Introduction

The methods involving plasma surfacing of different wear-resistant materials are commonly employed for the restoration or repair of forming rolls [1]. Among the surfacing materials, there is a specific interest in heat-resistant iron-based alloys with elevated tungsten levels (approximately 17 – 18 %) and carbon content (0.76 – 0.82 %), renowned for their high hardness and exceptional wear resistance [2]. To prevent the formation of cold cracks during the application of coatings made from such alloys, pre-heating and controlled slow cooling of the parts are utilized [3]. However, post this treatment, the coatings exhibit reduced hardness and wear resistance. A complex heat treatment process (involving annealing, hardening, and tempering) becomes necessary to enhance these properties. Yet, this significantly restricts the potential applications of these considered alloys [4]. Hence, finding plasma surfacing methods for heat-resistant alloys that circumvent crack formation while preserving high mechanical and tribological properties without the need for additional thermal treatments is crucial. Addressing this challenge requires an understanding of the mechanisms governing the formation of gradient structural-phase states in materials during surfacing. The crystallization processes of the materials play a pivotal role in shaping these states [5], ultimately determining the resulting structure and hence, the mechanical properties achieved during the surfacing process.

Currently, numerous papers [5 – 10] expound upon mechanisms and models pertaining to the crystallization of materials on surfaces of various geometries. Depending on the external factors such as cooling rate, rotation speed, ambient temperature, degree of supercooling, among others, the resulting structure can manifest as cellular, dendritic, or a combination of both structures concurrently [5; 6]. One of the main mechanisms driving their formation, as proposed in [7; 8], involves the instability of the crystallization front induced by the temperature decrease during the phase transition, caused by the expulsion of impurities into the melt and the phenomenon of solutal undercooling. Hence, the configuration of the interfacial boundary significantly influences the impurity distribution within the crystal [9]. The central consensus across all theories of morphological stability is that when a specific balance between temperature and concentration gradients is reached, the crystallization front becomes unstable to minor perturbations [10]. Consequently, intricate structural-phase states emerge, facilitating particle nucleation on dissolved impurities. This leads to the creation of an extended phase transition region preceding the crystallization front [11]. Notably, the models outlined in [5 – 11] solely consider the displacement of phase transition boundaries, disregarding the displacement of melt heating boundaries. Typically, it is assumed that this boundary either maintains a stabilized temperature distribution or extends infinitely [12; 13]. Recognizing the outcomes of resolving the thermal problem, where the heating boundary is not considered infinitely large [14], reveals that the growth rate of particles accelerates in comparison to scenarios where this aspect is disregarded. Consequently, in the development of mathematical models concerning the impact of plasma on material structure, it becomes imperative to consider not only the displacement of phase transition boundaries but also the shifting boundaries of heating. In the quest to understand the mechanisms governing the formation of gradient structures and phase compositions in heat-resistant alloys during plasma surfacing on rotating rolls, integrating the concepts associated with the emergence and progression of the Mullins–Sekerka instability [15] becomes crucial. Analyzing this instability aids in determining the conditions for the onset of these states, taking into consideration the shifts in the heating boundary. This instability is studied through several stages: initially determining the nature of interface perturbations and evaluating the impact of its curvature on the liquidus temperature. Subsequently, calculations are performed for the temperature and concentration fields within the solid and liquid phases. This analysis includes identifying the dependency of the perturbation growth rate on the conditions existing at the phase transition boundary.

The focal point of this paper revolves around resolving the thermal and diffusion Stefan problem involving two movable boundaries. This approach facilitates the monitoring of material solidification kinetics. Unlike conventional studies [16 – 18] that rely on the assumption of crystal growth being directly proportional to t1/2 this hypothesis is deliberately avoided in this investigation. Instead, the evolution of crystal growth over time is tracked by solving a system of kinetic equations derived from temperature and substance balance conditions at the phase transition boundaries.

Research methodology (setting up a problem)

Let us examine the process of directional solidification along the spatial axis r in a cylindrical front. Fig. 1 illustrates the geometry of the problem.

The initial phase occupies the region R2(t) < r < +∞ (where t represents time) and maintains a temperature of T0 . When the temperature T\(^{**}\) is attained, the second phase forms, which occupies the region R1(t) < r < R2(t). At temperature T\(^*\), the third phase emerges within the region 0 < r < R1(t). Let us formulate the thermal conductivity equation for each of these distinct regions:

where χ1 and χ2 , χ3 are thermal diffusivity coefficients in regions 1 ‒ 3.

The shift in phase transition boundaries is determined based on the balance conditions of temperature and heat flows, as expressed by equations (2):

where λ1 and λ2 , λ3 are thermal conductivity coefficients in the regions 1 ‒ 3; ΔH1 and ΔH2 represent the volumetric heat of phase transformations.

At r → 0, the temperature is Tw1 , and at r → ∞, the temperature is T0 . The initial conditions are defined as

where R0 and \(R_0^*\) are the initial radii of the phase transition boundaries.

The solution of this system of equations (1) ‒ (3) is determined as follows

where Ai , Bi are arbitrary constants; Ei (z) is an integral exponent; i = 1 ÷ 3.

Applying the boundary conditions (2) and initial conditions (3) to formula (4), the resulting solution is as follows:

The parameter R, introduced to prevent divergence at r → 0, is assigned a value of 10\(^–\)8 m to eliminate the singularity at this point in the radial coordinate.

Results and discussion

As equation (5) is integrated into the heat balance equations at the boundaries of phase transitions, the resulting kinetic equations are as follows:

The system (6) of ordinary differential equations is resolved using the high-order Runge-Kutta method. To simplify the mathematical computations, non-dimensional variables \({\tilde R_i} = \frac{R}{{{R_0}}}\) and \(\tau = t\frac{{{\chi _1}}}{{R_0^2}}\) (where τ is nondimensional time) were employed. Given that as at t → 0 the function \({E_i}\frac{{R_i^2}}{{4\chi t}} \to 0,\) the initial time value was set to 10\(^–\)9 s. The characteristics of the investigated material, specifically, iron – tungsten system, are outlined in the table provided.

In region 1 the initial melt temperature was set at T0 = 1811 K. At the R2 boundary, the liquidus temperature T\(^{**}\) is 1806 K, while at the R1 boundary, the solidus temperature T\(^*\) s 1803 K. These temperatures were determined based on the state diagram [19] for an 18 wt. % tungsten content. The temperature Tw1 was assumed to be lower than T\(^*\) and measured at 1790 K. It was considered that χ2 = χ3 and λ2 = λ3 , along with ΔН1 = ΔН2 . Fig. 2 displays the dependencies of the interface movements. At R0 = 1 μm, the boundary coordinate R1 exhibits nearly linear growth until τ = 0.028 (4.4118 ns), while R2 demonstrates non-monotonic changes, decreasing after τ > 0.028 (4.4118 ns). For R0 = 10 μm, a similar trend is observed yet the crystallization process takes place over a longer duration of 41.176 μs.

The results suggest that with a reduction in the size of nuclei, the duration of their steady growth significantly decreases by almost four orders of magnitude. The abrupt decline in the radial coordinate R2 , may indicate instability in the crystallization front, potentially due to interfacial surface tension and supercooling effects. The rapidity of the crystallization process in smaller regions is attributed to a high surface energy, which aims to decrease through size growth and interface configuration changes. In Fig. 3, the temperature dependencies plotted against the non-dimensional radial coordinate illustrate a notable trend: as time progresses, region 2 demonstrates a reduction in size (represented by curves 2 and 3), while conversely, region 3, displays growth.

In the scenario where T0 = 1790 K, T\(^{**}\) = 1803 K, T\(^*\) = 1803 K and Tw1 = 1811 K (Fig. 4). The coordinates of the phase transition boundaries exhibit a reduction (Fig. 4, a), with R1 decreasing linearly while R2 follows a parabolic trajectory. Temperature dependences (Fig. 4,  b) indicate a similar trend to the previous case, wherein regions 2 and 3 (curves 2 and 3) experience a decrease in size.

Conclusions

The theoretical investigation of the crystallization process within the iron – tungsten system, conducted by solving kinetic equations, revealed intriguing behavior. Specifically, it was observed that the movement of the liquidus boundary R2 follows a descending parabolic trajectory, contrary to the expected R ~ t1/2 law, whereas the solidus boundary R1 exhibits almost linear movement. Upon reaching a certain time threshold, the convergence of these boundaries occurs, indicating either the cessation of the crystallization process or the onset of crystallization front instability. The temperature dependencies derived from this study will serve as a foundational framework for further exploration of this observed instability. As the model progresses and adapts to simulate the plasma-arc surfacing process of rolls, future iterations will consider factors such as the rotation of one of the media involved and a more precise incorporation of the concentration of alloying elements.