An Improved Model for Prediction of Critical Velocity of Cold-Spray by First-Principles Calculations

Abstract

The first-principles calculation was applied to predict the critical velocity of Cu/Al cold-spray bonding for the first time. The bonding mechanism of cold-spray was clarified by analyzing the energy variation and atomic interaction during the cold-spray impact process. Our results showed that the shear deformation played a key role in the cold-spray bonding. The atomic interaction determined the effective absorption of impact kinetic energy and finally determined the successful bonding of the cold-spray. The heterogeneous atoms absorbed the impact kinetic energy by interatomic attraction to achieve cold-spray bonding, while the homogeneous atoms absorbed the impact kinetic energy by the deformation of interface layers. An excellent agreement between the predicted critical velocity and the experimental one could be obtained, especially for the heterogeneous material cold-spray. Our present method proved to be a simple and highly efficient computing method in critical velocity prediction. Most importantly, the critical velocity for cold-spray could be predicted without using any empirical or experimental parameters.

1. Introduction

Cold-spray is a manufacturing technology newly developed in recent years, which has been widely applied in coating preparation, surface repairing and additive manufacturing [1,2,3]. As compared with the traditional thermal spray, cold-spray has many advantages, such as low processing temperature, little oxidation and few defects in the deposited coatings [4]. In cold-spray, the spray particles undergo a series of physical and chemical variations, including acceleration, impact, severe plastic deformation, bonding of spray particles on substrates as well as the formation of an intermetallic in the bonding interfaces [5,6]. To achieve successful bonding in the cold-spray, the kinetic energy of particles is a key factor, which requires a sufficient particle velocity (typically 300–1200 m/s) [7]. Much research has proved that there exists a critical velocity for cold-spray [8,9,10]. Only when the particle velocity surpasses the critical value, a tight bonding between spray particles and substrates can be built successfully. Therefore, the critical velocity is the most important parameter for the cold-spray process. However, unfortunately, the critical velocity for most cold-sprayed systems is unknown due to the unclear bonding mechanism of cold-spray and the complex influence factors. Assadi et al. demonstrated that the particle temperature and particle size dominated the critical velocity [11]. Meanwhile, Palodhi et al. found that the mechanical properties of spray particles, such as yield strength and elastic modulus, also affected the critical velocity [12]. Therefore, establishing an effective prediction model for critical velocity without any physical parameters is a hot spot in the research field of cold-spray. Regarding the attempts to estimate the critical velocity, Assadi et al. proposed a simple method based on the bonding mechanism of adiabatic shear instability (ASI) [8]. Successful bonding is characterized by a viscous flow of spray particles, which corresponds to a jetting phenomenon during the high-speed impact. Thus, the critical velocity can be determined by the velocity threshold of strains transferring from the plastic type to the viscous type [8]. Based on the framework of ASI, the critical velocity can be easily predicted by using finite element analysis (FEA) [13,14,15]. It should be noted that the critical velocity predicted by FEA is usually not accurate enough since the calculated value greatly depends on the mesh size [8] and the numerical algorithm [10,16]. For example, Assadi et al. found that the predicted critical velocity decreased with the increase in the mesh size [8]. Schmidt et al. obtained a critical velocity of 500 m/s for the Cu_p/Cu_s case by using the Lagrangian method [10], while Wang et al. got the value of 300 m/s by using the Eulerian method [16]. Inducing massive calibration factors in the semi-empirical formulas of critical velocity creates difficulties in understanding the physical meaning of critical velocities [11]. Additionally, the recent new findings on the bonding mechanism of cold-spray have broken the framework of ASI. Hassani-Gangaraj et al. reported a new pressure-release (PR) mechanism to explain the jetting phenomenon [17]. If the pressure-release mechanism is correct, it implies we lost the criterion to identify the critical velocity again. It is worth noting that both ASI and PR mechanisms are established by analyzing the plastic deformation characteristic of spray particles on a macroscopic scale. Few studies involve the bonding mechanism of cold-spray on the microscopic scale, especially on the atomic scale. Nikravesh et al. investigated the cold-spray deposition of nanometric copper particles impacting a copper substrate by using molecular dynamics (MD) [18]. Their findings reveal that both ASI and jetting are not necessary for particle binding to the substrate. C.D. Reddy et al. found that interface bonding could occur at any particle velocity that belonged to three different bonding mechanisms [19]. In order to predict the critical velocity of cold-spray accurately, a new theoretical framework with few empirical parameters should be applied. Taking this into consideration, the first-principles calculation is a suitable method to solve the problem above, since it requires no input parameters but runs self-consistently. In fact, the bonding of cold-spray can be considered a process in which a spray atom squeezes into the crystalline lattice of substrate materials. The entrance of spray atoms will be hindered by the substrate atoms due to the intrinsic repulsive force between atoms. Thus, there exists an energy barrier for the bonding process of cold-spray. If the impact kinetic energy of spray atoms is greater than the energy barrier, the spray atoms could enter the substrate successfully. In other words, the energy barrier is the minimum kinetic energy for the entrance of spray atoms into substrates, by which the critical velocity can be obtained. The energy barrier is independent of the bonding mechanisms of cold-spray and can be calculated by the first-principles calculation.

In the present work, the first-principles calculation was employed to predict the critical velocity of cold-spray processes. The calculated critical velocity was compared with the reported experimental values and predicted values. The differences between them are discussed. The purpose of this work is to evaluate the reliability of predicting the critical velocity by first-principles calculations. It should be pointed out that this attempt is the first time to develop a simple and highly efficient method to predict the critical velocity of cold-spray processes without experimental measurements.

2. Calculation Methods

2.1. Model Description

In this work, copper/aluminum (Cu/Al) couples were taken as an example to check the reliability of first-principles calculations in predicting the critical velocity of cold-spray due to their sufficient experimental and simulated data. In total, four particle/substrate cases, i.e., Cu_p/Cu_s, Cu_p/Al_s, Al_p/Cu_s and Al_p/Al_s, were investigated (the subscript “p” stands for spray particles and “s” for substrates). An interfacial slab model was built to describe the bonding interface of Cu/Al cold-spray systems, as shown in Figure 1. To save the computational cost, a 1 × 1 slab model was applied, considering the periodicity of the interfacial structures.

As compared with FEA and MD methods, the most significant challenge for the first-principles method applied in cold-spray is to build an appropriate atomic model. To build a model close to the real cold-spray process, two issues need to be considered. The first one is how to determine the interfacial configurations of Cu/Al cold-spray systems. Since the impact of spray particles is random, there are many possible configurations of particle/substrate interfaces, and it is hard to monitor each track of the particles. But, taking into account the X-ray diffraction spectra of Cu and Al, the (111) plane is the most closely packed plane for the face-centered cubic (FCC) structure of Cu and Al, which dominates the surfaces of Cu and Al particles/substrates. The impact and bonding between particles and substrates are statistically most probable to occur on the (111) surface. Thus, four interfacial cases, i.e., Cu_p(111)/Cu_s(111), Cu_p(111)/Al_s(111), Al_p(111)/Cu_s(111) and Al_p(111)/Al_s(111), were considered in this work. Additionally, the impact site is also random on the (111) surfaces. There are three totally independent impact sites on the (111) surfaces that should be considered. They are two hollow sites and one bridge site marked as S1, S2 and S3, respectively, as shown in Figure 1. All of the interfacial configurations will be calculated to compare their differences in the energy barrier.

The second issue is how to determine the thickness of slab models. As is well known, severe plastic deformation (SPD) will occur in the bonding interfaces and result in grain refinement [20]. The SPD layer adsorbs most of the impact kinetic energies of spray particles and contributes to the bonding of sprayed coatings. The thickness of the SPD layer also indicates the rigidity of spray particles and substrates, which has a significant influence on the critical velocity of cold-spray. In order to fully describe the features of bonding interfaces, several slab models with different atomic layers were built, and the relationship between the energy barrier and the number of atomic layers was clarified. The calculated interfacial slab models were marked by a symbol of M/N, in which the “M” stands for the number of atomic layers for spray particles and the “N” stands for substrates. Thus, in the present work, seven slab models were calculated. They are 4/10, 6/12, 8/14, 10/16, 12/18, 14/20 and 16/22 cases.

2.2. Calculation Procedure and Set-Up

All calculations were performed with DS-PAW code from HZWTECH based on the density functional theory (DFT) [21,22]. The generalized gradient approximation (GGA) in the Perdew–Burke–Eruzerh (PBE) formula was applied to describe the exchange–correlation potential. The ion–electron interaction was evaluated by ultrasoft pseudopotentials of the Vanderbilt type. The cutoff energy of 500 eV was used for plane-wave expansions. The k-point meshes within the Monkhorst–Pack framework were set as 11 × 11 × 11 and 10 × 10 × 10 for the Cu and Al substrates, respectively. The completed geometry optimizations were considered to be converged when the energy change per atom and maximum stress were less than 1 × 10⁻⁴ eV and 0.05 GPa, respectively. After geometry optimizations for the Cu and Al substrates, the interfacial slab models were built by cleaving the (111) surface of the Cu and Al substrates. All models contained a vacuum layer of 20 Å. A full geometry optimization was conducted to obtain the equilibrium configuration of the particle/substrate interfaces. Figure 2 shows the atomic configurations on the interface during the impact process schematically. The volume of the interfacial models was changed by compression in the z direction and tension in the x and y directions to simulate the plastic deformation in cold-spray. Moving spray atoms towards substrate atoms stands for particle impact, while enlarging the atomic spacing in the x and y directions corresponds to the material jetting. The change in volume was represented by the compressive strain (ε_z) and tensile strain (ε_x and ε_y). The total energy of each configuration was calculated by geometry optimizations, and the energy variation with strains could be obtained. A successful bonding was determined by the atomic exchange between particles and substrates after the atomic relaxion of the interfacial model. The energy variation in the whole impact process could be monitored and the bonding path could be clarified. The total energy of the equilibrium configuration, E₀, was selected as a reference for energy barrier calculation. The energy barrier can be defined as the difference between the peak energy of unbonded configuration on the bonding path (E_p) and the energy of equilibrium configuration (E₀), i.e., $E_{barrier}=E_p-E_0$. For the geometry optimization of the interfacial models, the cutoff energy was set as 500 eV, the k-points mesh was set as 13 × 13 × 1 and the convergence criteria were the same as for the geometry optimization of Cu and Al bulks.

3. Results and Discussions

3.1. Energy Variation during Cold-Spray Bonding

When the spray atoms move close to the substrate, the energy of the cold-spray system will increase due to the repulsive nature of atoms. The energy barrier can be obtained by monitoring the energy variation during the compression process. Figure 3 shows the energy variations in Cu/Al systems with the plastic strain during the impact process. Taking the Al_p/Cu_s cases for example, as shown in Figure 3a–c, the characteristic of the energy variation can be well described. With the increase in compression strain in the z direction (ε_z), the energy barrier gradually increases before bonding occurs. When the bonding is successfully built, in some cases, the energy barrier will drop because of the strain relaxion, which is also called pressure-release (see Figure 3a,c). In most conditions, the bonding is accompanied with pressure-release. However, for the S2 site, the energy barrier will keep on increasing until the bonding is successfully built (see Figure 3b). By observing the bonding path, it can be found that the S3 site shows a smaller ε_z and relatively lower energy barrier than the other two impact sites, indicating a prior bonding path through the S3 site. Additionally, the interfacial tensile strain (ε_x and ε_y) plays a key role in successful bonding, which could significantly decrease the energy barrier along the bonding path. Taking the Al_p/Al_s-S3 case for instance (see Figure 3l), the enlarged ε_x or ε_y could decrease the energy barrier from 0.7 eV to 0.29 eV when the ε_z increases from 0.0328 to 0.0343. A large ε_x or ε_y implies an increased contact area, which also corresponds to the jetting phenomenon. By comparing the energy variation of each case, some rules can be summarized as follows. (1) Compared with the Al substrate, the spray materials are easier to deposit on the Cu substrate due to their lower energy barrier on the S1 and S2 sites. (2) In most cases, the bonding requires a large interfacial strain, which easily causes the material to jet around the spray particles. It is worth noting that the interfacial strain for the Cu_p/Al_s-S3 case is smaller than that for the Al_p/Cu_s-S3 case during the bonding process. The larger the interfacial strain is, the easier it is for jetting to occur. The material jetting mainly depends on the difference in the atomic size between the spray particle and the substrate. The squeezing of a big atom into a narrow lattice tends to cause a large interfacial strain, resulting in material jetting. This inference is confirmed by Fu’s experimental observations [23]. (3) The pressure-release bonding mechanism is more often observed in the heterogeneous material cold-spray process, such as the Al_p/Cu_s and Cu_p/Al_s cases.

Figure 4 shows the SEM images of bonding interfaces between spray particles and substrates. Here, we take the Al_p/Al_s and Cu_p/Al_s cases for example. It can be seen that there exists dense jetting on the interface of the Al_p/Al_s case while a slight jetting appears on the interface of the Cu_p/Al_s case. The experimental observation also verifies our calculated results.

3.2. Critical Velocity of Cu/Al Cold-Spray Systems

Further analyzing the energy barrier of interfacial models with different atomic layers, the minimum energy barrier can be obtained and the influence factors of critical velocity can be clarified. Figure 5 shows the relationships between the energy barrier and the number of atomic layers for the Cu/Al cold-spray system. It can be seen that the energy barrier greatly depends on substrate materials, spray materials, interfacial atomic layers as well as impact sites. For all cases, the energy barrier of S3 is significantly lower than that of the other two impact sites. This implies that the S3 site is the entrance for spray atoms squeezing into substrates. Due to the random impact, the atoms that impacted on the S1 or S2 sites should be moved to the S3 site by shear to enter the substrate. Therefore, the shear deformation is necessary for bonding in cold-spray. Our findings support Assadi’s view. In fact, the shear deformation of materials depends on their stacking fault energy (SFE). A small SFE is prone to induce shear deformation. The SFE of Cu and Al is 45 mJ/m² and 166 mJ/m², respectively [24]. When the Cu and Al are connected by cold-spray, the shear deformation will occur preferentially on the Cu side to fit the best connection site and to decrease the energy barrier of bonding. As shown in Figure 5a,b, the energy barriers for the Al_p/Cu_s case and the Cu_p/Al_s case are 0.067 eV and 0.075 eV, respectively, which are lower than 0.165 eV for the Al_p/Al_s case (see Figure 5d). It should be noted that the Cu_p/Cu_s case shows relatively high energy compared to the Al_p/Al_s case, although Cu has a lower SFE. This is because the twinning deformation between the Cu particle and the Cu substrate is difficult to coordinate. Gyuyeol Bae et al. pointed out that there is no obvious difference in the impact behavior of cold-spray systems if the same material was selected as the spray particle and the substrate [25]. Additionally, a large difference in atomic size between spray particles and substrates is beneficial for decreasing the energy barrier. This is consistent with the general understanding that small atoms are easy to squeeze into the large lattice. However, the difference in the energy barrier between the Al_p/Cu_s case and the Cu_p/Al_s case should be noted, although they have the same configuration on the interface. This is attributed to the layer thickness-dependent bonding characteristic between Al atoms and Cu atoms. After all calculations for the energy barrier are completed, the critical velocity (v_crit) can be calculated by the following equation, i.e., $v_{crit}=\sqrt{2E_{barrier}/M}$, where E_barrier is the minimum energy barrier obtained by optimizing the number of atomic layers and M is the mole mass of the spray materials. The calculated critical velocities for Cu/Al cold-spray systems are listed in

Table 1. The calculated critical velocity of Cu/Al cold-spray systems.

Particle/Substrate	Method	Critical Velocity (m/s)	Reference
Cu/Cu	Finite element analysis, Lagrangian method	570	[8]
	Finite element analysis, Lagrangian method, particle size-dependent	500	[10]
	Finite element analysis, Eulerian method	300	[16]
	Finite element analysis	445	[12]
	Finite element analysis	550	[25]
	Analytic method	531.5	[26]
	Analytic method, porosity-dependent	373–607	[27]
	Experiments	570	[28]
	In situ experiments	539–568	[29]
	First-principles calculation	739	Present work
Al/Al	Finite element analysis, Lagrangian method	750	[8]
	Finite element analysis, Lagrangian method, particle size-dependent	620	[10]
	Finite element analysis, Eulerian Method	410	[16]
	Finite element analysis	645	[12]
	Finite element analysis	775	[25]
	Molecular dynamics	353	[30]
	In situ experiments	797–824	[29]
	First-principles calculation	1083	Present work
Al/Cu	Molecular dynamics	301	[30]
	Experiment, deposition efficiency >0%.	640	[31]
	Finite element analysis	600–630	[32]
	First-principles calculation	690	Present work
Cu/Al	Experiment, deposition efficiency >0%.	490	[31]
	Finite element analysis	510–530	[32]
	First-principles calculation	479	Present work

. Other experimental and predicted values are also listed for comparison. The critical velocity predicted by finite element analysis is variable with the calculated method. Additionally, the analytic method involves massive empirical parameters, which makes the predicted value inaccurate. The critical velocity calculated by molecular dynamics also shows a significantly lower value than that obtained by experiments. Such low critical velocity is close to the condition of a ball with a diameter of 20 mm sprayed on the substrate [10]. It should be noted that the critical velocity calculated by the first-principles method in our present work also shows a difference with the experimental and other predicted values, especially for the homogeneous material sprayed cases. The calculated critical velocities of the Cu_p/Cu_s and Al_p/Al_s cases are 739 m/s and 1083 m/s, respectively, which is about 30% above the maximum experimental value. However, for the heterogeneous material sprayed cases, our calculated critical velocity agrees well with the experimental one, and the deviation is less than 8%. The great deviation between the calculated critical velocity and the experimental one mainly results from the insufficient thickness of the deformed atomic layer. For the Cu_p/Cu_s and Al_p/Al_s cases, the energy barrier gradually decreases with the increase in the thickness of the deformed atomic layer (see Figure 5c,d). This means that the influence of the deformed atomic layer on the energy barrier has not been completely eliminated. The atomic layer of calculated models should be increased to obtain the real minimum energy barrier for bonding. On the contrary, the minimum energy barrier for the Al_p/Cu_s and Cu_p/Al_s cases can be obtained by model 14/20 and model 12/18, respectively (see Figure 5a,b). This completely eliminates the effect of deformation layer thickness on the energy barrier. Therefore, an accurate critical velocity can be obtained. The difference between homogeneous material spraying and heterogeneous material spraying is mainly attributed to the bonding characteristic of contacted atoms. The bonding between Cu and Al atoms can improve the formation of the metallurgical interface and decrease the thickness of the deformed atomic layers.

3.3. Bonding Characteristics of Cu/Al Cold-Spray Systems

To clarify the difference between the heterogeneous material spraying and the homogeneous material spraying, the variations in the density of states (DOS) during the cold-spray process were analyzed. Figure 6 shows the evolution of the DOS of interfacial atoms for Cu/Al cold-spray cases during the impact process. For homogeneous material spraying, such as the Al_p/Al_s and Cu_p/Cu_s cases, the Fermi level (E_f) gradually decreases during the bonding process. The reduction in the Fermi level is due to the electronic transition after the absorption of the impact kinetic energy. Additionally, the complete overlap of the DOS for the homogeneous material spraying reveals a strong repulsive Coulomb force between electrons, which raises the energy barrier. In order to avoid the erosion of spray particles and achieve successful bonding, a thicker interfacial deformation layer is required to absorb the impact energy and reduce the repulsive Coulomb force. Meanwhile, for the heterogeneous material spraying, i.e., the Cu_p/Al_s case, the Fermi level hardly changes during the impact process. As compared with the Cu_p/Cu_s case, the DOS for the Cu_p/Al_s case shifts to a lower energy level, below the Fermi level, which indicates the formation of attractive bonds between Cu and Al atoms. These strong bonds enhance the absorption of the impact kinetic energy and decrease the energy barrier to prevent erosion from occurring during cold-spray. Li et al. investigated the cold-sprayed interfaces of Cu/Ni/Al mixtures by FEA. Their results show that there is a wider deformation region on the homogeneous material sprayed interface than that on the heterogeneous one (see effective plastic strain (PEEQ) map), which agrees well with our present calculation [33].

4. Conclusions

The critical velocities of Cu/Al cold-spray systems were predicted for the first time by employing the first-principles calculation. The bonding mechanisms of cold-spray technology were analyzed by investigating energy variation during the cold-spray process and atomic interaction on the sprayed interface. The results are summarized as follows:

(1)

A mixed bonding mechanism of shear deformation and pressure-release could be identified during the cold-spray process. The shear deformation played a key role in the successful cold-spray bonding. The pressure-release bonding mechanism occurred in the heterogeneous material cold-spray process.

(2)

A successful cold-spray bonding depended on the effective absorption of the impact kinetic energy. For the heterogeneous material cold-spray, the attractive bonds between atoms drove the formation of tightly bonded interfaces. Meanwhile, for the homogeneous material cold-spray, the repulsive Coulomb force between atoms increased the energy barrier of bonding, and a thicker interfacial deformation layer was required to achieve a successful bonding.

(3)

The critical velocity of the cold-spray could be accurately predicted by the first-principles method without any empirical or experimental parameters. This proved that the first-principles method was a simple and highly efficient computing method in critical velocity prediction.