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Analytical study of self-oscillatory pattern-forming crystal growth in two-parabola model

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Abstract

The directional solidification of a binary alloy was analytically studied within a phase field two-parabola model. The method of the Green’s function was used to find solutions to the equations of motion for the fields of the order parameter and impurity concentration because equations are piecewise linear in this model. Looking for periodic solutions with a small contribution of accelerated sections to the full displacement of the planar crystal-melt interface we have self-consistently derived the ordinary differential equation of motion for the interface position. This equation takes the form of the one of a nonlinear oscillator with the friction force and the mass both nonlinearly dependent on the velocity. Under typical experimental conditions this “friction” is negative therefore, there is a stable limit cycle. The self-oscillating dynamics of the interface and the spatial solute concentration profile were calculated without using the perturbation theory

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Notes

  1. The stability region of a plane IB on the plane of control parameters \(v-\nabla T\) was determined analytically in [12].

  2. The article [6] is devoted to the calculation of the transient regime.

  3. It should be noted that the instantaneous acceleration value can be even arbitrarily high, but only during a vanishingly small part \(\tau \) of the total oscillation period \(T_0\), so that the “accelerated” contributions to the displacement are small, \( {\ddot{Z}} \tau ^2\ll \dot{Z}T_0\) and similar inequalities should be valid for hyper-accelerations \(\frac{\partial ^3Z}{\partial t^3}, \ldots \)

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Acknowledgements

The work was supported by the Russian Science Foundation (project n. 19-19-00552)

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Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, V.O. Bolshoj pr., 61, St. Petersburg, 199178, Russia

    A. Chevrychkina & A. Korzhenevskii

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  1. A. Chevrychkina
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  2. A. Korzhenevskii
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Correspondence to A. Chevrychkina.

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Chevrychkina, A., Korzhenevskii, A. Analytical study of self-oscillatory pattern-forming crystal growth in two-parabola model. Eur. Phys. J. Spec. Top. 231, 1147–1152 (2022). https://doi.org/10.1140/epjs/s11734-022-00526-5

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  • DOI: https://doi.org/10.1140/epjs/s11734-022-00526-5

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