0\)

We integrate the basic undercooling equation which writes

$$\begin{aligned} \Delta T\left( x\right) = m^{\nu }\left( C_E - c\left( x\right) \right) + a/r\left( x\right) , \end{aligned}$$
(C1)

where \(\Delta T\) is the undercooling, m is the liquidus slope of the solid phase concerned, \(C_E\) is the eutectic composition, \(c\left( x\right) \) is the composition profile derived from the solution of the diffusion for a planar solidification front, a is the Gibbs–Thomson coefficient of the relevant solid–liquid interface, while \(r\left( x\right) \) is its curvature. The composition \(c\left( x\right) \) can be explicitly written as

$$\begin{aligned} c\left( x\right) = C_E + c_\infty + B_0 + \sum _{n=1}^{\infty } B_n \sin \left( \dfrac{n\pi x}{S_\alpha +S_\beta }\right) , \end{aligned}$$
(C2)

where \(B_0\) is the boundary layer composition, \(S_{\alpha ,\beta }\) are, respectively, the half widths of each of the lamellae. \(c_\infty \) is the departure of the far-field concentration from the equilibrium value, \(C_E\) is the eutectic composition, while each of the Fourier amplitudes \(B_n\) can be written as

$$\begin{aligned} B_n = \dfrac{2}{\left( n\pi \right) ^{2}}\left( S_\alpha + S_\beta \right) \dfrac{V}{D}C_0\sin \left( \dfrac{n\pi S_\alpha }{S_\alpha + S_\beta }\right) , \end{aligned}$$
(C3)

where \(C_0\) is \(|C_E - C_\alpha | + |C_\beta - C_E|\) and D is the diffusivity in the liquid.

The undercooling \(\Delta T\left( x\right) \) can in turn be expressed using the thermal gradient and the interface position as

$$\begin{aligned} \Delta T&= T_E - (T_0 + G\left( z - z_0\right) ) \nonumber \\&= (T_E - (T_0 + G z_0)) - Gz \nonumber \\&= \Delta T_0 - Gz \end{aligned}$$
(C4)

\(T_0\) and \(z_0\) being the temperature and z-ordinate of the triple point which is shared by both solid–liquid interfaces. The composite term \((T_E - (T_0 + G z_0))\) is expressed as \(\Delta T_0\) which by definition is the same for both the phases. With this Eqn. C1 becomes,

$$\begin{aligned} \Delta T_0 - Gz&= m^{\nu }\left( C_E - c\left( x\right) \right) + a/r\left( x\right) . \end{aligned}$$
(C5)

Integrating the preceding equation once over each of the solid–liquid interfaces, with the boundary conditions, that the center of each of the phases is flat and the tangent at the triple point is given by the balance of the surface tensions, we derive two relations, respectively, for each of the solid–liquid interfaces, which can be written as

$$\begin{aligned}&\Delta T_0 - G {\bar{z}}^{\alpha } -m^{\alpha }\left( -c_\infty - B_0 - 2\dfrac{\lambda V}{D}C_0P\left( \eta _\alpha \right) \right) \nonumber \\&\quad =a^{\alpha } \dfrac{\sin \left( \theta _{\alpha l}\right) }{\eta _\alpha \lambda } \nonumber \\&\Delta T_0 - G {\bar{z}}^{\beta } -m^{\beta }\left( -c_\infty - B_0 + 2\dfrac{\lambda V}{D}C_0P\left( \eta _\alpha \right) \right) \nonumber \\&\quad = a^{\beta }\dfrac{\sin \left( \theta _{\beta l}\right) }{\eta _\alpha \lambda }, \end{aligned}$$
(C6)

where \(\lambda = S_\alpha + S_\beta \), and \(\eta _\alpha =S_\alpha /(S_\alpha + S_\beta )\), while \({\bar{z}}^{\alpha }\) and \({\bar{z}}^{\beta }\) are average interface positions of the \(\alpha \) and the \(\beta \) interfaces, respectively. \(P\left( \eta _\alpha \right) \) is written as \(\sum _{n=1}^{\infty }\dfrac{\sin ^{2}\left( n\pi \eta _\alpha \right) }{\left( n\pi \right) ^{3}}\)

The previous expressions can now be simplified to derive \(\Delta T_0\) as well as \(B_0\) in terms of the average interface positions as

$$\begin{aligned} B_0&= G\dfrac{\left( {\bar{z}}^{\alpha } - {\bar{z}}^{\beta }\right) }{m^{\beta } - m^{\alpha }} \nonumber \\&\quad + \dfrac{1}{m^{\beta } - m^{\alpha }}\left( \dfrac{a^{\alpha }\sin \left( \theta _{\alpha l}\right) }{\eta _\alpha \lambda } - \dfrac{a^{\beta }\sin \left( \theta _{\beta l}\right) }{\left( 1-\eta _\alpha \right) \lambda }\right) \nonumber \\&\quad -\dfrac{2\lambda VC_0P}{D\left( m^{\beta } - m^{\alpha }\right) }\left( \dfrac{m^{\alpha }}{\eta _\alpha } - \dfrac{m^{\beta }}{1-\eta _\alpha }\right) , \end{aligned}$$
(C7)

while \(\Delta T_0\) is written as

$$\begin{aligned} \Delta T_0&= G{\bar{z}}^{\alpha } + m^{\alpha }\left( -c_\infty - B_0 - \dfrac{2\lambda VC_0P}{D\eta _\alpha }\right) \nonumber \\&\quad +\dfrac{a^{\alpha }\sin \left( \theta _{\alpha l}\right) }{\eta _\alpha \lambda } \end{aligned}$$
(C8)

Thereafter we set out to integrate Eqn. C5 using a finite difference approach where we discretize the curvature using central differences and the slopes as one-sided forward differences. The boundary conditions again involve imposing a zero slope at the mid-points of the lamellae which are at \(x=0\) and \(x= S_\alpha + S_\beta \). Also, the tangents at the triple point \(x= S_\alpha \) are given by the Young’s condition. In this discretized equation, the derived expressions for \(B_0\) and \(\Delta T_0\) which are expressed in Eqns. C7 and C8 are used. This gives us a set of N-2 non-linear equations (N being the points on a particular solid–liquid interface) for each of the solid–liquid interfaces, which are solved using the non-linear solution routine of “fsolve” which is in the optimization tool box of “octave-forge.” The result is the solution to the interface z-ordinates of each of the solid–liquid interface from which one can derive both the boundary layer composition \(B_0\) and \(\Delta T_0\) and thereby the temperature of the triple point \(T_0\) (since \(z_0\) is derived from the solution) along with the temperatures along each of the solid–liquid interface.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aramanda, S.K., Khanna, S., Salapaka, S.K. et al. Crystallographic and Morphological Evidence of Solid–Solid Interfacial Energy Anisotropy in the Sn-Zn Eutectic System. Metall Mater Trans A 51, 6387–6405 (2020). https://doi.org/10.1007/s11661-020-06007-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11661-020-06007-5