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ISSN 0031918X, The Physics of Metals and Metallography, 2013, Vol. 114, No. 10, pp. 799–820. © Pleiades Publishing, Ltd., 2013. Original Russian Text © M.D. Krivilyov, E.V. Kharanzhevskii, V.G. Lebedev, D.A. Danilov, E.V. Danilova, P.K. Galenko, 2013, published in Fizika Metallov i Metallovedenie, 2013, Vol. 114, No. 10, pp. 871–893.

THEORY OF METALS

Synthesis of Composite Coatings using Rapid Laser Sintering of Metallic Powder Mixtures M. D. Krivilyova, E. V. Kharanzhevskiia, V. G. Lebedeva, D. A. Danilovb, E. V. Danilovaa, c, and P. K. Galenkod aUdmurt

State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia Institut für Nanotechnologie, Karlsruhe Institut für Tecnhologie, HermannvonHelmholtzPlatz 1, EggensteinLeopoldshafen, 76344 Germany c Interdisciplinary Centre for Advanced Materials Simulation, Ruhr Universität Bochum, Stiepeler Str. 129, Bochum, 44801 Germany d FriedrichSchillerUniversität Jena, PhysikalischAstronomische Fakultät, Jena, D07743 Germany email: [email protected] b

Received January 11, 2013

Abstract—Presented here is a brief review of research of an authors’ collective dealing with laser treatment of materials, sintering metallic powders, and multiscale simulation. A theoretical analysis of the processes of structure formation upon the rapid laser synthesis of composite coatings has been performed. The experi mentally obtained structural and phase characteristics of the sintered layers have been explained based on an analytical and numerical simulation of the dynamics of thermal fields in the zone of treatment, processes of melting, and subsequent solidification of porous materials. Upon rapid sintering and solidification, the effect of impurity trapping has been taken into account, which determines the chemical composition of the powders under nonequilibrium conditions of their formation. It has been shown that rapid laser treatment retains the composite structure of the powder layer due to the high rates of local heating/cooling and high rate of solid ification comparable with the rate of diffusion of chemical components. The results obtained are applicable in the development of a wide class of functionalgradient composite materials. Keywords: composite coatings, rapid laser sintering, partitionless (diffusionless) solidification, phasefield model, multiscale simulation DOI: 10.1134/S0031918X13080073

1. INTRODUCTION The development of laser engineering in recent decades led to the creation of many promising indus trial technologies, such as laser synthesis of functional coatings based on composite metallic [1, 2] and met alloceramic [3] powder mixtures. The improved mechanical and electrochemical properties of these coatings explain their wide application in machine building, chemical industry, medicine, and power engineering [4, 5]. In recent years, coatings have been actively applied [6, 7] that are based on organic com ponents and are suitable in the production of high capacity power sources. The reviews [3, 8] of recent advances and methods in the field of producing com posite coatings show their high practical importance and acceptable cost. An important role in the further development of this technology belongs to the cre ation of the theoretical foundations of synthesis of composite materials. The method of selective laser sintering (SLS) of powder materials was developed in the late 1980s as a process for rapidly prototyping various parts [9]. The

SLS method is based on a laser treatment of powder mixtures consisting of components with different melting temperatures. Laser treatment leads to the synthesis of materials with a complex structure in which the highmelting ceramic or metallic particles become bound by the lowermelting organic or metal lic matrix [2, 9]. The uniqueness of the SLS method is that it makes it possible to use a wide spectrum of materials. The flexibility of the method is achieved due to the use of a direct computeraided control of the process; in contrast to the traditional methods of fab ricating parts that require an additional mechanical treatment, in the case of the SLS, the threedimen sional parts are produced directly by layerbylayer sintering of the powder. The metallic and metallocer amic composite articles produced by the SLS method are used in smallbatch and single production, e.g., for the fabrication of tools, casting molds (including molds for casting under pressure), and applying pro tective corrosionresistant and wearresistant coat ings. In medicine, the SLS method is used to create prostheses and implants.

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The selective laser sintering of powders is a repeated process that includes several stages: (a) the application of a powder layer, smoothing of its surface, and densification using a roll; (b) the laser treatment (scanning) of the powder layer with the full melting of the lowmelting compo nent of the powder mixture; (c) the cleaning of the obtained layer; (d) the shift of the table with the sample downward by a distance equal to the thickness of one layer; (e) repetition of the entire process, i.e., the applica tion of a next powder layer, laser scanning, etc. The treatment is carried out in a chamber that is blownthrough by an inert gas and is computercon trolled to obtain a desired 3D geometry of the parts. To ensure the high quality of the articles to be fab ricated and the high mechanical strength and wear resistance, a high density of the parts to be sintered is also necessary. The preliminary milling of the powders leads to a decrease in the porosity and roughness of the articles [5], but the achievement of high density requires that the powder layer be melted completely, which was implemented in the socalled technologies of laser melting [10]. Laser melting (LM) leads to the formation of significant residual stresses because of the existence of high temperature gradients in the surface layers of the material. These stresses can later lead to the failure of the parts or to a distortion of their geom etry, delamination, and cracking. In addition to ther mal stresses, it is the thermocapillary effects that can represent serious obstacles for both the formation of a desired geometry of the parts and for obtaining high quality interlayer bonding. Therefore, the successful production of dense articles by the LM method is so far restricted to a few materials, including stainless steel and highcarbon steels [11]. The common feature of the technologies of SLS and LM is a low rate of the introduction of thermal energy. Most frequently, either a continuous regime of the generation of laser radiation is used or a pulsed treatment with a frequency of a few hertz is carried out in these processes. An extremely important role in the formation of highquality articles is played by the energy characteristics of laser radiation; at the same time, parameters of laser treatment, such as the mode and direction of laser scanning can significantly decrease thermal deformations and the delamination of the articles [5]. The authors of [12, 13] suggested a method of rapid laser sintering (RLS) of ultradispersed metal powders, which is based on the ultrarapid laser heating of a local region of a powder layer and subsequent rapid removal of heat into the bulk of the sample; the time of one cycle of heating–cooling is in this case on the order of 10–5 s. The treatment is carried out using high temper ature gradients; therefore, the preparation of the pow ders requires special demands. A powder metallic or composite mixture is milled in a highenergy plane

tarytype ball mill until the size of the powder particles decreases to no more than 1 μm. This means that the thickness of each layer of the multilayer coating can be decreased as well. The high locality of the procedure of rapid laser treatment of ultradispersed materials makes it possible to avoid deficiencies inherent in the tradi tional SLS and LM technologies, such as thermal stresses, presence of coarse pores, surface irregularity, and high surface roughness. At the stage of cooling after exposure to a laser pulse, the nonequilibrium rapid solidification of the molten part of the powder occurs with the formation of a metastable structural state [14]. A specific feature of this structure is the existence of a system of coupled pores of various scales: from nanosized pores to voids with sizes up to several microns. The results of investigations of the structure are given in [15]; they show a complex dependence of the structural parameters of sintered layers on the regimes of laser irradiation. The final porosity of the coatings depends on both the regimes of laser treatment of the powder and the chemical composition of the powder mixture and size distribu tion of powder particles. The hardness and wear resis tance of the coatings can vary significantly depending on the energy characteristics of the laser radiation. The achievement of optimum values of the porosity and hardness of sintered coatings makes it possible to increase the wear resistance under conditions of both abrasive and normal oxidation wear, sometimes by more than two orders of magnitude as compared to those characteristic of the continuous surfaces of arti cles from the same alloys as the applied coating [16]. The RLS method makes it possible for sintered lay ers to inherit the nanostructured state of the initial composite powders. For example, it has been shown in [17] that the RLS of a composite iron powder in a nickel nanoshell with a nickel content of 3.2 wt % leads to the formation of a surface coating which, in its corrosion resistance, exceeds that of stainless iron– nickel alloys with a high percentage of nickel. Accord ing to the results of Xray photoelectron spectroscopy (XPS), a high percentage of nickel is retained in the surface layers; as a result, these layers have higher pas sivating properties as compared to the individual iron and nickel. The corrosion resistance, just as the struc ture and mechanical properties of the coatings, depend strongly on the regimes of laser treatment. A twofold increase in the laserradiation power during laser sintering of nanocomposite Fe–Ni powders leads to the deterioration of the corrosion resistance of the coatings and an increase in the corrosion current by an order of magnitude. This is explained by an increase in the time of residence of the melt at a high temperature, in which nickel has time to become completely dis solved in the iron matrix. A noticeable improvement in the corrosion resistance of the Fe–Ni layers is accom panied by an increase in the activity of these materials in the reactions of cathodic deposition of hydrogen [18]. This makes it possible to create materials, the

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cathodic activity of which is significantly higher than that of nickel, which is the most active metal of the iron subgroup.

into account both the morphological and phase char acteristics of the samples.

The empirical search for the energy regimes of laser treatment that can enable one to obtain the desired struc tural and phase states is quite difficult and laborious. In fact, layers of satisfactory quality can only be obtained in a very narrow range of conditions. To date, vast experi mental and theoretical data concerning the interrelation between the regimes of treatment and the microstructure of composite layers have been accumulated. It has been found [19] that an increase in the specific power W of the absorbed radiation leads to a sharp increase in the density of the coating at W  0.2 kJ/mm3. At 0.2  W  0.8 kJ/mm3, the density changes at a lower rate. At W  0.8 kJ/mm3, the density remains unaltered with increasing W, but the morphology changes, namely, ultradense composite layers with extended pores are formed. The morphology of the structure is also affected by the mode of scanning of the surface [19, 20]. The authors of [21] performed a three dimensional numerical simulation of the process of SLS using the socalled bilevel method as follows: the thermophysical parameters are assumed constant at small scales, but are functions of temperature and coordinates at the scale of the entire sample. This approach is applicable for simulating sintering of pow ders with large changes in the density. Although the adequacy of the model developed in [21] has been confirmed by comparison with the experiment, it ignores phase transformations and is restricted to organic systems. In [22], the authors suggested a model for an analysis of SLS based on the computa tion of thermal fields in the powder to be sintered. Along with the dependence of the thermophysical parameters on temperature, the dependence on the powder porosity was also introduced, which is a func tion of time and coordinates of the local volume in the powder layer under consideration. The adequacy of the simulation method developed in [22] was con firmed by comparing with the data of thermographing in an IR range. The application of this model is also restricted to systems in which no phase transitions are observed.

2. EXPERIMENTAL METHODS AND STRUCTURAL AND PHASE ANALYSES To compare the results of the investigation of the phase characteristics and structure of the coatings obtained by the SLS method with the results of com puter simulation, we prepared a nanocomposite ultradispersed Fe–Ni powder. As the initial materials for its preparation, we took carbonyl iron of grade R20 and highpurity nickel carbonate (NiCO3 ⋅ 6H2O) of grade KhCh. The preparation of the powder included the following stages [13]: (a) mechanical grinding of the carbonyl iron in a planetarytype mill for 10 min; (b) the addition of nickel carbonate to the powder in the amount necessary to obtain compositions with Fe–3.2 wt % Ni or Fe–10 wt % Ni and simultaneous milling for 10 min; (c) annealing in a hydrogen atmosphere at a tem perature of 450°C; (d) passivation of the powder by heptane. The general description of the processes that occur during mechanical activation has been presented most comprehensively in [23]. The highenergy action on the powder particles in the zone of the collision of balls with the vessel walls and between themselves leads to the repeated occurrence of the following processes: the plastic deformation of the particles, their destruc tion, and their backward bonding with each other via cold welding. The destruction of the powder particles leads to an increase in their dispersity; the cold weld ing leads to a coarsening of particles. Severe plastic deformation causes the evolution of the dislocation structure, which ultimately leads to the formation of a nanocrystalline state. A detailed description of the processes that occur during the preparation of a nanocomposite powder has been presented in [13]. The chemical deposition of crystalline nickel was carried out at stage (c) upon the annealing of the powder in a furnace at a temperature of 450°C in a hydrogen atmosphere. At this stage, the dehydration of the nickel carbonate occurs, as well as its dissociation and subsequent reduction by hydrogen according to the reactions

In this paper, we study the problem of a theoretical description of the RLS of twocomponent metallic powders that is characterized by ultrarapid rates of heating of local regions of a powder layer. As follows from the previous works [13, 15], The RLS leads to the formation of an inhomogeneous sintered layer, in which the distribution of chemical components is inherited from the initial multicomponent powder. The paper is aimed at a multiscale analysis using the phasefield method, model of a twophase zone, and analytical simulation of kinetic effects at the solidifi cation front. The suggested approach is novel in the physics of powders, since it makes it possible to take THE PHYSICS OF METALS AND METALLOGRAPHY

NiCO3 → NiO + CO 2, NiO + H 2 → Ni + H 2O. The atomic nickel that arises in the course of the reac tion is deposited on the surface of iron particles in the form of elemental metallic nickel. At the given anneal ing temperature, nickel diffuses into the iron particles only quite slowly and is distributed over their surface, forming a Fe–Ni composite. Figure 1 represents a general image of the Fe– 3.2 wt % Ni powder obtained in secondary electrons. Vol. 114

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2 μm

50 μm Fig. 1. Secondaryelectron image of surfaces of particles of a prepared composite powder of the Fe–3.2 wt % Ni alloy before RLS treatment.

(a)

(b) 50 nm

Fig. 2. Electronmicroscopic image of structure of Fe–Ni nanocomposite: (a) general appearance of structure of powder particles; and (b) electrondiffraction pattern taken from the region (a) [13].

The metallic nickel is distributed on the surface; there fore, its amount detected by the electron microprobe analysis (EMA) reaches 26 wt %. Figure 2 presents the electronmicroscopic image of the structure and the electrondiffraction pattern of the prepared powder after mechanical activation and chemical deposition of nickel, which were obtained by transmission electron microscopy (TEM). The dark regions in the micro graph represent metallic nickel located on the surface of the iron particle. The large number of nickelrelated reflections in the electrondiffraction pattern is due to nickel is distributed on curved surfaces of the iron parti cles, which leads to the angular divergence of crystallo graphic directions of neighboring crystallites. The results of Xray diffraction studies of the pow der (Fig. 3a) have shown that the powder contains αFe and metallic nickel. Apart from these phases, the Xray diffraction patterns reveal traces of the carbide phase Fe3C, which arises as a result of the presence of an insignificant content of carbon in the initial powder of the carbonyl iron. The identification of the Xray diffraction patterns of the powder makes it possible to estimate the average size of coherent domains. The estimate made based on the data for the nickel phase yields an average size of crystallites of about 30 nm. An analysis of the electron micrographs of the powder gives a comparable result. Thus, we can state that the finished powder consists of iron particles with nano sized nickel single crystals deposited on their surface; i.e., the powder particles are Fe–Ni nanocomposites. Taking into account the uniform distribution of nickel crystallites, this nanocomposite can be described as an iron particle in a nonsolid nickel nanoshell. To implement the RLS of a prepared powder, we used the radiation of an ytterbium fiberoptical laser

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(a)

Intensity, arb. units

Initial powder Fe + 10% Ni

αFe (A2, bcc) Ni (A1, fcc) Fe3C 20

30

40

50

60

70

80 2θ, deg

Intensity, arb. units

90

100

110

120

130

(b) Fe + 10 Ni

αFe (A2, bcc) FeNi (A1, fcc)

Fe3C 20

30

40

50

60

70 2θ, deg

80

90

100

110

120

Fig. 3. Xray diffraction studies of samples of the nanocomposite Fe–10 wt % Ni powder: (a) Xray diffraction pattern taken before the treatment; and (b) XRD pattern taken after RLS.

working in a pulsed regime of generation. The energy parameters of the laser processing are given in Table 1 in comparison with the regimes used in the SLS tech nology. The treatment was performed in a chamber with a forevacuum pumping and subsequent blowing by pure argon of OSCh grade. The surface coating was applied onto a steel substrate in the form of a cylinder 8 mm in diameter and 2 mm in height. The multilayer THE PHYSICS OF METALS AND METALLOGRAPHY

coating consisted of ten applied layers 0.08 mm thick. Thus, the total thickness of the coating was 0.8 mm. The external appearance of the surface formed upon the laser sintering of the powder containing 3.2 wt % Ni is shown in Fig. 4a. The image was obtained in backscattered electrons using an Auger electron spectrometer. It can be seen that the structure of the surface contains a branched system of coupled Vol. 114

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Table 1. Energy parameters of laser treatment Parameter

Rapid laser sintering

Pulse duration, µs

Laser sintering

0.1

Pulse frequency, Hz

3–5

32000

Scanning velocity, mm/s

3–10

100

20–200

Beam diameter in the focus, µm

30

100

Laser power, W

15

100–400

Average size of powder particles, µm

0.8

Thickness of powder layer, µm

80

anisomerous pores. On the surface, aggregates of vari ous structural levels (with a minimum size of ~100 nm) sintered between themselves are observed. The obtained surface coating reveals good plastic proper ties. The lapping of the coating leads to the healing of the pores and to the formation of a solid smooth sur face. For an example, Fig. 4b shows a photo of the interface between the lapped and nonlapped regions of the surface coating. This sample was prepared by lap ping at an angle of 7° to the sample surface. The results of Xray diffraction studies of the sin tered coatings are given in Fig. 3b. The Xray diffrac tion patterns of the Fe–Ni samples contain lines of two phases; the first of these is (with a high reliability) the αFe phase; the other group of lines belongs to either the intermetallic compound Fe3Ni or to the solid solution of nickel in γ iron. The difficulties in the phase identification are due to both a large width of the lines and to the presence of significant micros tresses in the crystal lattice, which are a result of the rapid nonequilibrium solidification. The low content of the γ phase detected is explained in [13] by the non uniform distribution of nickel after laser sintering.

(a)

3–5 100

The XPS results confirm the data obtained by Xray diffraction and suggest the preferential distribution of nickel near the surfaces of the sintered particles. In particular, the content of nickel detected by the XPS method after etching of the surface of the sintered Fe– 3.2 wt % Ni sample by argon ions to a depth of 40 nm is equal to 12 at %, which agrees with the data on the amount of nickel detected by the EMA, which yields 8 at % Ni. Thus, under the conditions of laser sinter ing, nickel has no time to segregate from the surface into the bulk of the sintered particles of iron and turns out to be distributed predominantly on their surfaces. This explains the high anticorrosion properties of the samples. To comprehensively explain this phenome non, we performed a mathematical simulation of the processes of heat and mass transfer during RLS of composite powders. 3. METHOD OF MULTISCALE SIMULATION The procedures of computational simulation are widely used in materials science due to the recent development of computer power and simulation methods, which permit one to rapidly test the theoret

100 μm

10 μm (b)

Fig. 4. Electronmicroscopic image of surface of coating produced by laser sintering of nanocomposite Fe–3.2 wt % Ni powders: (a) general appearance; and (b) the appearance of the boundary zone between the lapped and unlapped regions of the coating. THE PHYSICS OF METALS AND METALLOGRAPHY

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ical predictions or processing of complex experimen tal data with relatively low costs. However, many prob lems of materials science are concerned with an anal ysis of collective phenomena, which occur in the ranges of time and spatial scales that hardly can be analyzed in terms of the solution of a single set of equations and a single system of simulation [24]. Therefore, in recent decades, efficient methods of analysis have been developed that take into account a wide range of space–time scales and referring to the method of multiscale simulation [25–27]. This method is based on a synthesis of methods of atomistic simulation (molecular dynamics method, Monte Carlo method, densityfunctional theory), methods of mesoscopic simulation (phasefield models), and methods of describing the continuum (finiteelement analysis, cellularautomata models), which leads to a new class of hierarchical or hybrid models [28, 29]. As a result of this synthesis, the multiscale method per mits one to qualitatively describe and quantitatively estimate many details of a microstructure and proper ties of materials on both an atomistic and mesoscopic or even on a microscopic level. In particular, along with the use of recently emerging accelerated compu tational algorithms, the multiscale method makes it possible to simulate surface diffusion and grain boundary phenomena in metals, radiation damage in ceramic layers, strength of nanocrystalline alloys, propagation of cracks in composite briquettes, relax ation of polymer chains, and the nucleation of biom ass in macromolecular solutions [30]. To obtain an adequate theoretical description of the process of laser treatment, we used the method of mul tiscale simulation that was developed in the investiga tions concerning sintering powders [31] and solidifica tion of alloys [32, 33]. As is well known [14, 34, 35], upon the rapid solidification of metallic systems the motion of interphase boundaries is accompanied by the appearance of significant gradients of concentra tion and temperature caused by the limited solubility of solute components and finite rate of the diffusional removal of heat. In this case, the spatial scales lC and lT of the concentration and temperature inhomogene ities can be estimated by the expressions D/V and a/V, where D, a, and V are the diffusion coefficient, thermal diffusivity, and the velocity of motion of interphase boundaries, respectively. At values of these parameters characteristic of metallic systems, i.e., D ∼ 10–9 m2/s, a ∼ 10–6 m2/s, and V ∼ 10–2 m/s, the scales lC and lT are equal to 0.1 and 100 μm, respectively. The third char acteristic scale is the width lϕ of the interphase diffuse boundary within which the process of nonequilibrium trapping of the solute occurs. Assuming that this width is equal to 10–20 interatomic spacings, we obtain lϕ = 0.002–0.008 μm. Thus, the simultaneous analysis of the distributions of the concentration C, temperature T, and the variable ϕ (parameter for the description of the phase state) is difficult in view of the large differ THE PHYSICS OF METALS AND METALLOGRAPHY

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ences in the scales of lC, lT, and lϕ of the processes occurring. The description of the formation of a microstruc ture using the method of multiscale analysis consists in the decomposition of a complex problem into a set of problems of various scales that describe the main physical phenomena on corresponding space–time scales. We consider a submicron, mesoscopic, and macroscopic spatial scales with the characteristic val ues la < 1 μm, lµ = 1–10 μm, and lM > 10 μm, respec tively. The study of systems on a submicron scale makes it possible to predict effects of the impurity resistance and trapping impurities by the solid phase upon the motion of the interphase boundary. The mesoscopic scale is necessary to calculate the mor phology of the microstructure that is formed under the action of segregation of components and predict the phase composition of the samples. The macroscopic simulation is important when the conditions of mate rial treatment and calculation of thermal fields in the samples should be introduced into the model. The scheme that refines the connection between the mod els on different levels is given in Fig. 5. The macroscopic calculation of nonstationary tem perature fields in a layer of powder has been carried out in the quasiuniformmedium approximation, in which the thermophysical parameters of a nonuniform medium are replaced by effective coefficients obtained as a result of averaging over the porous medium. The description of the nonstationary heat transfer in the case of the occurrence of phase transitions has been per formed using the method of a twophase zone applied for the case of rapid solidification in a porous powder layer. The data obtained for the macroscopic level were extended to the mesoscopic level, where the regimes of motion of interphase boundaries and the segregation of chemical components were analyzed in terms of the phasefield method. The calculation of the nonequilib rium segregation coefficient on submicron scales was performed within the locally nonequilibrium phase field model [36]; the use of the locally nonequilibrium dynamics of the diffusion flux leads to partitionless solidification at rates equal to or greater than the diffu sion rates. Below, in this paper, we consider models that describe processes that occur on various scales and show their interrelation based on the example of the analysis of RLS. 4. ANALYSIS OF THERMAL FIELDS In [37], we suggested a macroscopic physicomath ematical model for calculating the thermal fields and characteristics of the zone of melting upon the laser sintering of powders. The mathematical model of heat transfer in porous media with phase transitions has been formulated for the description of processes of rapid melting of a powder mixture of two metallic components. The simulation of nonstationary thermal fields was carried out using the model of a twophase Vol. 114

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Macroscopic scale (10–2–10–4 m) Laser beam

Powder Matrix

Mesoscopic scale (10–5–10–6 m) Composite ultradispersed particles

Submicron scale (10–8–10–9 m) Segregation of components and kinetic effects

Analysis of the nonstationary heat transfer in the sample: • Allowance for the parameters and regimes of treatment (radiation power, rate of scanning, etc.) • Calculation of nonstationary thermal fields with allowance for the phase transition • Estimation of local temperature gradients and rates of cooling at various points of the powder layer

Simulation of processes of structure formation: • Mathematical model of melting and solidification of composite particles in terms of the phasefield theory • Allowance for the multicomponent and multiphase nature of the system • Simulation of the motion of phase boundaries • Calculation of the morphology and phase composition of the crystal structure

Allowance for the effect of impurity resistance and kinetic phase diagrams: • Analysis of kinetics of attachment of atoms and of locally nonequilibrium effects • Calculation of kinetic phase diagrams • Dependence of partition coefficient on the rate of solidification and chemical composition

Fig. 5. Multiscale model of analysis of crystalstructure formation and phase composition upon RLS of ultradispersed powders.

zone [35, 38] extended to the case of high rates of heating of the medium and heat transfer via conduc tive and radiative thermal conductivity. In this work, the model suggested in [37] is extended to the three dimensional case. To study alloys with a narrow tem

perature range of solidification and complex phase diagrams [39], we pass from the model with a single variable to a model with two variables, i.e., tempera ture T and specific enthalpy H. This permitted us to precisely allow for the temperature range of the phase

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transition and retain the absolute computational sta bility. The balance of the internal energy can be written in the form of a set of two equations ∂T = ∂T ∂H , (1) ∂t ∂H ∂t ∂H = k(ε )∇ 2T + F , (2) V ∂t where t is the time; k(εV ) = k0(1 − εV ) is the coefficient of thermal conductivity, which is dependent on the permeability of the medium caused by the porosity of the powder layer εV [40], k0 is the coefficient of ther mal conductivity of the metallic phase; and F is the intensity of the volume heat source, which takes into account the thermal effect due to the laser beam. The intensity of the volume source is written as F = ηqL,

where qL and η are the flux density of the energy of the laser radiation and the coefficient of light radiation in a local volume of the powder layer, respectively. The coefficient η depends on both the temperature and phase composition of the local volume; it determines in this model the change in the depth of penetration of the laser radiation upon the melting of the particles. The value of η was determined based on a series of experiments with the application of photodetectors and powder layers of various thicknesses [41]. The function ∂T ∂H is determined by the differen tiation of the known function T (H ), which is given by the equilibrium temperatures of the liquidus Tl and solidus Ts , as well as by the heat capacity C p and latent heat L of the phase transition, as follows:

⎧H ⎪C ρ , if H < C pρTs , ⎪ p ⎪ T − Ts T (H ) = ⎨Ts + l (H − Ts C pρ), if H ∈ [C pρTs , C pρTs + S m(d)L], S m(d)L ⎪ ⎪H S m(d)L ⎪C ρ + (Tl − Ts ) − C ρ , if H > C pρTs + S m(d)L, p ⎩ p where Sm(d) is the volume fraction of the liquid phase and d is the depth of the zone of melting. It is assumed that the function Sm(h) varies in the limits of 0 ≤ Sm(h) ≤ 1, since, in experiment [13], we revealed only partial melting of powder particles. The set of equations (1)–(3) was extended using additional equations that take into account the real conditions of laser treatment observed in experiment, namely, the heat transfer via the mechanisms of thermal conduc tivity, radiation, and evaporation of the metal under conditions of convective cooling by a flow of an inert gas. A detailed description of the model of heat trans fer and the control parameters of the process are given in [13, 37, 42]. A threedimensional simulation of thermal fields was carried out with allowance for the real regimes of treatment, the experimentally measured characteris tics of the Fe–Ni powder, and the characteristics of the penetration of the laser radiation into the powder layer (Tables 1, 2). The algorithm used for the calcula tion and analysis of the data includes the following: (1) the specification of the thermophysical proper ties of the powder and of the values of the parameters of the treatment; (2) the calculation of 3D thermal fields and of their evolution in time; (3) the estimation of the depth of the sintering zone based on the isotherm T = TL; (4) the determination of the temperature profiles from the data of the calculations of thermal fields, i.e., THE PHYSICS OF METALS AND METALLOGRAPHY

807

(3)

of the time dependence of the local temperature as a function of the distance to the surface. Then, these temperature profiles were used as the initial data for calculating the melting of composite Fe–Ni particles using the phasefield method. Figure 6 displays the isothermal surfaces at a fixed time moment t = 2 μs since the onset of a pulse for the case of a pulse duration of 0.1 μs. As follows from the figure, the zone of melting is still observed long after the end of irradiation. This indicates the existence of a significant period of relaxation ΔτT (several microsec onds; according to the data that will be given below, ΔτT ~ 5 μs), in which a sintering of separate powder particles occurs and a metallic matrix is formed in the powder layer. Nevertheless, this period of relaxation is much shorter than the period between the pulses, which was assumed to be 31 μs. Figure 7 shows the dynamics of the temperature evolution in time as a function of the distance z to the surface. A comparison of the temperature of the surface layer with the melting temperature indicates that the heating of the powder to the melting temperature occurs during the first one third of the pulse at the pulse duration equal to 100 ns. Based on these data, the rate of heating was estimated to be Vh ~ 109–1010 K/s. Then, the beginning of melting leads to a stabilization of the temperature that gradually decreases after the end of the pulse. The stage of cooling is characterized by a rate Vc ~ –108 K/s. This analysis of the real rates of heating and cooling in the powder mixture is of importance for a quantitative analysis of Vol. 114

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Table 2. Thermophysical parameters used in threedimensional simulation of thermal fields upon sintering of Fe–Ni pow der mixture Parameter

Designation

Magnitude

Molar heat capacity, J/(mol K)

Cp

25.19

Latent heat of solidification, J/mol

L

1 . 66 × 10

Thermal conductivity of the metallic phase, W/(m K)

k

80.83

ρ

7925

Temperature of solidification of the main element, K

Tm

1811

Nickel concentration, wt %

C0

3–25

Slope of the equilibrium liquidus line, K/wt %

me

–2.4

Equilibrium segregation coefficient

ke

0.9

Adiabatic temperature, K

θ

660.81

Thermal diffusivity, m2/s

a

2 . 27 × 10 − 5

Boiling temperature, K

Tboil

3023

Porosity

εV

0.62

Temperature of the matrix, K

Tmatr

293

Surface emittance

ε

0.4

Effective heatexchange coefficient, W/(m2 K);

heff

50

Heatexchange coefficient upon boiling, W/(m2 K)

hboil

5 × 10 5

Coefficient of absorption of laser radiation upon boiling

k

0.01

Coefficient of light absorption, 1/m

η

1 . 1 × 10

Volume fraction of the liquid phase upon partial melting

Sm

0.4

Density of the metallic phase,

kg/m3

the mechanism of the sintering of particles and segre gation of chemical components, as will be shown below. 5. MODEL OF PHASE FIELD FOR MELTING AND SOLIDIFYING A POWDER LAYER To describe the phase transition in the process of melting and subsequent solidification of a powder layer under the effect of laser radiation, we employed the method of phase field (PF), in which for each phase in a multiphase system there is introduced an order parameter ϕi (i is the index of the phase) that is a continuous function of spatial coordinates and time and also is called by the “phase field.” In the case of a twophase liquid–solid system, we will consider two order parameters, ϕ1 and ϕ2, corresponding to the solid and liquid phases, respectively. From the physical point of view, the parameter ϕi at each point of the sys tem describes the volume fraction of the ith phase in an infinitely small local volume. Thus, in the volume of phase i, we have the values of the order parameter ϕi = 1 and ϕ j ≠i = 0. In this case, the boundary between

4

5

the phases is described by a transition region between the phases in which the order parameters take on the values 0 < ϕ < 1. This transition region has a finite width and is therefore called a smeared or a diffuse interface. The concept applied to the description of the inter phase boundary as a diffuse interface having a finite width and characterized by a continuous (although sharp) variation of physical properties was used by van der Waals for the description of phase equilibria and the description of phase transformations by Cahn [43]. Using the Ginzburg–Landau functional, the concept of the diffuse boundary was then reformulated in the form of a set of the evolution Allen–Cahn and Cahn–Hilliard equations, which are the basic equa tions for one of the traditional formulations of the PF approach. At present, the PF method is a thermody namically consistent approach to the description of a wide class of phenomena (see, e.g., review [44] and monograph [45]. For a twophase solid–liquid metallic system with two order parameters, the evolution equations for the order parameter ϕi and the specific concentration cα of

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Fig. 6. Threedimensional isotherms at time moment t = 2 µs from moment of beginning of laser pulse on the scale of the entire sample (main figure) and directly in the zone of laser treatment (in the inset). Isotherms correspond to temperatures of 1805, 1470, 1130, 800, and 460 K.

the components of the binary system follow from the condition of the minimization of the Gibbs free energy functional (a detailed derivation of the equa tions is given in [46]). To facilitate writing, we will use indices S and L for the order parameters that corre spond to the solid and liquid phases, respectively: M S + M L ∂ϕS M S M L ∂t =

ε S2 ∇ 2ϕ S

−

ε 2L∇ 2ϕ L

− ( g + f )'ϕS + ( g + f )'ϕL ;

M S + M L ∂ϕ L M S M L ∂t =−

ε 2S ∇ 2ϕS

+

(4)

ε 2L∇ 2ϕ L

(5)

+ ( g + f )'ϕS − ( g + f )'ϕL ;

⎛ D D (∇g ' + ∇f c' ) AT ⎞ ∂ cα α (6) = ∇ ⋅ ⎜ α ϕ cα − jα ⎟ , ⎜ ⎟ ∂t '' '' g f + cα cα ⎝ ⎠ where α = A or B is the index of the chemical compo nent; M i is the mobility of the ith phase field in the dif fuse interface, which is proportional to the kinetic coefficient μ and inversely proportional to the bound ary width δ; ε i is the coefficient that takes into account the contribution to the freeenergy functional from the term that is quadratic in ϕi and is related to the sur face energy σ of the interphase boundary as ε i = 2σδ; Dα is the diffusion coefficient of the component α; and AT jα is the antitrapping flux, which will be defined THE PHYSICS OF METALS AND METALLOGRAPHY

below. To take into account the variation of the diffu sion mobility of atoms in different phases in Eq. (6), a function Dϕ = 1 − ϕS is used. The values of the order parameter ϕS = 1 and ϕS = 0 correspond to the solid and liquid phases, respectively. The function g(T , cα, ϕi ) describes the molar density of the free energy of the twophase system as a superposition of the contributions from the volumes of the phases; and the function f (ϕi ) determines the contribution from the interface to the free energy. The functions g and f are expressed through the densities of the free energy Gi of individual phases and the surface energy σ as follows:

g(T , cα, ϕi ) = GS (T , cα ) p(ϕS ) + G L(T , cα ) p(ϕ L );

(7)

f (ϕi ) = 9σ ϕ 2S ϕ 2L, δ where the interpolation function p(ϕ) is defined by a polynomial of degree 3, p(ϕ) = ϕ 2(3 − 2ϕ). The order parameters ϕS and ϕL and the concentra tions of the components cA and cB are linearly interde pendent, since the conditions of the balance ϕ S + ϕ L = 1 and cA + cB = 1 should be fulfilled. In spite of the linear interdependence of the order parameters and concentrations, the model (4)–(7) is written for the full set of the dependent variables ϕ S , ϕ L, c A, and cB, which makes it possible, if necessary, to extend it to the case of multiphase and multicomponent systems. Vol. 114

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demands on computer memory used in the calcula tions. Therefore, for many practically important prob lems of structure formation in the numerical solution of the equations of the PF model, the parameter δ (boundary width) is usually taken increased by two or three orders of magnitude [48]. In this case, however, the boundary accumulates the impurity in overequi librium amounts, even at a low rate of solidification. Therefore, the introduction of the antitrapping flux jαAT into the model in Eq. (6) is required to correct the chemical composition inside the wide diffuse bound ary. In this work, the allowance for jαAT was performed based on an extension of the PF model suggested by Karma as follows [49]:

z = 8 µm z = 10 µm z = 13 µm

2000 1800

Stage of cooling Temperature T, K

1600 1400 1200 1000 800

Stage of heating

600

AT jα = −a AT δ(1 − keα )cLαe

400

u(cα,ϕL

200 0

1

2 Time t, µs

3

4

5

Fig. 7. Local temperature T in focus of laser beam as a function of time t and distance z to sample surface (results of 3D calculations of thermal fields). Time is counted from moment of the beginning of the laser pulse.

(8)

G L(T , cα ) = RT (c A ln c A + cB ln cB ), Vm where R is the universal gas constant, T is the temper ature, and V m is the molar volume. The dimensionless functions F A and FB are determined from the condi tion of phase equilibrium and, for the equilibrium phase diagram with straight solidus and liquidus lines, they have the following form [47]: ⎛ 1 + (Tm − T ) me ⎞ F A (T ) = ln ⎜ ⎟, ⎝1 + keB (Tm − T ) me ⎠

∂ϕL ∇ϕL , ∂t ∇ϕL

(10)

where a AT = 1 (2 2) is a constant determined from the normalization conditions; cLα = cα0 keα is the concen tration of the chemical component α at the phase boundary; and the function u(cα, ϕ L ) is defined as 2cα cα0 (11) . 1 + keα − (1 − keα )ϕ L The flux (10) describes the diffusion of the solute out from the boundary that decreases its anomalous con centration obtained as a result of the anomalous trap ping of the solute by the wide diffuse boundary. Apart from antitrapping, we should also take into account a correction for the kinetics of motion of the wide boundary; namely, an increase in the boundary width requires that its mobility also be changed. The PF method supposes the fulfillment of scaling depen dences between the physical parameters (surface energy and width of the diffuse boundary) and the parameters of the phase field itself (mobility). The analysis performed [50] has shown that, for the model to be selfconsistent, the mobility should vary inversely proportional to the squared boundary width as follows: u(cα , ϕ L ) =

In the approximation of ideal solutions, the densities of the free energy G S and G L can be written as follows:

GS (T , cα ) RT = (c A ln c A + cB ln cB + c A FA (T ) + cB FB (T )), Vm

)

(9)

FB (T ) = − ln keB ,

where keB is the segregation coefficient of the impurity component B calculated from the equilibrium phase diagram (keB = cS cL , where cS and c L are the concen trations of the component B in the solid and liquid phases, respectively, at the interface); and me is the slope of the equilibrium line of liquidus. At present, the numerical solution of the equations of the PF model in the threedimensional or two dimensional space with the use of a real nanometric width of the diffuse boundary δ is difficult because of the large computational burden and anomalously large

M ϕ ∼ 1 δ2 .

(12)

Therefore, if, e.g., the boundary width increases by an order of magnitude, the mobility of the phase field should be decreased by two orders of magnitude. The model (4)–(12) of the phase transition in a twophase twocomponent system has been written in the isothermal approximation, in which the tempera ture T is assumed identical at various points of the sys tem on the scale under consideration, but variable in time. Indeed, the estimate of the characteristic scales of the inhomogeneity of the thermal field lT and of the temperature gradients in the powder layer show that this approximation will be valid on scales of an order of 1 µm at a general thickness of the powder layer of 40– 100 µm. The size of powder particles in the experi

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(13)

ϕi |t =0 = ϕi0(r), cα |t =0 = cα0 (r), where n is the unit vector of the normal to the bound ary ∂Ω and r is the position vector. To refine the parameters of the model, we per formed a comparison of the values calculated within the model with the data of the SGTE database of ther modynamic data for pure elements [51]. To this end, the expression for the term (RT V m )F A entering into Eq. (8) was calculated at cA = 1 and the data obtained were compared with the difference Δ G = G S − G L of the free energies G S and G L of the solid phase and the melt, respectively, for pure iron. It can be seen from Fig. 8a that, at the values of the system parameters chosen for the Fe–3.2 wt % Ni, i.e., keB = 0.65 and me = –7 K/wt %, a discrepancy is observed between the model expression for Δ G and the experimental data. The analysis performed has shown that this dis crepancy is due to the use of the model (8)–(9), which was formulated for a dilute solution with a phase dia gram with linear dependence of the solidus tempera ture on the composition, to the description of a phase diagram with a nonlinear solidus behavior. In reality, the solidus line has a bending at the nickel concentra tion of 2 wt % (Fig. 8b), which leads to a change in the value of the segregation coefficient from 0.8 to 0.65 in the model. The calculation of the Δ G function at keB = 0.8 and me = –7 K/wt % yields good agreement with the SGTE data. The complete data on the parameters used for the calculations are given in Table 3. Thus, the model (8)–(9) used to calculate thermodynamic potentials is sensitive to the nonlinearities in the phase diagram, which leads to an additional contribution to the error of the method upon calculations using the phasefield method. The PF model for concentrated alloys with allowance for the nonlinearity of the soli dus and liquidus lines has recently been discussed in terms of the grandpotential functional [52, 53]. The choice of the parameters of the phase field was carried out with allowance for the dependence (12) and for the antitrapping flux (10). However, a 2D simulation has shown a discrepancy between the calculated rates of melting and solidification and those experimentally observed for the Fe–Ni powder mixture. First, the cal culated values of the rates proved to be two to three orders of magnitude less than the rates obtained upon the sintering of the powder. This is related to the under estimated value of the mobility of the diffuse boundary THE PHYSICS OF METALS AND METALLOGRAPHY

Difference of Gibbs energies, J/mol

n ⋅ (∇ϕi ) |∂Ω = 0, n ⋅ (∇cα) |∂Ω = 0,

(a) 400 0 –400 –800 –1200

SGTE data for bcc Fe Model, ke = 0.8 Model, ke = 0.65

–1600 1720

1740

1760 T, K

1780

1800

(b) 1540

Bending in the solidus line T, °C

ments was about 1 µm. As the boundary conditions, the zero gradient of the order parameters and the absence of a flux of components at the boundary ∂Ω of the computational region were taken. At the initial time moment t = 0, there is an inhomogeneous distri bution of the variables ϕ1, ϕ 2 , c A , and c B in accordance with the experimentally observed distribution of com ponents and phases as follows:

811

1490 (δFe)

1440

0

(γFe, Ni)

4 1 2 3 5 Concentration of Ni, wt %

6

Fig. 8. (a) Difference ΔG = GS – GL of the free energies of the solid phase GS and the melt GL obtained via Eqs. (9) and (8) at different values of the segregation coefficient ke in comparison with the data of the SGTE database [51]. (b) Phase diagram of the Fe–Ni system in the region of small concentrations of Ni. A slight bending is seen in the solidus line at 2 wt % Ni.

with a thickness equal to 10–20 interatomic spacings compared to the physically reasonable value. This dis crepancy was explained by the fact that, at high veloc ities of the motion of a wide boundary, the suggested model of antitrapping compensates the anomalous trapping only in the case of small velocities (see [50]). This is confirmed by the results of calculations of the nonequilibrium trapping of solute and the rates of solidification performed for the Fe–Ni system (Sec tion 7). Thus, to better describe the rates of solidifica tion and melting and to use the model of antitrapping, we corrected the dependence (12) of the mobility of the phase field on the width of the diffuse boundary using its increased mobility (Table 3). Second, a com parison of the 2D calculations performed with and without allowance for the antitrapping flux jαAT has Vol. 114

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Table 3. Parameters of phasefield model used in calculations for Fe–3.2 wt % Ni alloy Parameter

Designation

Magnitude

Molar volume, m3/mol

Vm

1 . 25 × 10 − 5

Surface energy of the interface between the liquid and solid phases, J/m2

σ

0 . 204

Mobility of the diffuse interface:

ML, MS

in 2D calculations with allowance for scaling (12) and antitrapping flux jαAT

0 . 03

in 3D calculations without allowance for scaling and antitrapping

0 . 001 −9

Coefficient of nickel diffusion in the liquid phase, m2/s

D

Slope of the equilibrium liquidus line, K/wt %

me

− 4 . 33

Equilibrium segregation coefficient at the interface between the liquid and solid phases

keB

0 . 65

Width of the interface, m

δ

7 . 1 × 10

3 × 10 − 7

shown that the antitrapping flux defined by Eq. (10) exerts no significant influence on the chemical com position at high phasetransition rates observed in the zone of surface melting by a laser beam. Therefore, in the final calculations of the threedimensional com posite particle the term jαAT was ignored.

medium, the conglomerate of Fe and Ni particles can arbitrarily represented as a composite Fe–Ni particle with an iron core and nickel shell (Fig. 9b). In the cal culations, we studied the dynamics of the segregation of the components under the effect of rapid heating and subsequent cooling of such a composite particle with an inhomogeneous distribution of the nickel con centration over the particle volume (Fig. 9c).

6. ANALYSIS OF NICKEL MICROSEGREGATION

The melting most likely starts at the boundary between the core and shell, where at the initial time moment the liquid phase was assumed to appear as a result of the difference in the temperatures of melting of the components. Then, a series of computations was performed in which the variable parameter was the heating rate Vh, the values of which were assumed to be 1 × 108, 1 × 109, 3 × 109, 4 × 109, and 5 × 109 K/s. These values of the parameter Vh were chosen on the basis of the approximation of the temperature profiles at the stage of heating shown in Fig. 7. Thus, the value of Vh

An analysis of the distribution of chemical components in the Fe–3.2 wt % Ni powder mixture in the course of RLS was performed in terms of the model (4)–(13). It fol lows from the experimental data given in Section 2 (Figs. 1, 2) that the crystalline nickel particles are dis tributed on the surface of iron particles. This distribu tion of particles is shown schematically in Fig. 9a. After averaging the chemical composition over several particles and passing to the model of continuous

Ni

(a)

Fe

(b)

(c)

Liquid phase

Ni Fe

Fe Fe

Fe

Fe

Fe

Fe

Ni

Fe

Fe

Fe

Fe

Ni Ni

Ni Fe

Fig. 9. Model of the process of averaging the chemical composition of iron powder with nickel inclusions that was employed upon simulation: (a) distribution of Fe and Ni particles in the powder; (b) transition to a composite particle with an iron core and nickel shell; melting starts at the interface between the Fe and Ni phases; and (c) distribution of nickel in a threedimensional model specified as the initial conditions upon the simulation of the segregation of the components (Eq. (13)). THE PHYSICS OF METALS AND METALLOGRAPHY

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(a)

(b)

(c)

(d)

(e)

(f)

813 Ni, wt % 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 00

Fig. 10. Dynamics of the motion of interface surface and diffusion of nickel in time upon melting at a rate of Vh = 5 × 109 K/s. Position of the interface between the liquid L and solid S phases is shown in the figure by isosurfaces ϕS = 0.4 (white) and ϕS = 0.6 (black). Times and stages of melting are as follows: (a) t = 0, at the boundary between the iron core and nickel shell there is a spherical nucleus of the liquid phase; (b) t = 0.6 µs, onset of melting; (c) t = 1.2 µs, the closing stage of melting; (d) t = 2.4 µs, diffusion of components in the liquid state; (e) t = 3.6 µs, rapid solidification; and (f) t = 3.9 µs, final distribution of the compo nents after solidification.

unambiguously determines the distance from the composite particle to the sample surface and corre sponds to different conditions of heat treatment inside the powder layer. The cooling rate in the calculations was assumed to be Vc = 2 × 108 K/s. The diameter of a composite particles was taken to be 1 μm. The nickel concentration in the shell was chosen to be 5 wt %, which, upon the averaging over the entire volume of the particle, corresponded to 3.2 wt % Ni in the alloy under study. The results of the simulation are presented in Fig. 10. The motion of the interface at a high rate of heating Vh = 5 × 109 K/s is shown as the positions of the isosurfaces of the variable ϕS at different time moments in Fig. 10. At the stage of heating, gradual growth occurs in the region of the liquid phase (Figs. 10b, 10c), which ultimately leads to the com plete melting of the particle (Fig. 10d). Rapid solidifi cation occurs 2 μs after complete melting. This time of residence of the particle in the liquid state, which is determined by the magnitude of Vh, exerts a critical effect on the microsegregation of nickel, as is clearly shown in Fig. 11, where the final distribution of nickel THE PHYSICS OF METALS AND METALLOGRAPHY

after the complete solidification of the particle is dis played. The concentration profiles calculated along the radial direction indicate a decrease in the difference ΔcB of the nickel concentration at various points of the composite particle depending on Vh. At small Vh = 108–109 K/s, the duration of the stage of melting is insufficient for the equalization of the nickel concen tration via liquidphase diffusion. As a result, the chemically inhomogeneous structure of the composite particle is retained after sintering. These conditions are observed at a depth of 20–70 μm from the surface. On the contrary, at large Vh ~ 5 × 109 K/s, which cor respond to the surface layers up to 20 μm thick, the results of numerical simulation predict the equaliza tion of the nickel concentration inside the composite particle. The difference ΔcB decreases from the initial value of 5 wt % to values equal to 1–2 wt %. An impor tant result of the simulation is the estimation of the rate of solidification Vsol of the composite particle. For the case where Vh ~ 5 × 109 K/s (Fig. 10), the time of solidification was about 0.5 μs. With a particle size Vol. 114

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Concentration cB of nickel, wt %

5

Initial distribution Vh = 0.1 × 109 K/s Vh = 1 × 109 K/s Vh = 3 × 109 K/s Vh = 4 × 109 K/s Vh = 5 × 109 K/s

4 3 2 1 0 0

0.2 0.4 0.6 0.8 Distance x along the symmetry axis, µm

1.0

Fig. 11. Profiles of nickel concentration cB along the radial direction as a function of the heating rate Vh depending on the distance z to the sample surface.

equal to 1 μm, we obtain Vsol ~ 2 m/s. At such a high rate, a nonequilibrium trapping of the solute can take place by the solidification front, which will be ana lyzed quantitatively in the following section. 7. NONEQUILIBRIUM TRAPPING OF SOLUTES The nonequilibrium trapping of solutes is an effect that accompanies the rapid motion of the interface and is characterized by the formation of a solid phase with a nonequilibrium chemical composition. Upon the trap ping of a solute to concentrations exceeding equilib rium values, a supersaturated solid solution with a chemically homogeneous composition can arise. This occurs in the process of a completely chemically partitionless (segregationless) phase transformation [54–57]. For example, using various methods of quenching from the liquid state, chemically homoge neous alloys of the initial (nominal) composition can be obtained at cooling rates on the order of 105–106 K/s and greater [34, 58]. In the case of a nonequilibrium redistribution of atoms at the interface, the trapping of solutes is quan titatively estimated by the use of a segregation coeffi cient (local coefficient of chemical segregation)

solute concentration in crystal , (14) solute concentration in liquid interface where V is the velocity of motion of the interface. The deviation from the equilibrium composition at the interface is a result of (a) an increase in the chemical potential of the system [59]; (b) the balance of the dif fusion flux and the resulting atomic flux determined by the magnitude of the atomicinteraction potential [60]; and (c) the deviation of the segregation coeffi k(V ) =

cient (14) from its equilibrium value ke ≡ k(V = 0) up to unity (irrespective of the sign of the chemical poten tial) [61]. The reviews of the models of trapping solutes and the quantitative estimates of the nonequilibrium segregation coefficients for the trapping of solutes at a rapidly moving solidification front can be found in [47, 62–64]. Here, we present the results of calcula tions of the nonequilibrium trapping of impurities according to the phasefield model applied to the rapid solidification of a particle after it is melted by a laser beam in a powder of particles of the Fe–Ni alloy. To describe the trapping of solute atoms upon rapid solidification, we consider a binary system that con sists of atoms A (main component) with a small amount of atoms B (solute) at constant temperature T and pressure p. The requirement for the lack of growth of the free energy upon the relaxation of the system toward equilibrium leads to the following equations [47, 65]: 2  ⎡ ⎛ ∂2 f  ∂ 2 f  ⎞⎤ τ D ∂ C2 + ∂C = ∇ ⋅ ⎢M C ⎜ 2 ∇ C + ∇ϕ ; (15) ∂t ∂C ∂ϕ ⎟⎠⎦⎥ ∂t ⎣ ⎝ ∂C

τϕ

⎛ ∂ 2ϕ ∂ϕ ∂f ⎞ + = M ϕ ⎜ ε φ2∇ 2ϕ − ⎟ , 2 ∂t ∂ϕ ⎠ ∂t ⎝

(16)

where f is the density of free energy, C ≡ cB is the con centration of solute atoms (atoms B), τ D is the time of relaxation of the diffusion flux, M C is the mobility of atoms B, τ ϕ is the time of relaxation of the rate of vari ation of the phasefield parameter ∂ϕ ∂ t , and M ϕ is the mobility of the phase field. The set of hyperbolic equations (15)–(16) describes the solidification of the system in which the free energy does not increase with time [66, 67] at positive coefficients of atomic mobility M C and phasefield mobility M ϕ. To completely define the set (15)–(16), we should take the density of the free energy f in a local equilib rium. Choosing the EFKP model [68], we write the density of the free energy f for an ideal dilute binary system in the form

f (C, ϕ) = f A (T A ) − (T − T A )s(ϕ) + (ϕ)C + RT (C ln C − C) + Wg(ϕ), vm

(17)

A

where f (T ) is the density of the free energy of the sys tem of pure solvent (consisting only of atoms A); T A is the melting temperature of the solvent; R is the univer sal gas constant; v m is the molar volume, which is assumed to be equal for the A and B atoms; and W is the height of the potential barrier, which is simulated using the doublewell potential g(ϕ) = ϕ (1 − ϕ) . 2

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The density of entropy s(ϕ) and the density of the internal energy ε(ϕ) were obtained by the following approximation [68] developed for dilute alloys:

s(ϕ) =

s s + sl − ps (ϕ) L , ps (ϕ) = 1 − 2 p(ϕ), 2 2T A

 s + l − pε(ϕ) RT ln ke, 2 2v m pε(ϕ) = 2 ln [ke + p(ϕ)(1 − ke )] − 1, ln ke

(19)

ε(ϕ) =

max 〈Cl〉

(20) max 〈Cs〉

where L is the latent heat of solidification; ke is the equi librium coefficient of segregation; and the subscripts l and s refer to the liquid and solid phases, respectively. The interpolation function p(ϕ) is defined as

p(ϕ) = ϕ2(3 − 2ϕ), with the conditions

(21)

dp(ϕ) dp(ϕ) |ϕ=0 = |ϕ=1 = 0. (22) dϕ dϕ The function (21) interpolates two states: the liquid phase at ϕ = 1 and the solid phase at ϕ = 0, which are determined by the function g(ϕ) according to Eq. (18). The solution to Eqs. (15)–(16) was calculated via a special algorithm by the method described in [47]. As a result, we obtained the values of the phase field ϕ and concentration C in a onedimensional space for arbi trary values of the velocity V of the diffuse liquid–solid interface. Now, to find the value of the segregation coef ficient (14), we should have the definition of the con centration in the phases meeting at the diffuse interface, which requires a special approach to this definition. Thus, for example, the definition of the concentrations in the phases was given through the values at the asymp totic ends of the diffuse interface in the quasistationary regime of solidification as follows [47]:

Fig. 12. Definition of segregation coefficient of solute as ratio of maximum maxCs of solute concentration in the solid phase to maximum maxCl of its concentration in the liquid phase [36]. Solid line shows general profile of concentration C found from the solution to set of equations (15)–(22).

1 − p(ϕ) = p(1 − ϕ),

C s ≡ C(ϕ → 0) . Cl ≡ C(ϕ → 1) This definition makes it possible to obtain the com plete trapping of the solute in the calculations via the hyperbolic model at a finite rate of solidification, which agrees with the experimental data (see discus sion in [47]). However, this definition yields the com plete trapping of the solute at a rate that is lower than the rate of diffusion in the liquid phase, which contra dicts the definition of the transition to the diffusionless solidification [69]. Therefore, for the diffuse interface, we introduced a special dependence of the segregation coefficient on the velocity such that the concentration at the interface corresponded to the solution of equa tions of the phase field in equilibrium, with the approximation of this solution to the nonequilibrium quasistationary regime of solidification [36]. Then, the segregation coefficient is defined as the ratio of the maximum concentration of the solute atoms in the k(V ) =

THE PHYSICS OF METALS AND METALLOGRAPHY

solid phase to its maximum concentration in the liquid (Fig. 12) in the form

k(V ) =

max〈C s 〉 , max〈Cl 〉

(23)

where

C s (x) = [1 − h [ p(ϕ,V )]] C(x), Cl (x) = h[ p(ϕ,V )]C(x), with the function h(p) defined as

(24) (25)

p(ϕ) (26) . k(V ) + [1 − k(V )]p(ϕ) The large velocities of heating/cooling and the sig nificant temperature gradients upon laser treatment of materials create conditions for the occurrence of rapid phase transitions [70–72]. The macroscopic calcula tions of thermal fields and the dynamics of melting of powder particles that were described in the preceding sections have shown that upon the melting of ultradis persed particles the melting rate reaches maximum values exceeding 10 m/s. During solidification, the rate of the process changes from 0.1 to 10 m/s depend ing on the position of the point in the zone of melting. Thus, the conditions for sintering of particles are achieved, each of which solidifies after laser melting via the diffusionless mechanism, i.e., without segrega tion of chemical components [63]. Taking into account the sharp chemical inhomogeneity of the ini tial particles of the composite Fe–3.2 wt % Ni powder, which represent spherical iron particles coated by dis continuous nickel shells with an average thickness of 30 nm, the diffusionless mechanism of solidification leads to the retention of the high nickel concentration on the surface of the coating and on the internal sur face of pores, which is confirmed by XPS results. This h( p,V ) =

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Table 4. Parameters of Fe–3.2 wt % Ni alloy used in calculations of nonequilibrium trapping of solute (nickel) upon rapid solidification. Definitions of parameters and relationships between them are given in [36, 47, 69] Parameter

Designation

Magnitude

Melting temperature of the pure component A (Fe), K

TA

1811

Slope of the liquidus line in an equilibrium phase diagram, K

me

− 433

Equilibrium segregation coefficient

ke

0 . 65

Molar volume, m3

vm

1 . 2 × 10

Ratio of the diffusion coefficients (at T = 1784 K)

DS/DL

3 . 11 × 10 − 3

Energy of the interface, J/m2

σ

0 . 204

Diffusion parameter of the phase field, m /s

ν

1 . 22 × 10

Thickness of the interface, m

δ

1 . 875 × 10 − 9

Universal gas constant, J/(mol K)

R

8 . 31

Time scale of the rate of variation of the phase field, s

τϕ

1 . 0 × 10

Time of relaxation of the flux of solute, s

τD

6 . 8 × 10 − 10

Rate of diffusion in a diffuse interface, m/s

VD

I

0 . 65

Diffusion rate in the bulk of phases, m/s

VDB

1 . 31

Velocity of the phase field in the diffuse interface, m/s

V ϕI

90. 5

Velocity of phase field in bulk of phases, m/s

VϕB

43 . 7

2

high nickel concentration imparts anticorrosion prop erties to these coatings that are comparable to the alloy iron–chromium–nickel stainless steels. The study of the phenomenon of the nonequilib rium trapping of nickel (as a solute component) was performed for the system Fe–3.2 wt % Ni, the param

1

Concentration C, wt %

4.0

2

3.5 3.0 3

2.5 2.0

1, 2, 3

1.5

Solute concentration Concentration in the solid phase CS Concentration in liquid CL

1.0 0.5 0 –3

–2

–1 0 1 2 3 Dimensionless coordinate x

4

Fig. 13. Concentration profiles inside the phases and in the diffuse interface depending on the rate of crystal growth: (1) V = 0.5, (2) 1.0, and (3) 1.5 m/s. Calculation using the hyperbolic model [36] via Eqs. (15)–(26).

5

−5

−8

− 11

eters of which are given in Table 4. The diffusion rates VDB and V DI were calculated with allowance for the tem perature dependence of the diffusion coefficient via the Arrhenius law. The calculations were carried out for the motion of the solidification front with a con stant velocity V using the method described in [36] for two analytical models of rapid solidification of the binary alloy. The first, parabolic model takes into account the deviation from the local equilibrium in the diffuse interface between the liquid and solid phases. The second, hyperbolic, model, which demonstrates better agreement with the experiment and with the results of moleculardynamics simulation [64], takes into account the locally nonequilibrium effects of relaxation of the diffusion flux and phase field in both the diffuse interface and in the bulk of the melt. Figure 13 demonstrates the changes in the concen tration profile of the solute before the solidification front with increasing velocity of its motion from 0.5 to 1.5 m/s. As the velocity of the solidification front increases to about 1 m/s, the difference in the concen trations in the liquid and solid phases decreases, which indicates a significant trapping of nickel above the equilibrium concentration. Indeed, the calculations of kinetic phase diagrams, which take into account the positions of the lines of coexisting phases with allow ance for the nonequilibrium effects [73], have shown a shift of the kinetic lines of liquidus and solidus toward

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(a)

Temperature T, K

1812 1810 1808

Equilibrium diagram Liquidus Solidus

1806 1804 1802 1800 1798

Kinetic diagram V = 0.51 m/s Liquidus Solidus

1796 1794 1792

Temperature T, K

0.95 0.90 0.85 0.80 Fe–3.2 wt % Ni Parabolic EFKP model Hyperbolic EFKP model Hyperbolic CGM model

0.75 0.70

0.5 1.0 1.5 Velocity of interface V, m/s

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Concentration C, wt % (b) Equilibrium diagram Liquidus Solidus Kinetic diagram V = 2.01 m/s Liquidus and solidus

817

1.00

0 0

1812 1810 1808 1806 1804 1802 1800 1798 1796 1794 1792 1790

Segregation coefficient of solute k(V)

SYNTHESIS OF COMPOSITE COATINGS USING RAPID LASER SINTERING

2.0

Fig. 15. Coefficient of nonequilibrium trapping of Ni upon solidification of Fe–3.2 wt % Ni alloy depending on veloc ity V of front of solidification. Parabolic and hyperbolic phasefield models are compared with the hyperbolic con tinuous growth model (CGM) [63].

transition to the complete trapping of the solute (if it occurs) is observed at 5–20 m/s [14]. Thus, based on the results of calculations using the phasefield model, we obtain that the trapping of nickel atoms by a rapidly moving the solidification front and the transition to the chemically partitionless (diffusionless) solidification at V ≥ V D in a powder mixture of composition Fe–3.2 wt % Ni leads to the 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 incorporation of nickel into the crystal lattice of iron Concentration C, wt % in amounts that correspond to the nonequilibrium chemical composition of the composite particle. This Fig. 14. Kinetic phase diagram constructed for different rates transition leads to the formation of a crystal structure of crystal growth: (a) V = 0.51 m/s < VD; (b) V = 2.01 m/s > of the initial chemical composition, i.e., of composi VD, where VD is the rate of diffusion of nickel in iron. Calcu lations using hyperbolic model [36] via Eqs. (15)–(26). tion existing prior to the treatment, of nominal com position, which favors the retention of the chemical homogeneity of domains of iron and nickel in the sin smaller values of the solute (nickel) concentration as tered layer. However, upon the transition from an iron compared to the positions of equilibrium lines in the domain to a nickel domain, a change is observed in the phase diagram (Fig. 14). Upon a subsequent increase concentration of nickel in the composite Fe–Ni parti in the velocity, the decrease in the temperature range cle. This result agrees with the experimental data of solidification observed at a velocity of 0.5 m/s obtained, as was discussed in Section 2 and was shown (Fig. 14a) leads to the confluence of the lines of liqui in Fig. 2. The method of RLS makes it possible to dus and solidus into a single line (Fig. 14b). achieve the complete trapping of solutes with the for The confluence of the kinetic lines of liquidus and mation of chemically uniform composite materials on solidus from the physical point of view corresponds to the the scale of separate composite particles, but the transition to the diffusionless solidification. Figure 15 porous structure with a chemical inhomogeneity on displays the calculated kinetic coefficient k(V) of seg the scale of the entire composite layer is retained. regation as a function of the interface velocity V. It can be seen from Fig. 15 that the hyperbolic model pre 8. CONCLUSIONS dicts the transition to the diffusionless solidification at velocities that are greater than the diffusion velocity In this work, we have analyzed the problem of syn VD, which for nickel in iron was estimated to be thesis of composite coatings by RLS of metallic pow 1.25 m/s. This value is smaller than the experimental ders used to produce a surface with desired properties values found for other metallic systems, in which the under controlled parameters of its treatment. The THE PHYSICS OF METALS AND METALLOGRAPHY

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experimental analysis of coatings based on Fe–Ni alloys with a nickel concentration of 3.2 and 10 wt % has revealed the formation of a composite layer con taining structural elements with crystallite dimensions of about 30 nm. The data obtained by TEM and XPS confirmed the existence of local inhomogeneities in the distribution of chemical components arising due to the sintering of ultradispersed particles of iron and nickel. A theoretical analysis performed by the method of multiscale simulation has shown that the retention of the composite structure of the powder layer upon sin tering is caused by two factors. First, the high density of the energy of irradiation upon pulsed laser treat ment leads to ultrahigh rates of the heating and cool ing of the laseraffected zone, which results in only the partial melting of the powder particles and, as a conse quence, in the inheritance of the initial distribution of chemical components. Second, the calculations of the characteristic rates of solidification show that these rates are comparable with or even exceed the rate of nickel diffusion in the melt. As a result, we observed the effect of the anomalous trapping of solutes and the tran sition to the regime of partitionless (diffusionless) solid ification, in which the segregation of chemical compo nents at the moving interface between the solid and liq uid phases is absent. This favors the solidification of the liquid phase with the freezing of the microstructure of the initial (nominal) chemical composition. The suggested method of synthesis of composite coatings (Russian Federation patent no. RU 2010 113 121 “Method of Producing Massive Articles or Coatings on Articles”) and the method of the theoretical analy sis (program package “ComputerAided Optimization of Processes of Laser Treatment of Powders,” state certificate no. 2010614748) can be employed to pro duce coatings based on other metallic mixtures. ACKNOWLEDGMENTS This work was supported in part by the Russian Foundation for Basic Research, project no. 0902 12110 ofim, and Federal Target Program “Person nel,” campaign 1.5, grant application no. 20091.5 507007; by the Ministry of Education and Science of the Russian Federation, Federal Target Program “Sci entific and Pedagogical Personnel of Innovative Rus sia,” project no 14.A18.21.0858 (2009–2013), and “Applied Research in the Field of Education,” project no. 2.947.2011; and by the Russian Space Agency under the Peritectic Space Experiment. We are grateful to S.M. Reshetnikov for a critical discussion of the results and to V.E. Ankudinov and G.A. Gordeev for assistance in calculations. One of us (M.D.K.) thanks Central Research Institute of Machine Building and, personally, E.G. Lavrenko for his assistance with our experiments.

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