A relaxation time of secondary dendritic branches to their steady-state growth

Abstract

Abstract. In the present article, we use the selection theory to estimate the non-stationarity time of evolving dendrites to their steady-state growth. IntroductionThe microstructure predictions in solidification processes have the deep scientific and practical roots (see [1][2][3][4][5] and references therein). A development of experimental methods in solidification processing made possible to reach a much broader range of measurable crystal growth velocities, temperature gradients and cooling rates [6]. For instance, splats and films can be quenched with the cooling rate of the order of 10 7 K/s, the temperature gradients can have the order of 10 8 K/s in laser annealing of sample surfaces, and containerless methods of droplets processing provide the deep undercoolings having the values of 200-400 K prior to the primary crystallization. A large driving force for transformation, arising in such methods, leads to fast solidification from a liquid metastable state as well as the high-speed solid-state transformations of metastable crystalline phases [7]. For instance, the experimentally measured solidification velocity has the order of 10 −1 − 10 2 m/s in droplets processed by the electromagnetic levitation facility [7,8]. As such, the total duration of primary solidification in small droplets is rather short and estimated as [8]: 10 −5 − 10 −3 s.For the analytical calculations of rapid solidification regimes and theoretical estimations of microstructural parameters of dendritic and eutectic crystals, different models of non-equilibrium crystallization are used, as a rule, formally developed for the steady state scenario [9][10][11]. However, the steady state approximation for conditions of rapid solidification of small samples (films, splats, droplets) is questionable and such approach is often criticized. A very common view is that a steady state scenario may be expected to exist but not in rapidly solidifying small samples [12]: the stationary regime of solidification can be reached at long times that is possible in usual experimental circumstances (directional solidification) or well-known classic technologies (continuous casting and low intensive brazing). In other words, during the short periods of time, the steady state is not achieved and, consequently, the quasi-stationary approximation in modeling of rapid solidification does fail. The detailed calculations of solidification regimes lead, however, to the remarkable behavior of the transient time between the non-stationary and stationary regimes of solidification: the non-stationary time sharply depends on the interface