Formation of the gradient of structural-phase states of high-speed steel during surfacing. Part 2. The role of the Mullins–Sekerka instability in formation of crystallization structures

Introduction

High-speed steels are increasingly used as wear-resistant coating materials, applied by plasma spraying onto the working surfaces of mining and metallurgical equipment subjected to abrasive wear conditions [1]. These steels exhibit high mechanical properties such as hardness and wear resistance. However, the spraying process can lead to the formation of structures that cause cracks and reduce hardness, preventing the full utilization of the high-performance characteristics of high-alloy, heat-resistant alloys [2]. To maintain the high mechanical properties of the resulting coating, additional heat treatments [3] or adjustments to the spraying parameters [4] are necessary. Optimal spraying parameters require an understanding of the material crystallization processes and the associated structural-phase transformations. The formation of specific structures (cellular or dendritic) during the action of concentrated energy fluxes is explained by the morphological instability of the crystallization front, known as Mullins–Sekerka instability [5; 6].

Currently, various authors are studying this instability [7 – 10]. In [7], the instability was examined for binary alloys. Criteria for absolute and relative stability of a spherical crystallization nucleus were formulated for these alloys, and it was demonstrated that as the particle size increases, the initial concentration in the diluted binary melt initially suppresses and then enhances the morphological stability of the particle. The critical concentration at which this effect begins was also determined. The work described in [8] focuses on studying the influence of an incoming melt flow on the crystallization front of supercooled liquids with a two-phase layer. It was found that the incoming melt flow plays a crucial role in the changes in the parameters of the two-phase layer and its internal structure. The proportion of the solid phase in this layer and its thickness significantly increase, while its permeability and average distance between dendrites decrease with increasing intensity of the incoming melt flow. In [9], the verification of results from linear stability analysis using the phase field method in multi-component melts with a planar solidification front demonstrated that, despite using a unified set of equations for linear stability analysis under various solidification conditions, theoretical differences between directional and isothermal solidification should be noted. Specifically, in the case of directional solidification, if considering the steady-state solution of the planar problem, the equilibrium compositions at the interface are invariant with respect to the choice of the diffusion coefficient matrix, thus allowing the change in concentration gradients ahead of the interface, which affect instability behavior, to be easily determined since the interface velocity is known [9]. Conversely, in isothermal solidification, the planar problem does not have a steady-state solution characterized by a constant system velocity. The growth rate of the interface and the equilibrium compositions depend on the choice of the diffusion coefficient matrix as well as the alloy composition. These characteristics of phase transition affect the growth of morphological perturbations and thus influence the choice of length scales of the microstructure. In [10], similar investigations were conducted on a particle with spherical geometry, taking into account non-stationary terms in the diffusion equations and a complete representation of the diffusion coefficient matrix. This allowed the establishment that stability criteria are not reduced to low velocities as previously thought [11]. The stability of the growing sphere can be considered at high perturbation growth rates, which correspond to high supersaturation. The results clearly show that the threshold values for the destabilization of the growing sphere interface strongly depend on the growth rate. It was also found that the degree of spherical harmonics at which stability is not maintained increases with increasing growth rate. The role of surface tension in the melt is not sufficiently addressed in the presented works [7 – 11], while it can significantly shift the values of the wavelength corresponding to the maximum mode of perturbations [12; 13]. In [12], the dependence of this wavelength on surface tension and crystal growth mechanism is established. It is shown that for most binary compositions considered by the author, the growth mechanism occurs through screw dislocations, and the dependence of λ on V\(^\rm{–1/2}\) is practically linear and coincides with experimental data. Surface tension, according to the data from [13], has a significant influence on the kinetic coefficient in the growth model through screw dislocations. Thus, when constructing mathematical models of crystallization of materials under plasma exposure and formulating stability criteria for the interface between the melt and crystal, in addition to concentration supercooling, it is necessary to take into account the role of surface tension and crystal growth mechanisms. As indicated in [14], the instability of Mullins–Sekerka should be studied in several stages: 1 – determine the nature of perturbations of the interface and assess the influence of its curvature on the liquidus temperature; 2 – calculate the temperature and concentration fields in the solid and liquid phases; 3 – find the dependencies of the growth rate of perturbations from conditions at the phase transition boundary. In this study, attention is given to the first and third stages of research, assuming a cylindrical shape of the crystallization front. To verify the obtained results, investigations of the structure of coatings made of high-speed steels after surfacing were conducted using scanning electron microscopy.

Materials and methods

Plasma coating of high-speed steel R18Yu was performed in reverse polarity within a nitrogen protective and alloying environment using non-conductive additive powder wire, as established in [15]. The chemical composition of the steel (wt. %) includes C 0.87; Cr 4.41; W 17.00; Mo 0.10; V 1.50; Ti 0.35; Al 1.15; N 0.06. This composition ensured optimal conditions for wetting the surface of the product with the deposited metal and defect-free formation of the deposited layer. Samples were taken from the upper parts of the deposited layer and subsequently sectioned on an electric spark cutting machine using kerosene for metallographic studies. The samples were then mechanically leveled using fine emery paper and diamond paste, followed by etching of the deformed layer and leveling by an electrolytic method. Studies were conducted using a KYKY-EM6900 scanning electron microscope with a thermionic tungsten cathode equipped with a microprobe attachment. The operating parameters were an accelerating voltage of 20 kV, an emission current of 150 μA, and a filament saturation point of 2.4 A. The working distance between the sample and the objective lens was set at 15 mm. The sizes of the structural elements were determined using the random sectioning method [16].

Results of the experiment

Fig. 1 illustrates the microstructure of the surface layer of the plasma-coated high-speed steel, revealing two morphological components: cellular and dendritic. The grain sizes range from 3 to 45 µm, with the most common sizes between 10 to 15 µm, (Fig. 1, b).

The analysis of the histogram (Fig. 1, b) highlights that the instability of the crystallization front exhibits a single peak, corresponding to the wavelength that matches the most probable grain size. The presence of two distinct morphological components indicates two types of instabilities: “soft”, associated with the cellular structure, and “hard” linked to the dendritic structure.

Formulation of the problem

Let’s examine the stability of the cylindrical crystallization front concerning small harmonic disturbances (Fig. 2).

For the sake of further calculations’ convenience, we introduce the following dimensionless variables, as in [12]:

\[\begin{array}{c}{T_l} = \frac{{{T_{lr}}}}{{{T_0}}},{\rm{ }}{T_s} = \frac{{{T_{sr}}}}{{{T_0}}},{\rm{ }}{C_l} = \frac{{{C_{lr}}}}{{{C_0}}},{\rm{ }}r = \frac{{{r_r}}}{a},{\rm{ }}z = \frac{{{z_r}}}{a},{\rm{ }}\\t = \frac{{{\chi _0}}}{{{a^2}}}{t_r},{\rm{ }}{D_l} = \frac{{{D_{lr}}}}{{{\chi _0}}},{\rm{ }}{\chi _l} = \frac{{{\chi _{lr}}}}{{{\chi _0}}},{\rm{ }}{\chi _s} = \frac{{{\chi _{sr}}}}{{{\chi _0}}},\end{array}\]

where T_lr , T_sr, C_lr , r_r, z_r, D_lr , χ_lr , χ_sr are the dimensional temperatures of the liquid and solid phases, impurity concentrations in the liquid phase, radial and longitudinal coordinates, diffusion coefficient of the impurity in the liquid, thermal diffusivity of the liquid and solid phases, respectively; T₀ is the phase transition temperature (assumed equal to the liquidus temperature); C₀ is the initial impurity concentration, a is the initial radius of the cylindrical nucleus (~1 μm); χ₀ is the characteristic value of the thermal diffusivity coefficient (~10\(^–\)⁵ m²/s).

Let us express the latent heat of phase transition ΔH in dimensionless form \({\rm{\varepsilon }} = \frac{{\Delta H}}{{{c_0}{T_0}}}\) (where c₀ is the heat capacity of the substance under study at the phase transition temperature). We will now formulate the equations of thermal conductivity and diffusion in dimensionless form for the solid and liquid phases:

Boundary conditions:

To analyze the stability of the crystallization front, we express solution (1) as a sum of stationary and disturbed components:

For the stationary component, we have:

Considering Eq. (3), the boundary conditions for Eq. (4) will be as follows:

The interface perturbation equations will be as follows:

Accordingly, boundary conditions (2) will be as follows:

Analyzing linear stability requires knowledge of the analytical form of the unperturbed temperature gradients included in equation (6). To obtain them, we need to solve the boundary value problem (4), (5). Let’s present the solutions to this problem in the form:

Let us proceed to solve the boundary value problem (6) and (7) for perturbations of temperature, concentration, and velocity of the crystallization front. We will seek its solution in the following form:

where ω = ω₁ + iω₂; k = k₁ + ik₂.

Then, equations (6) will take the form:

Accordingly, the boundary conditions (7) take the form (where the prime denotes the derivative with respect to the radial coordinate):

As in equation [1], the amplitude of interphase boundary velocity disturbances as a function of maximum temperature and concentration disturbances is as follows:

where \(\theta = \frac{{\partial V}}{{\partial T}};{\rm{ }}\gamma = \frac{{\partial V}}{{\partial C}}.\)

Dispersion equation of perturbations of phase transition boundary

Equations (10) are degenerate inhomogeneous hypergeometric equations whose solutions are Kummer functions. Obtaining and analyzing the dispersion equation, which includes these functions, is a complex and non-trivial task. Therefore, we will limit ourselves to considering special cases. In the first case, we neglect the convective term in equation (1). Then, stationary solutions are as follows:

Accordingly, equation (10) are as follows:

where \(S_1^2 = \frac{\omega }{{{\chi _l}}} - {k^2};{\rm{ }}S_2^2 = \frac{\omega }{{{\chi _s}}} - {k^2};{\rm{ }}S_3^2 = \frac{\omega }{{{D_l}}} - {k^2}.\)

Solutions (14) are as follows:

Substituting equations (15) into boundary conditions (11) and subsequent transformations, taking into account equation (12), lead to the following dispersion equation:

At an impurity concentration approximately equal to the eutectic, the value (1 – k_s) is close to zero, therefore, these terms in dependence (16) can be neglected. As a result, we have:

The value of θ, as in [1], is considered equal to \(\frac{{\Lambda \omega }}{{\Lambda \Gamma {k^2} - \omega }}\) (where \(\Lambda = \frac{{\partial V}}{{\partial \Delta T}}\) is a coefficient depending on the crystal growth mechanism; Г = αГ_r; Г_r is the ratio of the product of surface tension and phase transition temperature to the volumetric latent energy of phase transformation; α = 1/(a_T0)).

In the case of short waves, where S_1, 2 \( \gg \) 1, the approximate values of the Bessel functions can be represented as

\[{I_0}({S_2}a) \approx {I_1}({S_2}a) \approx \frac{{\exp ({S_2}a)}}{{\sqrt {2\pi {S_2}a} }}\]

and

\[{K_0}({S_1}a) \approx {K_1}({S_1}a) \approx \frac{{\pi \exp ( - {S_1}a)}}{{\sqrt {2\pi {S_1}a} }}.\]

Then, equation (17) is as follows

We assume that k₁ = 0 and ω₂ = 0, then \({S_1} = \sqrt {k_2^2 + \frac{{{\omega _1}}}{{{\chi _l}}}} ,\) \({S_2} = \sqrt {k_2^2 + \frac{{{\omega _1}}}{{{\chi _s}}}} .\) Let us substitute as follows: \({\omega _1} = \frac{{\delta V_s^2}}{{{D_l}}},\) \(k_2^2 = \frac{{V_s^2}}{{D_l^2}}Y.\) Then, \({S_1} = \frac{{{V_s}}}{{{D_l}}}\sqrt {Y + \frac{{\delta {D_l}}}{{{\chi _l}}}} ,\) \({S_2} = \frac{{{V_s}}}{{{D_l}}}\sqrt {Y + \frac{{\delta {D_l}}}{{{\chi _s}}}} .\) Provided that \(Y \gg \frac{{\delta {D_l}}}{{{\chi _s}}}\) and at δ ~ Y the maximum growth rate will be observed at the wavelength

Thus, it can be concluded that equation (19) completely coincides with the dependency obtained in [12]. This allows us to infer that in the short-wave approximation, considering only diffusion terms, the problem of finding the wavelength at which the growth rate maximum occurs simplifies for cylindrical geometry to a problem on a plane.

In the second case, we neglect the diffusion term in equation (6). Then the equation takes the form:

Substitution of solutions (22) into boundary conditions (11), subsequent transformations taking into account (12) lead to the following dispersion equation:

The solutions are as follows:

Substituting solutions (22) into boundary conditions (11), subsequent transformations taking into account (12), lead to the following dispersion equation:

Also, as in the previous case, in equation (23) we neglect the terms that contain (k_s – 1). As a result, we get

The maximum growth rate of disturbances is observed at a wavelength

During normal crystal growth, the crystallization front velocity is directly proportional to the degree of undercooling V_s = hΔT [12; 13] (where h is the proportionality coefficient; ΔT is undercooling). Then Λ = h. According to model [13], the dimensional value of the coefficient h is determined as

where M is the molar weight; R is the universal gas constant; Δl is the disturbance amplitude of the interface (approximately 0.1 nm); T_rL is the liquidus temperature; β is the coefficient that accounts for the difference between the mean free path of molecules in the liquid phase and the period of the crystalline lattice, as well as the symmetry of the molecules (for symmetric molecules β ~ 10).

According to the data from the Table in equation (26), it follows that with an initial nucleus size of 1 µm, the value of the coefficient h_r is 0.558 m/(s·K).

The transition to dimensional variables in equation (19) gives

The crystallization front velocity is determined based on the data from reference [14] as follows \({V_{sr}} = \frac{{\Delta a}}{{\Delta t}},\) knowing the undercooling, we determine h_r. Reference [14] indicates that Δa = 10\(^–\)⁸ m, and Δτ = 4.4118 ns, then V_sr = 2.27 m/s and h_r = 0.757 m/(s·K) for an undercooling of 3 K. The coefficient Γ_r is determined as \({\Gamma _r} = \frac{{\gamma {T_{Lr}}}}{{\Delta H\rho }} = \) 1.71·10\(^–\)⁶ K·m. Calculation according to equation (27) shows that in the case when h_r is 0.558 m/(s·K), λ = 0.382 µm. With h_r = 0.757 m/(s·K), calculation according to equation (27) leads to λ = 0.291 µm. Calculation based on equation (25) shows that for h_r = 0.558 m/(s·K), λ = 0.324 µm, and for h_r = 0.757 m/(s·K), λ = 0.242 µm. Comparison of the obtained results with the grain sizes in Fig. 1, b shows that both convective and diffusive approximations provide an explanation for grain size formation through the mechanism of normal growth up to 5 µm, although the maximum is found at sizes of 10 – 15 µm. This suggests that under these conditions, the normal growth model is not adequate. Let us consider the growth mechanism through screw dislocations. The crystallization front velocity in this case is directly proportional to the square of the undercooling. In this case \(\Lambda = 2\sqrt {h{V_s}} \) [12], equation (19) will take the form:

Returning to the dimensional variables in equation (28), we obtain

For the growth mechanism through screw dislocations, the value of the kinetic coefficient h_r is determined as

where \(g = 2{\pi ^4}{n^3}\exp \left( { - \frac{{{\pi ^2}n}}{2}} \right);\) n is the number of molecular layers [16]; V_m is the molar volume.

For metallic materials, when n ~ 6, g ~ 5.99·10\(^–\)⁹. The value of the kinetic coefficient h_r, calculated from the table data for the growth model through screw dislocations, is 433 m/(s·K²). The value of the wavelength, calculated using equation (30) with this value of h_r and a crystallization front velocity of about 10\(^–\)⁷ m/s, is 14.8 µm, which coincides with the most probable grain size values in Fig. 1, b. In the convective approximation, calculation using equation (25) shows that the wavelength takes values of the order of 10⁴ µm. This suggests a predominance of diffusion processes at these crystallization front velocities. When the crystallization velocity is increased by three orders of magnitude, λ = 14.9 µm, which matches the experimental data. Thus, the convective approximation is significant for crystallization velocities greater than 10\(^–\)⁴ m/s. Based on the above, it can be concluded that the Mullins–Sekerka instability provides an adequate explanation for the formation of cellular structures with sizes of about 10 µm at V_s < 1 m/s and undercooling degrees of about ~10\(^–\)⁵ K. When a volumetric heat source acts on the surface of the irradiated material, as shown in [17; 18], obtained using the phase-field method, the transformation front velocity can range from 10\(^–\)⁷ to 10³ m/s, depending on the power density of the source and the characteristics of the medium. These works [19; 20] indicate that the action of this source leads to the occurrence of large temperature gradients in the surface layers of the material and, as a result, to the emergence of thermocapillary effects. This allows us to conclude that to build a crystallization model, in addition to the Mullins–Sekerka morphological instability, it is necessary to take into account other instabilities (thermocapillary and concentration-capillary) that occur in the molten material. Analysis of the dispersion equations obtained in [21; 22] for these instabilities showed that for the Fe – W system under consideration, the wavelength corresponding to the maximum disturbances is 12 µm, which also coincides with experimental data.

Conclusions

The theoretical study conducted on the formation of cellular structures during the crystallization process of the iron-tungsten system, by analyzing the dispersion equation characterizing the morphological instability of the crystallization front (the Mullins–Sekerka instability), revealed that the mechanism of normal crystal growth provides an adequate explanation for the formation of cells with sizes up to 5 µm when the convective term can be neglected. Additionally, the growth mechanism through screw dislocations leads to λ = 14.8 µm, coinciding with experimental data under the condition that the crystallization front velocity is less than 1 m/s and the undercooling degree is about ~10\(^–\)⁵ K. Further development of the presented model involves incorporating thermocapillary and concentration-capillary effects.