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Exact transport theory of dendritic competition and arrest

Beñat Gurrutxaga-Lerma, Matthew Hughes, and Nils Warnken

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Phys. Rev. E 114, 035505 – Published 8 September, 2026

DOI: https://doi.org/10.1103/mg2q-7w4v

Abstract

We solve exactly the transient two-dimensional advection diffusion problem generated by one and two semi-infinite internal Dirichlet lines, representing slender dendrites advancing into a supersaturated liquid while fixing the solute concentration on their surfaces. For a single dendrite, the mixed boundary value problem is reduced to a Wiener-Hopf equation, factorized explicitly, and inverted to the time domain by the Cagniard-de Hoop method, yielding the single-layer density in closed form in terms of the complex error function. For two parallel dendrites separated by a lateral distance h, the interaction enters through the transcendental kernel (1+e−κh)−1, whose lower Wiener-Hopf factor is represented by a logarithmic Cauchy integral, and the resulting finite distance feeding field is reduced to a regularized oscillatory integral. The exact two-dendrite solution reveals a structural obstruction to purely local growth laws: the leading inverse square-root tip singularity is universal, with identical amplitudes for both dendrites independently of spacing, even though the neighbor modifies the feeding field at every finite distance behind the tip. Competition therefore cannot be encoded in any local tip amplitude alone. This obstruction forces a nonlocal coupling variable, which we identify as a finite tip zone functional that retains the subleading near tip structure responsible for competitive depletion. The interaction further resolves into a discrete hierarchy of exponentially screened modes with dominant screening length h/π, which sets the natural scale of competitive suppression. These exact transport results provide the analytical backbone for a reduced theory of dendrite arrest and winner selection, and are validated against independent finite-difference computations.

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Wall-enhanced dendritic growth: An exact Wiener-Hopf-Hankel solution for oblique reflection

Beñat Gurrutxaga-Lerma, Matthew Hughes, and Nils Warnken
Phys. Rev. E 114, 035506 (2026)

Article Text

References (53)

  1. G. P. Ivantsov, Temperature field around a spheroidal, cylindrical and acicular crystal growing in a supercooled melt, Dokl. Akad. Nauk SSSR 58, 567 (1947).
  2. D. A. Kessler, J. Koplik, and H. Levine, Pattern selection in fingered growth phenomena, Adv. Phys. 37, 255 (1988).
  3. M. Ben Amar and Y. Pomeau, Theory of dendritic growth in a weakly undercooled melt, Europhys. Lett. 2, 307 (1986).
  4. A. Barbieri, D. C. Hong, and J. S. Langer, Velocity selection in the symmetric model of dendritic crystal growth, Phys. Rev. A 35, 1802 (1987).
  5. M. Ben Amar and E. Brener, Theory of pattern selection in three-dimensional nonaxisymmetric dendritic growth, Phys. Rev. Lett. 71, 589 (1993).
  6. E. A. Brener, Needle-crystal solution in three-dimensional dendritic growth, Phys. Rev. Lett. 71, 3653 (1993).
  7. G. Horvay and J. W. Cahn, Dendritic and spheroidal growth, Acta Metall. 9, 695 (1961).
  8. D. Walton and B. Chalmers, The origin of the preferred orientation in the columnar zone of ingots, Trans. AIME 215, 447 (1959).
  9. M. H. Burden and J. D. Hunt, Cellular and dendritic growth, J. Cryst. Growth 22, 109 (1974).
  10. J. D. Hunt, Solidification and Casting of Metals (The Metals Society, London, 1979), Vol. 192, pp. 3–9.
  11. W. Kurz and D. J. Fisher, Dendrite growth at the limit of stability: Tip radius and spacing, Acta Metall. 29, 11 (1981).
  12. R. Trivedi, Interdendritic spacing: Part II. A comparison of theory and experiment, Metall. Trans. A 15, 977 (1984).
  13. J. A. Warren and J. S. Langer, Stability of dendritic arrays, Phys. Rev. A 42, 3518 (1990).
  14. J. A. Warren and J. S. Langer, Prediction of dendritic spacings in a directional-solidification experiment, Phys. Rev. E 47, 2702 (1993).
  15. S. H. Han and R. Trivedi, Primary spacing selection in directionally solidified alloys, Acta Metall. Mater. 42, 25 (1994).
  16. S. McFadden and D. J. Browne, A front-tracking model to predict solidification macrostructures and columnar to equiaxed transitions in alloy castings, Appl. Math. Model. 33, 1397 (2009).
  17. D. Tourret, Y. Song, A. J. Clarke, and A. Karma, Grain growth competition during thin-sample directional solidification of dendritic microstructures: A phase-field study, Acta Mater. 122, 220 (2017).
  18. J. Li, J. Wang, and G. Yang, Phase-field simulation of microstructure development involving nucleation and crystallographic orientations in alloy solidification, J. Cryst. Growth 309, 65 (2007).
  19. C.-A. Gandin and M. Rappaz, A 3D cellular automaton algorithm for the prediction of dendritic grain growth, Acta Mater. 45, 2187 (1997).
  20. D. Tourret and A. Karma, Growth competition of columnar dendritic grains: A phase-field study, Acta Mater. 82, 64 (2015).
  21. D. Tourret and A. Karma, Three-dimensional dendritic needle network model for alloy solidification, Acta Mater. 120, 240 (2016).
  22. D. Tourret, A. Karma, A. J. Clarke, P. J. Gibbs, and S. D. Imhoff, Three-dimensional dendritic needle network model with application to Al-Cu directional solidification experiments, IOP Conf. Ser. 84, 012082 (2015).
  23. D. Tourret, M. M. Francois, and A. J. Clarke, Multiscale dendritic needle network model of alloy solidification with fluid flow, Comput. Mater. Sci. 162, 206 (2019).
  24. B. J. Spencer and H. E. Huppert, The relationship between dendrite tip characteristics and dendrite spacings in alloy directional solidification, J. Crystal Growth 200, 287 (1999).
  25. B. J. Spencer and H. E. Huppert, On the solidification of dendritic arrays An asymptotic theory for the directional solidification of slender needle crystals, Acta Mater. 45, 1535 (1997).
  26. B. Noble, Methods Based on the Wiener-Hopf Technique (Pergamon Press, London, UK, 1958).
  27. S. G. Springer and T. J. Pedley, The solution of heat-transfer problems by the Wiener-Hopf technique, Proc. R. Soc. London A 333, 347 (1973).
  28. S. G. Springer, The solution of heat-transfer problems by the Wiener-Hopf technique. II. Trailing edge of a hot film, Proc. R. Soc. London A 337, 395 (1974).
  29. W. Koch, Laminar boundary-layer heat transfer from a semi-infinite flat plate with arbitrary surface temperature or heat flux, Int. J. Heat Mass Transf. 18, 1409 (1975).
  30. M. L. Shendeleva, Reflection and refraction of a transient temperature field at a plane interface using Cagniard-de Hoop approach, Phys. Rev. E 64, 036612 (2001).
  31. W. J. Parnell, V.-H. Nguyen, R. Assier, S. Naili, and I. D. Abrahams, Transient thermal mixed boundary value problems in the half-space, SIAM J. Appl. Math. 76, 845 (2016).
  32. N. Gorbushin and W. J. Parnell, Transient thermal boundary value problems in the half-space with mixed boundary conditions, J. Eng. Math. 114, 141 (2019).
  33. A. T. de Hoop, A modification of Cagniard's method for solving seismic pulse problems, Appl. Sci. Res. B 8, 349 (1960).
  34. A. T. de Hoop and M. L. Oristaglio, Application of the modified Cagniard technique to transient electromagnetic diffusion problems, Geophys. J. Internat. 94, 387 (1988).
  35. M. L. Williams, On the stress distribution at the base of a stationary crack, J. Appl. Mech. 24, 109 (1957).
  36. G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, J. Appl. Mech. 24, 361 (1957).
  37. B. Gurrutxaga-Lerma, M. Hughes, and N. Warnken, following paper, Wall-enhanced dendritic growth: An exact Wiener-Hopf-Hankel solution for oblique reflection, Phys. Rev. E 114, 035506 (2026).
  38. G. B. Folland, Introduction to Partial Differential Equations (Princeton University Press, Princeton, NJ, 1995), Vol. 102.
  39. L. C. Evans, Partial Differential Equations (American Mathematical Society, Providence, RI, 2022), Vol. 19.
  40. A. I. Markusevich, Theory of Functions of a Complex Variable (American Mathematical Society, Providence, RI, 2005).
  41. L. Cagniard, Réflexion et réfraction des ondes séismiques progressives (Gauthiers-Villars, Paris, France, 1939).
  42. K. Watanabe, Cagniard-de Hoop Technique (Springer, Cham, 2015), pp. 153–204.
  43. F. Oberhettinger and L. Badii, Tables of Laplace Transforms (Springer Science & Business Media, New York, 2012).
  44. A. M. Haubold, H. J. Mathai, and R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math. 2011, 298628 (2011).
  45. D. Tourret and A. Karma, Multiscale dendritic needle network model of alloy solidification, Acta Mater. 61, 6474 (2013).
  46. K. R. Stromberg, An Introduction to Classical Real Analysis (American Mathematical Society, Providence, RI, 2015), Vol. 376.
  47. J. R. Rice, Path independent integral and approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35, 379 (1968).
  48. J. D. Eshelby, The elastic energy-momentum tensor, J. Elast. 5, 321 (1975).
  49. L. B. Freund, Dynamic Fracture Mechanics (Cambridge University Press, Cambridge, UK, 1998).
  50. K. B. Broberg, Cracks and Fracture (Academic Press, New York, NY, 1999).
  51. W. Kurz and D. J. Fisher, Fundamentals of Solidification, 4th ed. (CRC Press, Boca Raton, FL, 2005).
  52. A. Talbot, The accurate numerical inversion of Laplace transforms, IMA J. Appl. Math. 23, 97 (1979).
  53. J. A. C. Weideman, Optimizing Talbot's contours for the inversion of the Laplace transform, SIAM J. Numer. Anal. 44, 2342 (2006).

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Phys. Rev. E 114, 035505– Published 8 September, 2026
  • Received 14 April 2026
  • Revised 5 August 2026
  • Accepted 14 August 2026
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